"Testing integral"

clear

-- basic integrals

check(integral(a) = a x)
check(integral(x) = 1/2 x^2)
check(integral(a x) = 1/2 a x^2)
check(integral(x^2) = 1/3 x^3)
check(integral(a x^2) = 1/3 a x^3)
check(integral(x^n) = x^(n + 1) / (n + 1))
check(integral(a x^n) = a x^(n + 1) / (n + 1))
check(infixform(integral(f())) == "integral(f(),x)")
check(infixform(integral(f(),x)) == "integral(f(),x)")
check(infixform(integral(f(x))) == "integral(f(x),x)")
check(infixform(integral(f(x),x)) == "integral(f(x),x)")
check(infixform(integral(f(y))) == "x f(y)")
check(infixform(integral(f(y),x)) == "x f(y)")
check(infixform(integral(f(a,b,c),x)) == "x f(a,b,c)")
check(infixform(integral(f(a,b,c,x))) == "integral(f(a,b,c,x),x)")
check(infixform(integral(d(f()))) == "f()")
check(infixform(integral(d(f(t),t),t)) == "f(t)")
check(infixform(d(integral(f(),x),y)) == "integral(d(f(),y),x)")

-- check classifier false positives

check(integral(a^b,x) = a^b x)
check(integral(x^exp(y),x) = x^(exp(y) + 1) / (exp(y) + 1))
check(integral(x^log(y),x) = x^(log(y) + 1) / (log(y) + 1))
check(integral(x^sin(y),x) = x^(sin(y) + 1) / (sin(y) + 1))

-- x^n log(a x + b)

f = log(a x)
g = integral(f)
h = d(g)
check(f == h)
f = log(a x + b)
g = integral(f)
h = d(g)
check(f == h)
f = x log(a x)
g = integral(f)
h = d(g)
check(f == h)
f = x log(a x + b)
g = integral(f)
h = d(g)
check(f == h)
f = x^2 log(a x)
g = integral(f)
h = d(g)
check(f == h)
f = x^2 log(a x + b)
g = integral(f)
h = d(g)
check(f == h)
f = log(a x)^2
g = integral(f)
h = d(g)
check(f == h)
f = log(a x + b)^2
g = integral(f)
h = d(g)
check(f == h)
f = log(a x) / x^2
g = integral(f)
h = d(g)
check(f == h)
f = log(a x + b) / x^2
g = integral(f)
h = d(g)
check(f == h)
--
T = integral(x^2+x)
check(T = 1/3 x^3 + 1/2 x^2)
#1
T = integral(A,X)
check(T = A*X)
#4
T = integral(A+B,X)-(A*X+B*X)
check(T = 0)
#9
T = integral(1/X,X)
check(T = log(X))
#14
T = integral(log(X),X)-X*log(X)+X
check(T = 0)
#15
T = integral(3^X*log(3),X)
check(T = 3^X)
#16
f = 1 / (3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / (3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / (-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / (-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#17
f = 1 / sqrt((3 X^2 + 7))
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / sqrt((3 X^2 - 7))
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / sqrt(-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / sqrt(-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#18
f = X / sqrt((3 X^2 + 7))
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X / sqrt((3 X^2 - 7))
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X / sqrt(-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X / sqrt(-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#27
T = integral(1/(a+b*x),x)-(log(a+b*x)/b)
check(T = 0)
#28
T = integral(1/(A+B*X)^2,X)+1/B*1/(A+B*X)
check(T = 0)
#29
T = integral(1/(a+b*x)^3,x)+1/2*1/b*(a+b*x)^(-2)
check(T = 0)
#30
T = integral(X/(A+B*X),X)+A*B^(-2)*log(A+B*X)-X/B
check(T = 0)
#31
T = integral(X/(A+B*X)^2,X) - log(A+B*X)/B^2 - A/(B^2*(A+B*X))
check(T = 0)
#33
f = integral(X^2 / (A + B X),X)
g = (2 A^2 log(A + B X) + B X (B X - 2 A)) / (2 B^3)
check(f = g)
#34
f = integral(X^2 / (A + B X)^2,X)
g = (-A^2 / (A + B X) - 2 A log(A + B X) + B X) / B^3
check(f = g)
#35
T = integral(X^2/(A+B*X)^3,X)-1/B^3*(log(A+B*X)+2*A/(A+B*X)-1/2*A^2/(A+B*X)^2)
check(T = 0)
#37
T = integral(1/X*1/(A+B*X),X)+1/A*log((A+B*X)/X)
check(T = 0)
#38
T = integral(1/X*1/(A+B*X)^2,X)-1/A*1/(A+B*X)+1/A^2*log((A+B*X)/X)
check(T = 0)
#39
T = integral(1/X*1/(A+B*X)^3,X)-1/A^3*(1/2*((2*A+B*X)/(A+B*X))^2+log(X/(A+B*X)))
check(T = 0)
#40
T = integral(1/X^2*1/(A+B*X),X)+1/(A*X)-B/A^2*log((A+B*X)/X)
check(T = 0)
#41
T = integral(1/X^3*1/(A+B*X),X)-(2*B*X-A)/(2*A^2*X^2)-B^2/A^3*log(X/(A+B*X))
check(T = 0)
#42
T = integral(1/X^2*1/(A+B*X)^2,X)+(A+2*B*X)/(A^2*X*(A+B*X))-2*B/A^3*log((A+B*X)/X)
check(T = 0)
#63
T = integral(X/(A+B*X^2),X)-1/2*1/B*log(A+B*X^2)
check(T = 0)
#64
f = X^2 / (3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^2 / (3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^2 / (-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^2 / (-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)

