# Maple integration test file: "4 Trig functions\4.4 Cotangent\4.4.1.2 (d csc)^m (a+b cot)^n.txt"

lst:=[

# Integrands of the form (d Csc[e+f x])^m (a+b Cot[e+f x])^n

# Integrands of the form (d Csc[e+f x])^m (a+b Cot[e+f x])^n with a^2+b^2=0

# Integrands of the form Csc[e+f x]^m (a+a I Cot[e+f x])^n

# n>0

# n<0
[sin(x)^4/(I+cot(x)),x,4,-5/16*I*x+1/32/(I-cot(x))^2+(-1/8*I)/(I-cot(x))+(-1/24*I)/(I+cot(x))^3+(-3/32)/(I+cot(x))^2+3/16*I/(I+cot(x))],
[sin(x)^3/(I+cot(x)),x,3,4/5*I*cos(x)-4/15*I*cos(x)^3+1/5*I*sin(x)^3/(I+cot(x))],
[sin(x)^2/(I+cot(x)),x,4,-3/8*I*x+(-1/8*I)/(I-cot(x))+(-1/8)/(I+cot(x))^2+1/4*I/(I+cot(x))],
[sin(x)/(I+cot(x)),x,2,2/3*I*cos(x)+1/3*I*sin(x)/(I+cot(x))],
[csc(x)/(I+cot(x)),x,1,I*csc(x)/(I+cot(x))],
[csc(x)^2/(I+cot(x)),x,2,-I*x+log(sin(x))],
[csc(x)^3/(I+cot(x)),x,2,I*arctanh(cos(x))-csc(x)],
[csc(x)^4/(I+cot(x)),x,2,I*cot(x)-1/2*cot(x)^2],
[csc(x)^5/(I+cot(x)),x,3,1/2*I*arctanh(cos(x))+1/2*I*cot(x)*csc(x)-1/3*csc(x)^3],
[csc(x)^6/(I+cot(x)),x,3,I*cot(x)-1/2*cot(x)^2+1/3*I*cot(x)^3-1/4*cot(x)^4],
[csc(x)^7/(I+cot(x)),x,4,3/8*I*arctanh(cos(x))+3/8*I*cot(x)*csc(x)+1/4*I*cot(x)*csc(x)^3-1/5*csc(x)^5],

# Integrands of the form (d Csc[e+f x])^m (a+b Cot[e+f x])^n

# Integrands of the form Csc[e+f x]^m (a+b Cot[e+f x])^n

# n>0

# n<0
[csc(x)^6/(a+b*cot(x)),x,3,a*(a^2+2*b^2)*cot(x)/b^4-1/2*(a^2+2*b^2)*cot(x)^2/b^3+1/3*a*cot(x)^3/b^2-1/4*cot(x)^4/b-(a^2+b^2)^2*log(a+b*cot(x))/b^5],
[csc(x)^4/(a+b*cot(x)),x,3,a*cot(x)/b^2-1/2*cot(x)^2/b-(a^2+b^2)*log(a+b*cot(x))/b^3],
[csc(x)^2/(a+b*cot(x)),x,2,-log(a+b*cot(x))/b],
[sin(x)^2/(a+b*cot(x)),x,7,1/2*a*(a^2+3*b^2)*x/(a^2+b^2)^2-b^3*log(b*cos(x)+a*sin(x))/(a^2+b^2)^2-1/2*(b+a*cot(x))*sin(x)^2/(a^2+b^2)],
[sin(x)^4/(a+b*cot(x)),x,8,1/8*a*(3*a^4+10*a^2*b^2+15*b^4)*x/(a^2+b^2)^3-b^5*log(b*cos(x)+a*sin(x))/(a^2+b^2)^3-1/8*(4*b^3+a*(3*a^2+7*b^2)*cot(x))*sin(x)^2/(a^2+b^2)^2-1/4*(b+a*cot(x))*sin(x)^4/(a^2+b^2)],
[csc(x)^5/(a+b*cot(x)),x,9,1/2*a*arctanh(cos(x))/b^2+a*(a^2+b^2)*arctanh(cos(x))/b^4+(a^2+b^2)^(3/2)*arctanh((b-a*cot(x))*sin(x)/sqrt(a^2+b^2))/b^4-(a^2+b^2)*csc(x)/b^3+1/2*a*cot(x)*csc(x)/b^2-1/3*csc(x)^3/b],
[csc(x)^3/(a+b*cot(x)),x,5,a*arctanh(cos(x))/b^2-csc(x)/b+arctanh((b-a*cot(x))*sin(x)/sqrt(a^2+b^2))*sqrt(a^2+b^2)/b^2],
[csc(x)/(a+b*cot(x)),x,2,-arctanh((-b+a*cot(x))*sin(x)/sqrt(a^2+b^2))/sqrt(a^2+b^2)],
[sin(x)/(a+b*cot(x)),x,5,b^2*arctanh((b-a*cot(x))*sin(x)/sqrt(a^2+b^2))/(a^2+b^2)^(3/2)-a*cos(x)/(a^2+b^2)-b*sin(x)/(a^2+b^2)],
[sin(x)^3/(a+b*cot(x)),x,9,b^4*arctanh((b-a*cot(x))*sin(x)/sqrt(a^2+b^2))/(a^2+b^2)^(5/2)-a*b^2*cos(x)/(a^2+b^2)^2-a*cos(x)/(a^2+b^2)+1/3*a*cos(x)^3/(a^2+b^2)-b^3*sin(x)/(a^2+b^2)^2-1/3*b*sin(x)^3/(a^2+b^2)],
[csc(x)^2/(a+b*cot(x))^2,x,2,1/(b*(a+b*cot(x)))],

# Integrands of the form Csc[e+f x]^m (a+b Cot[e+f x])^(n/2)

# Integrands of the form Csc[e+f x]^(m/2) (a+b Cot[e+f x])^n

# Integrands of the form Csc[e+f x]^(m/2) (a+b Cot[e+f x])^(n/2)

# Integrands of the form (d Csc[e+f x])^m (a+b Cot[e+f x])^n with n symbolic
[(a+b*cot(x))^n*csc(x)^2,x,2,-(a+b*cot(x))^(1+n)/(b*(1+n))]]:

# Integrands of the form Sin[e+f x]^(m/2) (a+b Cot[e+f x])^n

# Integrands of the form Sin[e+f x]^(m/2) (a+b Cot[e+f x])^(n/2)
