Integrand size = 15, antiderivative size = 103 \[ \int \csc ^3(a+b x) \tan (c+b x) \, dx=-\frac {\csc (a+b x) \sec ^2(a-c)}{b}+\frac {\text {arctanh}(\sin (c+b x)) \sec ^3(a-c)}{b}-\frac {\text {arctanh}(\cos (a+b x)) \sec ^3(a-c) (-9 \sin (a-c)+\sin (3 (-a+c)))}{8 b}+\frac {\cot (a+b x) \csc (a+b x) \tan (a-c)}{2 b} \] Output:
-csc(b*x+a)*sec(a-c)^2/b+arctanh(sin(b*x+c))*sec(a-c)^3/b-1/8*arctanh(cos( b*x+a))*sec(a-c)^3*(-9*sin(a-c)-sin(3*a-3*c))/b+1/2*cot(b*x+a)*csc(b*x+a)* tan(a-c)/b
Result contains complex when optimal does not.
Time = 6.76 (sec) , antiderivative size = 322, normalized size of antiderivative = 3.13 \[ \int \csc ^3(a+b x) \tan (c+b x) \, dx=-\frac {2 i \arctan \left (\frac {(i \cos (c)+\sin (c)) \left (\cos \left (\frac {b x}{2}\right ) \sin (c)+\cos (c) \sin \left (\frac {b x}{2}\right )\right )}{\cos (c) \cos \left (\frac {b x}{2}\right )-i \cos \left (\frac {b x}{2}\right ) \sin (c)}\right )}{\frac {1}{4} b \cos (3 a-3 c)+\frac {3}{4} b \cos (a-c)}+\frac {\log \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \sec ^3(a-c) (-\sin (3 a-3 c)-9 \sin (a-c))}{8 b}+\frac {\log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \sec ^3(a-c) (\sin (3 a-3 c)+9 \sin (a-c))}{8 b}+\frac {\csc \left (\frac {a}{2}\right ) \csc \left (\frac {a}{2}+\frac {b x}{2}\right ) \sec ^2(a-c) \sin \left (\frac {b x}{2}\right )}{2 b}-\frac {\sec \left (\frac {a}{2}\right ) \sec ^2(a-c) \sec \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {b x}{2}\right )}{2 b}+\frac {\csc ^2\left (\frac {a}{2}+\frac {b x}{2}\right ) \tan (a-c)}{8 b}-\frac {\sec ^2\left (\frac {a}{2}+\frac {b x}{2}\right ) \tan (a-c)}{8 b} \] Input:
Integrate[Csc[a + b*x]^3*Tan[c + b*x],x]
Output:
((-2*I)*ArcTan[((I*Cos[c] + Sin[c])*(Cos[(b*x)/2]*Sin[c] + Cos[c]*Sin[(b*x )/2]))/(Cos[c]*Cos[(b*x)/2] - I*Cos[(b*x)/2]*Sin[c])])/((b*Cos[3*a - 3*c]) /4 + (3*b*Cos[a - c])/4) + (Log[Sin[a/2 + (b*x)/2]]*Sec[a - c]^3*(-Sin[3*a - 3*c] - 9*Sin[a - c]))/(8*b) + (Log[Cos[a/2 + (b*x)/2]]*Sec[a - c]^3*(Si n[3*a - 3*c] + 9*Sin[a - c]))/(8*b) + (Csc[a/2]*Csc[a/2 + (b*x)/2]*Sec[a - c]^2*Sin[(b*x)/2])/(2*b) - (Sec[a/2]*Sec[a - c]^2*Sec[a/2 + (b*x)/2]*Sin[ (b*x)/2])/(2*b) + (Csc[a/2 + (b*x)/2]^2*Tan[a - c])/(8*b) - (Sec[a/2 + (b* x)/2]^2*Tan[a - c])/(8*b)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \csc ^3(a+b x) \tan (b x+c) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \csc ^3(a+b x) \tan (b x+c)dx\) |
Input:
Int[Csc[a + b*x]^3*Tan[c + b*x],x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 742, normalized size of antiderivative = 7.20
| method | result | size |
| risch | \(\frac {i \left ({\mathrm e}^{i \left (3 b x +7 a \right )}-8 \,{\mathrm e}^{i \left (3 b x +5 a +2 c \right )}-{\mathrm e}^{i \left (3 b x +3 a +4 c \right )}+{\mathrm e}^{i \left (b x +5 a \right )}+8 \,{\mathrm e}^{i \left (b x +3 a +2 c \right )}-{\mathrm e}^{i \left (b x +a +4 c \right )}\right )}{b \left ({\mathrm e}^{2 i a}+{\mathrm e}^{2 i c}\right )^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {8 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) {\mathrm