#65
f = 1 / (3 X^2 + 7)^2
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / (3 X^2 - 7)^2
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / (-3 X^2 + 7)^2
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / (-3 X^2 - 7)^2
g = integral(f,X)
h = d(g,X)
check(f == h)

#66 (same as 62)
f = 1 / (A - B X^2)
F = integral(f,X)
check(f == d(F,X))

#67 (m=1) (same as 65)
f = 1 / (A X^2 + B)^2
F = integral(f,X)
check(f == d(F,X))
#67 (m=2)
f = 1 / (A X^2 + B)^3
F = integral(f,X)
check(f == d(F,X))
#67 (m=3)
f = 1 / (A X^2 + B)^4
F = integral(f,X)
check(f == d(F,X))

#68 (m=1)
f = X / (A X^2 + B)^2
F = integral(f,X)
check(f == d(F,X))
#68 (m=2)
f = X / (A X^2 + B)^3
F = integral(f,X)
check(f == d(F,X))
#68 (m=3)
f = X / (A X^2 + B)^4
F = integral(f,X)
check(f == d(F,X))

#69 (m=1)
f = X^2 / (A X^2 + B)^2
F = integral(f,X)
check(f == d(F,X))
#69 (m=2)
f = X^2 / (A X^2 + B)^3
F = integral(f,X)
check(f == d(F,X))
#69 (m=3)
f = X^2 / (A X^2 + B)^4
F = integral(f,X)
check(f == d(F,X))

#70
T = integral(1/X*1/(A+B*X^2),X)-1/2*1/A*log(X^2/(A+B*X^2))
check(T = 0)

#71
f = 1 / X^2 * 1 / (3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X^2 * 1 / (3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X^2 * 1 / (-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X^2 * 1 / (-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#74
f = 1 / (A X^3 + B)
F = integral(f,X)
check(f == d(F,X))
#75
f = X / (A X^3 + B)
F = integral(f,X)
check(f == d(F,X))
#76
f = X^2 / (A X^3 + B)
F = integral(f,X)
check(f == d(F,X))
#77 78
f = 1 / (A X^4 + B)
F = integral(f,X)
check(f == d(F,X))
#79 80
f = X / (A X^4 + B)
F = integral(f,X)
check(f == d(F,X))
#81 82
f = X^2 / (A X^4 + B)
F = integral(f,X)
check(f == d(F,X))
#83
f = X^3 / (A X^4 + B)
F = integral(f,X)
check(f == d(F,X))

