\[ \int \sin (a+b x) \sin ^6(2 a+2 b x) \, dx \]
Optimal antiderivative \[ -\frac {64 \left (\cos ^{7}\left (b x +a \right )\right )}{7 b}+\frac {64 \left (\cos ^{9}\left (b x +a \right )\right )}{3 b}-\frac {192 \left (\cos ^{11}\left (b x +a \right )\right )}{11 b}+\frac {64 \left (\cos ^{13}\left (b x +a \right )\right )}{13 b} \]
command
integrate(sin(b*x+a)*sin(2*b*x+2*a)**6,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {1084 \sin {\left (a + b x \right )} \sin ^{5}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{3003 b} - \frac {64 \sin {\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{143 b} - \frac {512 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {835 \sin ^{6}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{3003 b} - \frac {2776 \sin ^{4}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {2944 \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {1024 \cos {\left (a + b x \right )} \cos ^{6}{\left (2 a + 2 b x \right )}}{3003 b} & \text {for}\: b \neq 0 \\x \sin {\left (a \right )} \sin ^{6}{\left (2 a \right )} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________