\[ \int e^{2 (a+b x)} \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx \]
Optimal antiderivative \[ \frac {{\mathrm e}^{-4 b x -4 a}}{256 b}-\frac {3 \,{\mathrm e}^{4 b x +4 a}}{256 b}+\frac {{\mathrm e}^{8 b x +8 a}}{512 b}+\frac {3 x}{64} \]
command
integrate(exp(2*b*x+2*a)*cosh(b*x+a)**3*sinh(b*x+a)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {3 x e^{2 a} e^{2 b x} \sinh ^{6}{\left (a + b x \right )}}{64} - \frac {3 x e^{2 a} e^{2 b x} \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{32} - \frac {3 x e^{2 a} e^{2 b x} \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{64} + \frac {3 x e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{16} - \frac {3 x e^{2 a} e^{2 b x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{64} - \frac {3 x e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{32} + \frac {3 x e^{2 a} e^{2 b x} \cosh ^{6}{\left (a + b x \right )}}{64} + \frac {3 e^{2 a} e^{2 b x} \sinh ^{6}{\left (a + b x \right )}}{32 b} - \frac {15 e^{2 a} e^{2 b x} \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{64 b} + \frac {13 e^{2 a} e^{2 b x} \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{32 b} - \frac {15 e^{2 a} e^{2 b x} \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{64 b} + \frac {3 e^{2 a} e^{2 b x} \cosh ^{6}{\left (a + b x \right )}}{32 b} & \text {for}\: b \neq 0 \\x e^{2 a} \sinh ^{3}{\left (a \right )} \cosh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________