f = 1 / (A X^5 + B)
F = integral(f,X)
check(f == d(F,X))
#124
f = sqrt(3 X + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(3 X - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(-3 X + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(-3 X - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#125
T = integral(X*sqrt(A+B*X),X)+2*(2*A-3*B*X)*sqrt((A+B*X)^3)/15*B^(-2)
check(T = 0)
#126
T = integral(X^2*sqrt(A+B*X),X)-2*(8*A^2-12*A*B*X+15*B^2*X^2)*sqrt((A+B*X)^3)/105*B^(-3)
check(T = 0)
#128
f = sqrt(3 X + 7) / X
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(3 X - 7) / X
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(-3 X + 7) / X
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(-3 X - 7) / X
g = integral(f,X)
h = d(g,X)
check(f == h)
#129
f = sqrt(3 X + 7) / X^2
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(3 X - 7) / X^2
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(-3 X + 7) / X^2
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(-3 X - 7) / X^2
g = integral(f,X)
h = d(g,X)
check(f == h)
#131
f = 1 / sqrt(3 X + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / sqrt(3 X - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / sqrt(-3 X + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / sqrt(-3 X - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#132
T = integral(X/sqrt(A+B*X),X)+2/3*(2*A-B*X)*sqrt(A+B*X)/B^2
check(T = 0)
#133
T = integral(X^2/sqrt(A+B*X),X)-2/15*(8*A^2-4*A*B*X+3*B^2*X^2)*sqrt(A+B*X)/B^3
check(T = 0)
#134
f = 1 / X / sqrt(3 X + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X / sqrt(3 X - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X / sqrt(-3 X + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X / sqrt(-3 X - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#137
f = 1 / X^2 / sqrt(3 X + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X^2 / sqrt(3 X - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X^2 / sqrt(-3 X + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X^2 / sqrt(-3 X - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#156
T = integral(sqrt(X^2+A),X)-1/2*(X*sqrt(X^2+A)+A*log(X+sqrt(X^2+A)))
check(T = 0)
#157
T = integral(1/sqrt(X^2+A),X)-log(X+sqrt(X^2+A))
check(T = 0)
#158
f = 1 / X / sqrt(3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X / sqrt(3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X / sqrt(-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X / sqrt(-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#138
f = sqrt(3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#160
f = sqrt(3 X^2 + 7) / X
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(3 X^2 - 7) / X
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(-3 X^2 + 7) / X
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(-3 X^2 - 7) / X
g = integral(f,X)
h = d(g,X)
check(f == h)
#163
f = X sqrt(3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X sqrt(3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X sqrt(-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X sqrt(-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#164
T = integral(sqrt((X^2+A)^3),X)-1/4*(X*sqrt((X^2+A)^3)+3/2*A*X*sqrt(X^2+A)+3/2*A^2*log(X+sqrt(X^2+A)))
check(T = 0)
#165
f = (3 X^2 + 7)^(-3/2)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = (3 X^2 - 7)^(-3/2)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = (-3 X^2 + 7)^(-3/2)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = (-3 X^2 - 7)^(-3/2)
g = integral(f,X)
h = d(g,X)
check(f == h)
#166
f = X (3 X^2 + 7)^(-3/2)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X (3 X^2 - 7)^(-3/2)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X (-3 X^2 + 7)^(-3/2)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X (-3 X^2 - 7)^(-3/2)
g = integral(f,X)
h = d(g,X)
check(f == h)
#167
T = integral(X*sqrt((X^2+A)^3),X)-1/5*sqrt((X^2+A)^5)
check(T = 0)
#168
f = X^2 sqrt(3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^2 sqrt(3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#169
f = X^3 sqrt(3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^3 sqrt(3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^3 sqrt(-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^3 sqrt(-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#171
f = X^2 / sqrt(3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^2 / sqrt(3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^2 / sqrt(-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^2 / sqrt(-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#172
f = X^3 / sqrt(3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^3 / sqrt(3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^3 / sqrt(-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = X^3 / sqrt(-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#173
f = 1 / X^2 / sqrt(3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X^2 / sqrt(3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X^2 / sqrt(-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X^2 / sqrt(-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#174