e}^{3 i \left (a +c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}-\frac {i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{6 i a}}{2 b \left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right )}-\frac {9 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{2 i \left (2 a +c \right )}}{2 b \left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right )}+\frac {9 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{2 i \left (a +2 c \right )}}{2 b \left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right )}+\frac {i \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) {\mathrm e}^{6 i c}}{2 b \left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right )}+\frac {i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{6 i a}}{2 b \left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right )}+\frac {9 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i \left (2 a +c \right )}}{2 b \left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right )}-\frac {9 i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i \left (a +2 c \right )}}{2 b \left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right )}-\frac {i \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right ) {\mathrm e}^{6 i c}}{2 b \left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right )}-\frac {8 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) {\mathrm e}^{3 i \left (a +c \right )}}{\left ({\mathrm e}^{6 i a}+3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}+{\mathrm e}^{6 i c}\right ) b}\) | \(742\) |
Input:
int(csc(b*x+a)^3*tan(b*x+c),x,method=_RETURNVERBOSE)
Output:
I/b/(exp(2*I*a)+exp(2*I*c))^2/(exp(2*I*(b*x+a))-1)^2*(exp(I*(3*b*x+7*a))-8 *exp(I*(3*b*x+5*a+2*c))-exp(I*(3*b*x+3*a+4*c))+exp(I*(b*x+5*a))+8*exp(I*(b *x+3*a+2*c))-exp(I*(b*x+a+4*c)))+8*ln(exp(I*(b*x+a))+I*exp(I*(a-c)))/(exp( 6*I*a)+3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))+exp(6*I*c))/b*exp(3*I*(a+c))- 1/2*I/b/(exp(6*I*a)+3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))+exp(6*I*c))*ln(e xp(I*(b*x+a))+1)*exp(6*I*a)-9/2*I/b/(exp(6*I*a)+3*exp(2*I*(2*a+c))+3*exp(2 *I*(a+2*c))+exp(6*I*c))*ln(exp(I*(b*x+a))+1)*exp(2*I*(2*a+c))+9/2*I/b/(exp (6*I*a)+3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))+exp(6*I*c))*ln(exp(I*(b*x+a) )+1)*exp(2*I*(a+2*c))+1/2*I/b/(exp(6*I*a)+3*exp(2*I*(2*a+c))+3*exp(2*I*(a+ 2*c))+exp(6*I*c))*ln(exp(I*(b*x+a))+1)*exp(6*I*c)+1/2*I/b/(exp(6*I*a)+3*ex p(2*I*(2*a+c))+3*exp(2*I*(a+2*c))+exp(6*I*c))*ln(exp(I*(b*x+a))-1)*exp(6*I *a)+9/2*I/b/(exp(6*I*a)+3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))+exp(6*I*c))* ln(exp(I*(b*x+a))-1)*exp(2*I*(2*a+c))-9/2*I/b/(exp(6*I*a)+3*exp(2*I*(2*a+c ))+3*exp(2*I*(a+2*c))+exp(6*I*c))*ln(exp(I*(b*x+a))-1)*exp(2*I*(a+2*c))-1/ 2*I/b/(exp(6*I*a)+3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))+exp(6*I*c))*ln(exp (I*(b*x+a))-1)*exp(6*I*c)-8*ln(exp(I*(b*x+a))-I*exp(I*(a-c)))/(exp(6*I*a)+ 3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))+exp(6*I*c))/b*exp(3*I*(a+c))
Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (99) = 198\).
