\[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx \]
Optimal antiderivative \[ \frac {32 \left (\sin ^{9}\left (b x +a \right )\right )}{9 b}-\frac {64 \left (\sin ^{11}\left (b x +a \right )\right )}{11 b}+\frac {32 \left (\sin ^{13}\left (b x +a \right )\right )}{13 b} \]
command
integrate(sin(b*x+a)**3*sin(2*b*x+2*a)**5,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {1366 \sin ^{3}{\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{3003 b} - \frac {4960 \sin ^{3}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{9009 b} - \frac {256 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{1287 b} - \frac {271 \sin ^{2}{\left (a + b x \right )} \sin ^{5}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{3003 b} - \frac {48 \sin ^{2}{\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{143 b} - \frac {640 \sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {1388 \sin {\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{3003 b} - \frac {2944 \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {512 \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{1001 b} + \frac {2234 \sin ^{5}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )}}{9009 b} + \frac {4544 \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{9009 b} + \frac {256 \sin {\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{1001 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} \sin ^{5}{\left (2 a \right )} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________