f = 1 / X^3 / sqrt(3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X^3 / sqrt(3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X^3 / sqrt(-3 X^2 + 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
f = 1 / X^3 / sqrt(-3 X^2 - 7)
g = integral(f,X)
h = d(g,X)
check(f == h)
#204
T = integral(X/sqrt(A-X^2),X)+sqrt(A-X^2)
check(T = 0)
#210
f = integral(X^2 sqrt(7 - X^2),X)
g = 1/4 X^3 (-X^2 + 7)^(1/2) - 7/8 X (-X^2 + 7)^(1/2) - 49/8 i log(-X + i (-X^2 + 7)^(1/2))
check(f = g)
#215
T = integral(1/X^2*1/sqrt(7-X^2),X)+sqrt(7-X^2)/7/X
check(T = 0)
#216
f = sqrt(3 X^2 + 7) / X^2
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(3 X^2 - 7) / X^2
g = integral(f,X)
h = d(g,X)
check(f == h)
#217
f = sqrt(3 X^2 + 7) / X^3
g = integral(f,X)
h = d(g,X)
check(f == h)
f = sqrt(-3 X^2 + 7) / X^3
g = integral(f,X)
h = d(g,X)
check(f == h)
#290
T = integral(sin(A*X),X)+cos(A*X)/A
check(T = 0)
#291
T = integral(cos(A*X),X)-sin(A*X)/A
check(T = 0)
#292
T = integral(tan(A*X),X)+log(cos(A*X))/A
check(T = 0)
#293
T = integral(1/tan(A*X),X)-log(sin(A*X))/A
check(T = 0)
#294
T = integral(1/cos(A*X),X)-log(tan(pi/4+A*X/2))/A
check(T = 0)
#295
T = integral(1/sin(A*X),X)-log(tan(A*X/2))/A
check(T = 0)
#296
T = integral(sin(A*X)^2,X)-X/2+sin(2*A*X)/(4*A)
check(T = 0)
#297
T = integral(sin(A*X)^3,X)+cos(A*X)*(sin(A*X)^2+2)/(3*A)
check(T = 0)
#298
T = integral(sin(A*X)^4,X)-3/8*X+sin(2*A*X)/(4*A)-sin(4*A*X)/(32*A)
check(T = 0)
#302
T = integral(cos(A*X)^2,X)-X/2-sin(2*A*X)/(4*A)
check(T = 0)
#303
T = integral(cos(A*X)^3,X)-sin(A*X)*(cos(A*X)^2+2)/(3*A)
check(T = 0)
#304
T = integral(cos(A*X)^4,X)-3/8*X-sin(2*A*X)/(4*A)-sin(4*A*X)/(32*A)
check(T = 0)
#308
T = integral((1/sin(A*X))^2,X)+1/A*1/tan(A*X)
check(T = 0)
#312
T = integral((1/cos(A*X))^2,X)-tan(A*X)/A
check(T = 0)
#318
T = integral(sin(A*X)*cos(A*X),X)-sin(A*X)^2/(2*A)
check(T = 0)
#320
T = integral(sin(A*X)^2*cos(A*X)^2,X)+sin(4*A*X)/(32*A)-X/8
check(T = 0)
#326
T = integral(sin(A*X)/cos(A*X)/cos(A*X),X)-1/(A*cos(A*X))
check(T = 0)
#328
T = integral(cos(A*X)/sin(A*X)^2,X)+1/(A*sin(A*X))
check(T = 0)
#329
T = integral(1/sin(A*X)/cos(A*X),X)-log(tan(A*X))/A
check(T = 0)
#330
T = integral(1/sin(A*X)/cos(A*X)^2,X)-(1/cos(A*X)+log(tan(A*X/2)))/A
check(T = 0)
#332
T = integral(1/sin(A*X)^2/cos(A*X),X)-(log(tan(pi/4+A*X/2))-1/sin(A*X))/A
check(T = 0)
#333
T = integral(1/sin(A*X)^2/cos(A*X)^2,X)+2/(A*tan(2*A*X))
check(T = 0)
#335
T = integral(sin(A+B*X),X)+cos(A+B*X)/B
check(T = 0)
#336
T = integral(cos(A+B*X),X)-sin(A+B*X)/B
check(T = 0)
#337+
T = integral(1/(1+sin(A*X)),X)+tan(pi/4-A*X/2)/A
check(T = 0)
#337b+
T = integral(1/(B+B*sin(A*X)),X)+tan(pi/4-A*X/2)/A/B
check(T = 0)
#337-
T = integral(1/(1-sin(A*X)),X)-tan(pi/4+A*X/2)/A
check(T = 0)
#337b-
T = integral(1/(B-B*sin(A*X)),X)-tan(pi/4+A*X/2)/A/B
check(T = 0)
#338
T = integral(1/(1+cos(A*X)),X)-tan(A*X/2)/A
check(T = 0)
#339
T = integral(1/(1-cos(A*X)),X)+1/(A*tan(A*X/2))
check(T = 0)
#340
T = integral(1/(A+B*sin(X)),X)-1/sqrt(B^2-A^2)*log((A*tan(X/2)+B-sqrt(B^2-A^2))/(A*tan(X/2)+B+sqrt(B^2-A^2)))
check(T = 0)
#341
T = integral(1/(A+B*cos(X)),X)-1/sqrt(B^2-A^2)*log((sqrt(B^2-A^2)*tan(X/2)+A+B)/(sqrt(B^2-A^2)*tan(X/2)-A-B))
check(T = 0)
#389
T = x*sin(A*x)-d(integral(x*sin(A*x)))
check(T = 0)
#390
T = x^2*sin(A*x)-d(integral(x^2*sin(A*x)))
check(T = 0)
#393
T = x*cos(A*x)-d(integral(x*cos(A*x)))
check(T = 0)
#394
T = x^2*cos(A*x)-d(integral(x^2*cos(A*x)))
check(T = 0)
#441
T = integral(arcsin(A*X),X)-X*arcsin(A*X)-sqrt(1-A^2*X^2)/A
check(T = 0)
#442
T = integral(arccos(A*X),X)-X*arccos(A*X)-sqrt(1-A^2*X^2)/A
check(T = 0)
#443
T = integral(arctan(A*X),X)-X*arctan(A*X)+log(1+A^2*X^2)/(2*A)
check(T = 0)
#493
T = integral(1/X*1/log(A*X),X)-log(log(A*X))
check(T = 0)
#554
T = integral(sinh(X),X)-cosh(X)
check(T = 0)
#555
T = integral(cosh(X),X)-sinh(X)
check(T = 0)
#556
T = integral(tanh(X),X)-log(cosh(X))
check(T = 0)
#560
T = integral(X*sinh(X),X)-X*cosh(X)+sinh(X)
check(T = 0)
#562
T = integral(X*cosh(X),X)-X*sinh(X)+cosh(X)
check(T = 0)
#566
T = integral(sinh(X)^2,X)-sinh(2*X)/4+X/2
check(T = 0)
#569
T = integral(tanh(X)^2,X)-X+tanh(X)
check(T = 0)
#572
T = integral(cosh(X)^2,X)-sinh(2*X)/4-X/2
check(T = 0)
T = integral(1/x*1/(a+log(x)),x)-log(a+log(x))
check(T = 0)
T = integral(x^2*(1-x^2)^(3/2))-(x*sqrt(1-x^2)*(-8*x^4+14*x^2-3)+3*arcsin(x))/48
check(T = 0)
T = integral(x^4*(1-x^2)^(3/2))-(-x*sqrt(1-x^2)*(16*x^6-24*x^4+2*x^2+3)+3*arcsin(x))/128
check(T = 0)
T = integral(x^2*(1-x^2)^(5/2))-(x*sqrt(1-x^2)*(48*x^6-136*x^4+118*x^2-15)+15*arcsin(x))/384
check(T = 0)
T = integral(cos(x)^2*sin(x))
check(T = -cos(x)^3/3)
T = integral((1+cos(x)^2)*sin(x))
check(T = -cos(x)-cos(x)^3/3)
T = defint((1+cos(theta)^2)*sin(theta),theta,0,pi,phi,0,2*pi)
check(T = 16*pi/3)
T = integral(sin(A X) / (cos(A X) - 1)^2,X)
check(T == 1/A * 1/(cos(A X) - 1))
T = integral(sin(A X) / (1 - cos(A X))^2,X)
check(T == 1/A * 1/(cos(A X) - 1))
check(integral(d(arcsin(x))) == arcsin(x))
check(integral(sqrt(1 + x^2 / (1 - x^2))) == arcsin(x))