Time = 0.12 (sec) , antiderivative size = 420, normalized size of antiderivative = 4.08 \[ \int \csc ^3(a+b x) \tan (c+b x) \, dx=\frac {2 \, {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 5\right )} \cos \left (b x + a\right )^{2} - \cos \left (-2 \, a + 2 \, c\right ) - 5\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (-2 \, a + 2 \, c\right ) + {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 5\right )} \cos \left (b x + a\right )^{2} - \cos \left (-2 \, a + 2 \, c\right ) - 5\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac {4 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right )^{2} - \cos \left (-2 \, a + 2 \, c\right ) - 1\right )} \log \left (\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) - 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) + 1}\right )}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} + 8 \, {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \sin \left (b x + a\right )}{4 \, {\left ({\left (b \cos \left (-2 \, a + 2 \, c\right )^{2} + 2 \, b \cos \left (-2 \, a + 2 \, c\right ) + b\right )} \cos \left (b x + a\right )^{2} - b \cos \left (-2 \, a + 2 \, c\right )^{2} - 2 \, b \cos \left (-2 \, a + 2 \, c\right ) - b\right )}} \] Input:
integrate(csc(b*x+a)^3*tan(b*x+c),x, algorithm="fricas")
Output:
1/4*(2*(cos(-2*a + 2*c) + 1)*cos(b*x + a)*sin(-2*a + 2*c) - ((cos(-2*a + 2 *c) + 5)*cos(b*x + a)^2 - cos(-2*a + 2*c) - 5)*log(1/2*cos(b*x + a) + 1/2) *sin(-2*a + 2*c) + ((cos(-2*a + 2*c) + 5)*cos(b*x + a)^2 - cos(-2*a + 2*c) - 5)*log(-1/2*cos(b*x + a) + 1/2)*sin(-2*a + 2*c) + 4*sqrt(2)*((cos(-2*a + 2*c) + 1)*cos(b*x + a)^2 - cos(-2*a + 2*c) - 1)*log((2*cos(b*x + a)^2*co s(-2*a + 2*c) - 2*cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - 2*sqrt(2)*(( cos(-2*a + 2*c) + 1)*sin(b*x + a) + cos(b*x + a)*sin(-2*a + 2*c))/sqrt(cos (-2*a + 2*c) + 1) - cos(-2*a + 2*c) - 3)/(2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - cos(-2*a + 2*c) + 1))/sqr t(cos(-2*a + 2*c) + 1) + 8*(cos(-2*a + 2*c) + 1)*sin(b*x + a))/((b*cos(-2* a + 2*c)^2 + 2*b*cos(-2*a + 2*c) + b)*cos(b*x + a)^2 - b*cos(-2*a + 2*c)^2 - 2*b*cos(-2*a + 2*c) - b)
\[ \int \csc ^3(a+b x) \tan (c+b x) \, dx=\int \tan {\left (b x + c \right )} \csc ^{3}{\left (a + b x \right )}\, dx \] Input:
integrate(csc(b*x+a)**3*tan(b*x+c),x)
Output:
Integral(tan(b*x + c)*csc(a + b*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 116810 vs. \(2 (99) = 198\).
Time = 8.14 (sec) , antiderivative size = 116810, normalized size of antiderivative = 1134.08 \[ \int \csc ^3(a+b x) \tan (c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^3*tan(b*x+c),x, algorithm="maxima")
Output:
1/2*(16*(((sin(6*a) + 3*sin(4*a + 2*c) + sin(6*c))*cos(3*a + 3*c) - (cos(6 *a) + 3*cos(4*a + 2*c) + cos(6*c))*sin(3*a + 3*c) - 3*cos(2*a + 4*c)*sin(3 *a + 3*c) + 3*cos(3*a + 3*c)*sin(2*a + 4*c))*cos(4*b*x + 8*a)^2 + 4*((sin( 6*a) + 3*sin(4*a + 2*c) + sin(6*c))*cos(3*a + 3*c) - (cos(6*a) + 3*cos(4*a + 2*c) + cos(6*c))*sin(3*a + 3*c) - 3*cos(2*a + 4*c)*sin(3*a + 3*c) + 3*c os(3*a + 3*c)*sin(2*a + 4*c))*cos(4*b*x + 6*a + 2*c)^2 + ((sin(6*a) + 3*si n(4*a + 2*c) + sin(6*c))*cos(3*a + 3*c) - (cos(6*a) + 3*cos(4*a + 2*c) + c os(6*c))*sin(3*a + 3*c) - 3*cos(2*a + 4*c)*sin(3*a + 3*c) + 3*cos(3*a + 3* c)*sin(2*a + 4*c))*cos(4*b*x + 4*a + 4*c)^2 + 4*((sin(6*a) + 3*sin(4*a + 2 *c) + sin(6*c))*cos(3*a + 3*c) - (cos(6*a) + 3*cos(4*a + 2*c) + cos(6*c))* sin(3*a + 3*c) - 3*cos(2*a + 4*c)*sin(3*a + 3*c) + 3*cos(3*a + 3*c)*sin(2* a + 4*c))*cos(2*b*x + 6*a)^2 + 16*((sin(6*a) + 3*sin(4*a + 2*c) + sin(6*c) )*cos(3*a + 3*c) - (cos(6*a) + 3*cos(4*a + 2*c) + cos(6*c))*sin(3*a + 3*c) - 3*cos(2*a + 4*c)*sin(3*a + 3*c) + 3*cos(3*a + 3*c)*sin(2*a + 4*c))*cos( 2*b*x + 4*a + 2*c)^2 + 4*((sin(6*a) + 3*sin(4*a + 2*c) + sin(6*c))*cos(3*a + 3*c) - (cos(6*a) + 3*cos(4*a + 2*c) + cos(6*c))*sin(3*a + 3*c) - 3*cos( 2*a + 4*c)*sin(3*a + 3*c) + 3*cos(3*a + 3*c)*sin(2*a + 4*c))*cos(2*b*x + 2 *a + 4*c)^2 + ((sin(6*a) + 3*sin(4*a + 2*c) + sin(6*c))*cos(3*a + 3*c) - ( cos(6*a) + 3*cos(4*a + 2*c) + cos(6*c))*sin(3*a + 3*c) - 3*cos(2*a + 4*c)* sin(3*a + 3*c) + 3*cos(3*a + 3*c)*sin(2*a + 4*c))*sin(4*b*x + 8*a)^2 + ...
