\[ \int \csc ^3(a+b x) \csc ^4(2 a+2 b x) \, dx \]
Optimal antiderivative \[ -\frac {105 \arctanh \left (\cos \left (b x +a \right )\right )}{256 b}+\frac {105 \sec \left (b x +a \right )}{256 b}+\frac {35 \left (\sec ^{3}\left (b x +a \right )\right )}{256 b}-\frac {21 \left (\csc ^{2}\left (b x +a \right )\right ) \left (\sec ^{3}\left (b x +a \right )\right )}{256 b}-\frac {3 \left (\csc ^{4}\left (b x +a \right )\right ) \left (\sec ^{3}\left (b x +a \right )\right )}{128 b}-\frac {\left (\csc ^{6}\left (b x +a \right )\right ) \left (\sec ^{3}\left (b x +a \right )\right )}{96 b} \]
command
integrate(csc(b*x+a)^3*csc(2*b*x+2*a)^4,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {285 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {21 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {\frac {18 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {225 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - \frac {2966 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac {3513 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - \frac {660 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {1155 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} - 1}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}}\right )}^{3}} - 1260 \, \log \left (-\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1}\right )}{6144 \, b} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________