f = sin(x) exp(a x)
check(d(integral(f)) == f)
f = sin(x) exp(a x + b)
check(d(integral(f)) == f)
f = sin(x) exp(i x)
check(d(integral(f)) == f)
f = sin(x) exp(-i x)
check(d(integral(f)) == f)
f = sin(x) exp(i x + b)
check(d(integral(f)) == f)
f = sin(x) exp(-i x + b)
check(d(integral(f)) == f)
f = sin(x) exp(i a x)
check(d(integral(f)) == f)
f = sin(x) exp(-i a x)
check(d(integral(f)) == f)
f = sin(x) exp(i a x + b)
check(d(integral(f)) == f)
f = sin(x) exp(-i a x + b)
check(d(integral(f)) == f)

f = cos(x) exp(a x)
check(d(integral(f)) == f)
f = cos(x) exp(a x + b)
check(d(integral(f)) == f)
f = cos(x) exp(i x)
check(d(integral(f)) == f)
f = cos(x) exp(-i x)
check(d(integral(f)) == f)
f = cos(x) exp(i x + b)
check(d(integral(f)) == f)
f = cos(x) exp(-i x + b)
check(d(integral(f)) == f)
f = cos(x) exp(i a x)
check(d(integral(f)) == f)
f = cos(x) exp(-i a x)
check(d(integral(f)) == f)
f = cos(x) exp(i a x + b)
check(d(integral(f)) == f)
f = cos(x) exp(-i a x + b)
check(d(integral(f)) == f)