Leaf count of result is larger than twice the leaf count of optimal. 6012 vs. \(2 (99) = 198\).
Time = 0.43 (sec) , antiderivative size = 6012, normalized size of antiderivative = 58.37 \[ \int \csc ^3(a+b x) \tan (c+b x) \, dx=\text {Too large to display} \] Input:
integrate(csc(b*x+a)^3*tan(b*x+c),x, algorithm="giac")
Output:
-1/4*(4*(tan(1/2*a)^7*tan(1/2*c)^7 - tan(1/2*a)^7*tan(1/2*c)^6 + tan(1/2*a )^6*tan(1/2*c)^7 + 3*tan(1/2*a)^7*tan(1/2*c)^5 + tan(1/2*a)^6*tan(1/2*c)^6 + 3*tan(1/2*a)^5*tan(1/2*c)^7 - 3*tan(1/2*a)^7*tan(1/2*c)^4 + 3*tan(1/2*a )^6*tan(1/2*c)^5 - 3*tan(1/2*a)^5*tan(1/2*c)^6 + 3*tan(1/2*a)^4*tan(1/2*c) ^7 + 3*tan(1/2*a)^7*tan(1/2*c)^3 + 3*tan(1/2*a)^6*tan(1/2*c)^4 + 9*tan(1/2 *a)^5*tan(1/2*c)^5 + 3*tan(1/2*a)^4*tan(1/2*c)^6 + 3*tan(1/2*a)^3*tan(1/2* c)^7 - 3*tan(1/2*a)^7*tan(1/2*c)^2 + 3*tan(1/2*a)^6*tan(1/2*c)^3 - 9*tan(1 /2*a)^5*tan(1/2*c)^4 + 9*tan(1/2*a)^4*tan(1/2*c)^5 - 3*tan(1/2*a)^3*tan(1/ 2*c)^6 + 3*tan(1/2*a)^2*tan(1/2*c)^7 + tan(1/2*a)^7*tan(1/2*c) + 3*tan(1/2 *a)^6*tan(1/2*c)^2 + 9*tan(1/2*a)^5*tan(1/2*c)^3 + 9*tan(1/2*a)^4*tan(1/2* c)^4 + 9*tan(1/2*a)^3*tan(1/2*c)^5 + 3*tan(1/2*a)^2*tan(1/2*c)^6 + tan(1/2 *a)*tan(1/2*c)^7 - tan(1/2*a)^7 + tan(1/2*a)^6*tan(1/2*c) - 9*tan(1/2*a)^5 *tan(1/2*c)^2 + 9*tan(1/2*a)^4*tan(1/2*c)^3 - 9*tan(1/2*a)^3*tan(1/2*c)^4 + 9*tan(1/2*a)^2*tan(1/2*c)^5 - tan(1/2*a)*tan(1/2*c)^6 + tan(1/2*c)^7 + t an(1/2*a)^6 + 3*tan(1/2*a)^5*tan(1/2*c) + 9*tan(1/2*a)^4*tan(1/2*c)^2 + 9* tan(1/2*a)^3*tan(1/2*c)^3 + 9*tan(1/2*a)^2*tan(1/2*c)^4 + 3*tan(1/2*a)*tan (1/2*c)^5 + tan(1/2*c)^6 - 3*tan(1/2*a)^5 + 3*tan(1/2*a)^4*tan(1/2*c) - 9* tan(1/2*a)^3*tan(1/2*c)^2 + 9*tan(1/2*a)^2*tan(1/2*c)^3 - 3*tan(1/2*a)*tan (1/2*c)^4 + 3*tan(1/2*c)^5 + 3*tan(1/2*a)^4 + 3*tan(1/2*a)^3*tan(1/2*c) + 9*tan(1/2*a)^2*tan(1/2*c)^2 + 3*tan(1/2*a)*tan(1/2*c)^3 + 3*tan(1/2*c)^...
Timed out. \[ \int \csc ^3(a+b x) \tan (c+b x) \, dx=\text {Hanged} \] Input:
int(tan(c + b*x)/sin(a + b*x)^3,x)
Output:
\text{Hanged}
\[ \int \csc ^3(a+b x) \tan (c+b x) \, dx=\int \csc \left (b x +a \right )^{3} \tan \left (b x +c \right )d x \] Input:
int(csc(b*x+a)^3*tan(b*x+c),x)
Output:
int(csc(a + b*x)**3*tan(b*x + c),x)