\[ \int \frac {\cosh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {\left (a +2 b \right ) \sinh \! \left (d x +c \right )}{\left (a +b \right )^{2} d}+\frac {\sinh ^{3}\left (d x +c \right )}{3 \left (a +b \right ) d}+\frac {b^{2} \arctan \! \left (\frac {\sinh \left (d x +c \right ) \sqrt {a +b}}{\sqrt {a}}\right )}{\left (a +b \right )^{\frac {5}{2}} d \sqrt {a}} \]
command
integrate(cosh(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 \, {\left (a b^{2} + \sqrt {-a b} b^{2}\right )} \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (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 )}}{a^{6} + 3 \, a^{5} b + 2 \, a^{4} b^{2} - 2 \, a^{3} b^{3} - 3 \, a^{2} b^{4} - a b^{5} + 2 \, {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \sqrt {-a b}} + \frac {24 \, {\left (a^{2} b^{2} - 3 \, a b^{3} - {\left (3 \, a b^{2} - b^{3}\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^{5} + 2 \, a^{4} b - 2 \, a^{2} b^{3} - a b^{4} - 2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a 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 )} + 21 \, 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 )} + 30 \, a b e^{\left (d x + 22 \, c\right )} + 21 \, 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} \]