f = sin(x) cos(x) exp(a x)
check(d(integral(f)) == f)

f = cos(x)^3 sin(x)
check(d(integral(f)) == f)

f = cos(a x)^5
check(d(integral(f)) == f)

f = cos(a x)^5 / sin(a x)^2
check(d(integral(f)) == f)

f = cos(a x)^3 / sin(a x)^2
check(d(integral(f)) == f)

f = cos(a x)^5 / sin(a x)
g = d(integral(f))
check(g == f)

f = cos(a x)^3 / sin(a x)
g = d(integral(f))
check(g == f)

f = cos(a x) sin(a x)^3
check(d(integral(f)) == f)

f = cos(a x)^3 sin(a x)^2
check(d(integral(f)) == f)

f = cos(a x)^2 sin(a x)^3
check(d(integral(f)) == f)

f = cos(a x)^4 sin(a x)
check(d(integral(f)) == f)

f = cos(a x)^7 / sin(a x)^2
check(d(integral(f)) == f)

-- sin(a x)^n

f = sin(a x)^2
check(d(integral(f)) == f)

f = sin(a x)^3
check(d(integral(f)) == f)

f = sin(a x)^4
check(d(integral(f)) == f)

f = sin(a x)^5
check(d(integral(f)) == f)

f = sin(a x)^6
check(d(integral(f)) == f)

-- cos(a x)^n

f = cos(a x)^2
check(d(integral(f)) == f)

f = cos(a x)^3
check(d(integral(f)) == f)

f = cos(a x)^4
check(d(integral(f)) == f)

f = cos(a x)^5
check(d(integral(f)) == f)

f = cos(a x)^6
check(d(integral(f)) == f)

-- cos(a x)^n / sin(a x)

f = cos(a x)^2 / sin(a x)
check(d(integral(f)) == f)

f = cos(a x)^4 / sin(a x)
check(d(integral(f)) == f)

f = cos(a x)^6 / sin(a x)
check(d(integral(f)) == f)

-- x^n exp(a x + b)

f = x^0 exp(a x)
check(d(integral(f)) == f)
f = x^0 exp(x)
check(d(integral(f)) == f)
f = x^0 exp(-x)
check(d(integral(f)) == f)
f = x^0 exp(-a x)
check(d(integral(f)) == f)
f = x^0 exp(a x + b)
check(d(integral(f)) == f)

f = x^1 exp(a x)
check(d(integral(f)) == f)
f = x^1 exp(x)
check(d(integral(f)) == f)
f = x^1 exp(-x)
check(d(integral(f)) == f)
f = x^1 exp(-a x)
check(d(integral(f)) == f)
f = x^1 exp(a x + b)
check(d(integral(f)) == f)

f = x exp(a x)
check(d(integral(f)) == f)
f = x exp(x)
check(d(integral(f)) == f)
f = x exp(-x)
check(d(integral(f)) == f)
f = x exp(-a x)
check(d(integral(f)) == f)
f = x exp(a x + b)
check(d(integral(f)) == f)

f = x^3 exp(a x)
check(d(integral(f)) == f)
f = x^3 exp(x)
check(d(integral(f)) == f)
f = x^3 exp(-x)
check(d(integral(f)) == f)
f = x^3 exp(-a x)
check(d(integral(f)) == f)
f = x^3 exp(a x + b)
check(d(integral(f)) == f)

