\[ \int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx \]
Optimal antiderivative \[ \frac {128 \left (\sin ^{9}\left (b x +a \right )\right )}{9 b}-\frac {384 \left (\sin ^{11}\left (b x +a \right )\right )}{11 b}+\frac {384 \left (\sin ^{13}\left (b x +a \right )\right )}{13 b}-\frac {128 \left (\sin ^{15}\left (b x +a \right )\right )}{15 b} \]
command
integrate(sin(b*x+a)*sin(2*b*x+2*a)**7,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {3838 \sin {\left (a + b x \right )} \sin ^{6}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{6435 b} - \frac {1648 \sin {\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{1287 b} - \frac {768 \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{715 b} - \frac {2048 \sin {\left (a + b x \right )} \cos ^{7}{\left (2 a + 2 b x \right )}}{6435 b} + \frac {1241 \sin ^{7}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{6435 b} + \frac {376 \sin ^{5}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{715 b} + \frac {640 \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{1287 b} + \frac {1024 \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{6}{\left (2 a + 2 b x \right )}}{6435 b} & \text {for}\: b \neq 0 \\x \sin {\left (a \right )} \sin ^{7}{\left (2 a \right )} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________