\[ \int \sinh ^5(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx \]
Optimal antiderivative \[ \frac {\left (a +b \right )^{3} \cosh \left (d x +c \right )}{d}-\frac {2 \left (a +b \right )^{2} \left (a +4 b \right ) \left (\cosh ^{3}\left (d x +c \right )\right )}{3 d}+\frac {\left (a +b \right ) \left (a^{2}+17 a b +28 b^{2}\right ) \left (\cosh ^{5}\left (d x +c \right )\right )}{5 d}-\frac {4 b \left (3 a^{2}+15 a b +14 b^{2}\right ) \left (\cosh ^{7}\left (d x +c \right )\right )}{7 d}+\frac {b \left (3 a^{2}+45 a b +70 b^{2}\right ) \left (\cosh ^{9}\left (d x +c \right )\right )}{9 d}-\frac {2 b^{2} \left (9 a +28 b \right ) \left (\cosh ^{11}\left (d x +c \right )\right )}{11 d}+\frac {b^{2} \left (3 a +28 b \right ) \left (\cosh ^{13}\left (d x +c \right )\right )}{13 d}-\frac {8 b^{3} \left (\cosh ^{15}\left (d x +c \right )\right )}{15 d}+\frac {b^{3} \left (\cosh ^{17}\left (d x +c \right )\right )}{17 d} \]
command
integrate(sinh(d*x+c)**5*(a+b*sinh(d*x+c)**4)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {a^{3} \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 a^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {8 a^{3} \cosh ^{5}{\left (c + d x \right )}}{15 d} + \frac {3 a^{2} b \sinh ^{8}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 a^{2} b \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {48 a^{2} b \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {192 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {128 a^{2} b \cosh ^{9}{\left (c + d x \right )}}{105 d} + \frac {3 a b^{2} \sinh ^{12}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {12 a b^{2} \sinh ^{10}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {24 a b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{d} - \frac {192 a b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{7 d} + \frac {128 a b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{7 d} - \frac {512 a b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{77 d} + \frac {1024 a b^{2} \cosh ^{13}{\left (c + d x \right )}}{1001 d} + \frac {b^{3} \sinh ^{16}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {16 b^{3} \sinh ^{14}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {224 b^{3} \sinh ^{12}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{15 d} - \frac {128 b^{3} \sinh ^{10}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{5 d} + \frac {256 b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{9 d} - \frac {2048 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{99 d} + \frac {4096 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{13}{\left (c + d x \right )}}{429 d} - \frac {16384 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{15}{\left (c + d x \right )}}{6435 d} + \frac {32768 b^{3} \cosh ^{17}{\left (c + d x \right )}}{109395 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{3} \sinh ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________