\[ \int \cos ^2(a+b x) \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx \]
Optimal antiderivative \[ -\frac {4 \left (\cos ^{5}\left (b x +a \right )\right )}{5 b}+\frac {8 \left (\cos ^{7}\left (b x +a \right )\right )}{7 b}-\frac {4 \left (\cos ^{9}\left (b x +a \right )\right )}{9 b} \]
command
integrate(cos(b*x+a)**2*sin(b*x+a)**3*sin(2*b*x+2*a)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {8 \sin ^{5}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{315 b} + \frac {16 \sin ^{4}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{315 b} - \frac {16 \sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{315 b} + \frac {44 \sin ^{3}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{315 b} - \frac {113 \sin ^{2}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )}}{315 b} + \frac {8 \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{315 b} - \frac {88 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{4}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{315 b} - \frac {2 \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{5}{\left (a + b x \right )}}{63 b} - \frac {32 \cos ^{5}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{315 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} \sin ^{2}{\left (2 a \right )} \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________