f = x^4 exp(a x)
check(d(integral(f)) == f)
f = x^4 exp(x)
check(d(integral(f)) == f)
f = x^4 exp(-x)
check(d(integral(f)) == f)
f = x^4 exp(-a x)
check(d(integral(f)) == f)
f = x^4 exp(a x + b)
check(d(integral(f)) == f)

f = x^5 exp(a x)
check(d(integral(f)) == f)
f = x^5 exp(x)
check(d(integral(f)) == f)
f = x^5 exp(-x)
check(d(integral(f)) == f)
f = x^5 exp(-a x)
check(d(integral(f)) == f)
f = x^5 exp(a x + b)
check(d(integral(f)) == f)

f = x^6 exp(a x)
check(d(integral(f)) == f)
f = x^6 exp(x)
check(d(integral(f)) == f)
f = x^6 exp(-x)
check(d(integral(f)) == f)
f = x^6 exp(-a x)
check(d(integral(f)) == f)
f = x^6 exp(a x + b)
check(d(integral(f)) == f)

f = x^7 exp(a x)
check(d(integral(f)) == f)
f = x^7 exp(x)
check(d(integral(f)) == f)
f = x^7 exp(-x)
check(d(integral(f)) == f)
f = x^7 exp(-a x)
check(d(integral(f)) == f)
f = x^7 exp(a x + b)
check(d(integral(f)) == f)

f = x^8 exp(a x)
check(d(integral(f)) == f)
f = x^8 exp(x)
check(d(integral(f)) == f)
f = x^8 exp(-x)
check(d(integral(f)) == f)
f = x^8 exp(-a x)
check(d(integral(f)) == f)
f = x^8 exp(a x + b)
check(d(integral(f)) == f)

f = x^9 exp(a x)
check(d(integral(f)) == f)
f = x^9 exp(x)
check(d(integral(f)) == f)
f = x^9 exp(-x)
check(d(integral(f)) == f)
f = x^9 exp(-a x)
check(d(integral(f)) == f)
f = x^9 exp(a x + b)
check(d(integral(f)) == f)

f = x^10 exp(a x)
check(d(integral(f)) == f)
f = x^10 exp(x)
check(d(integral(f)) == f)
f = x^10 exp(-x)
check(d(integral(f)) == f)
f = x^10 exp(-a x)
check(d(integral(f)) == f)
f = x^10 exp(a x + b)
check(d(integral(f)) == f)

f = x^11 exp(a x)
check(d(integral(f)) == f)
f = x^11 exp(x)
check(d(integral(f)) == f)
f = x^11 exp(-x)
check(d(integral(f)) == f)
f = x^11 exp(-a x)
check(d(integral(f)) == f)
f = x^11 exp(a x + b)
check(d(integral(f)) == f)

-- x^n exp(a x^2 + b)

f = x^0 exp(a x^2)
check(d(integral(f)) == f)
f = x^0 exp(x^2)
check(d(integral(f)) == f)
f = x^0 exp(-x^2)
check(d(integral(f)) == f)
f = x^0 exp(-a x^2)
check(d(integral(f)) == f)
f = x^0 exp(a x^2 + b)
check(d(integral(f)) == f)

f = x^1 exp(a x^2)
check(d(integral(f)) == f)
f = x^1 exp(x^2)
check(d(integral(f)) == f)
f = x^1 exp(-x^2)
check(d(integral(f)) == f)
f = x^1 exp(-a x^2)
check(d(integral(f)) == f)
f = x^1 exp(a x^2 + b)
check(d(integral(f)) == f)

f = x^2 exp(a x^2)
check(d(integral(f)) == f)
f = x^2 exp(x^2)
check(d(integral(f)) == f)
f = x^2 exp(-x^2)
check(d(integral(f)) == f)
f = x^2 exp(-a x^2)
check(d(integral(f)) == f)
f = x^2 exp(a x^2 + b)
check(d(integral(f)) == f)

f = x^3 exp(a x^2)
check(d(integral(f)) == f)
f = x^3 exp(x^2)
check(d(integral(f)) == f)
f = x^3 exp(-x^2)
check(d(integral(f)) == f)
f = x^3 exp(-a x^2)
check(d(integral(f)) == f)
f = x^3 exp(a x^2 + b)
check(d(integral(f)) == f)

