\[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {a \cosh \! \left (d x +c \right )}{\left (a +b \right )^{2} d}+\frac {\cosh ^{3}\left (d x +c \right )}{3 \left (a +b \right ) d}+\frac {a \arctanh \! \left (\frac {\mathrm {sech}\left (d x +c \right ) \sqrt {b}}{\sqrt {a +b}}\right ) \sqrt {b}}{\left (a +b \right )^{\frac {5}{2}} d} \]
command
integrate(sinh(d*x+c)^3/(a+b*tanh(d*x+c)^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {\frac {24 \, \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} {\left (a b - \sqrt {-a b} a\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a^{3} e^{\left (2 \, c\right )} + a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )} + \sqrt {{\left (a^{3} e^{\left (2 \, c\right )} + a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )}^{2} - {\left (a^{3} e^{\left (4 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}\right )} {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}}}{a^{3} e^{\left (4 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5} + 2 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sqrt {-a b}} + \frac {24 \, {\left (3 \, a^{2} b - a b^{2} + {\left (a^{2} - 3 \, a b\right )} \sqrt {-a b}\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a^{3} e^{\left (2 \, c\right )} + a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )} - \sqrt {{\left (a^{3} e^{\left (2 \, c\right )} + a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )}^{2} - {\left (a^{3} e^{\left (4 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}\right )} {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}}}{a^{3} e^{\left (4 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{4} + 2 \, a^{3} b - 2 \, a b^{3} - b^{4} - 2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {-a b}\right )} \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}}} + \frac {{\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b e^{\left (2 \, d x + 2 \, c\right )} - a - b\right )} e^{\left (-3 \, d x\right )}}{a^{2} e^{\left (3 \, c\right )} + 2 \, a b e^{\left (3 \, c\right )} + b^{2} e^{\left (3 \, c\right )}} - \frac {a^{2} e^{\left (3 \, d x + 24 \, c\right )} + 2 \, a b e^{\left (3 \, d x + 24 \, c\right )} + b^{2} e^{\left (3 \, d x + 24 \, c\right )} - 9 \, a^{2} e^{\left (d x + 22 \, c\right )} - 6 \, a b e^{\left (d x + 22 \, c\right )} + 3 \, b^{2} e^{\left (d x + 22 \, c\right )}}{a^{3} e^{\left (21 \, c\right )} + 3 \, a^{2} b e^{\left (21 \, c\right )} + 3 \, a b^{2} e^{\left (21 \, c\right )} + b^{3} e^{\left (21 \, c\right )}}}{24 \, d} \]