f = x^4 exp(a x^2)
check(d(integral(f)) == f)
f = x^4 exp(x^2)
check(d(integral(f)) == f)
f = x^4 exp(-x^2)
check(d(integral(f)) == f)
f = x^4 exp(-a x^2)
check(d(integral(f)) == f)
f = x^4 exp(a x^2 + b)
check(d(integral(f)) == f)

f = x^5 exp(a x^2)
check(d(integral(f)) == f)
f = x^5 exp(x^2)
check(d(integral(f)) == f)
f = x^5 exp(-x^2)
check(d(integral(f)) == f)
f = x^5 exp(-a x^2)
check(d(integral(f)) == f)
f = x^5 exp(a x^2 + b)
check(d(integral(f)) == f)

f = x^6 exp(a x^2)
check(d(integral(f)) == f)
f = x^6 exp(x^2)
check(d(integral(f)) == f)
f = x^6 exp(-x^2)
check(d(integral(f)) == f)
f = x^6 exp(-a x^2)
check(d(integral(f)) == f)
f = x^6 exp(a x^2 + b)
check(d(integral(f)) == f)

f = x^7 exp(a x^2)
check(d(integral(f)) == f)
f = x^7 exp(x^2)
check(d(integral(f)) == f)
f = x^7 exp(-x^2)
check(d(integral(f)) == f)
f = x^7 exp(-a x^2)
check(d(integral(f)) == f)
f = x^7 exp(a x^2 + b)
check(d(integral(f)) == f)

f = x^8 exp(a x^2)
check(d(integral(f)) == f)
f = x^8 exp(x^2)
check(d(integral(f)) == f)
f = x^8 exp(-x^2)
check(d(integral(f)) == f)
f = x^8 exp(-a x^2)
check(d(integral(f)) == f)
f = x^8 exp(a x^2 + b)
check(d(integral(f)) == f)

f = x^9 exp(a x^2)
check(d(integral(f)) == f)
f = x^9 exp(x^2)
check(d(integral(f)) == f)
f = x^9 exp(-x^2)
check(d(integral(f)) == f)
f = x^9 exp(-a x^2)
check(d(integral(f)) == f)
f = x^9 exp(a x^2 + b)
check(d(integral(f)) == f)

-- test collect_coeffs()

clear

f = exp(-i (En - omega - i epsilon) t / hbar)
check(integral(f,t) == i hbar f / (En - omega - i epsilon))

f = exp(a x^2 + b + c)
g = integral(f,x)
check(infixform(g) == "-i pi^(1/2) erf(i a^(1/2) x) exp(b + c) / (2 a^(1/2))")
check(f == d(g,x))

check(infixform(integral(a(t),t)) == "integral(a(t),t)")

check(infixform(integral(f(x,y),x,y)) == "integral(integral(f(x,y),x),y)")
check(infixform(integral(f(x,y),y,x)) == "integral(integral(f(x,y),x),y)")

-- integral of integral

check(infixform(integral(f(x),x,x)) == "integral(integral(f(x),x),x)")
check(infixform(integral(f(x,y),x,y)) == "integral(integral(f(x,y),x),y)")
check(infixform(integral(f(x,y),y,x)) == "integral(integral(f(x,y),x),y)")

-- integral of derivative

check(infixform(integral(d(f(x),x),x)) == "f(x)")
check(infixform(integral(d(f(x),x),y)) == "y d(f(x),x)")
check(infixform(integral(d(f(x),y),x)) == "0")
check(infixform(integral(d(f(x),y),y)) == "0")

check(infixform(integral(d(f(x),x,x),x,x)) == "f(x)")

check(infixform(integral(d(f(x),x),x,x)) == "integral(f(x),x)")
check(infixform(integral(d(f(x),x,x),x)) == "d(f(x),x)")

check(infixform(integral(d(f(x,y),x,y),x)) == "d(f(x,y),y)")
check(infixform(integral(d(f(x,y),x,y),y)) == "d(f(x,y),x)")

check(infixform(integral(d(f(x,y),x,y),x,y)) == "f(x,y)")
check(infixform(integral(d(f(x,y),x,y),y,x)) == "f(x,y)")

check(infixform(integral(d(f(x,y),x),x,y)) == "integral(f(x,y),y)")
check(infixform(integral(d(f(x,y),x),y,x)) == "integral(f(x,y),y)")
