\[ \int \frac {\text {sech}^5(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\sinh \! \left (d x +c \right )}{4 a d \left (a +\left (a +b \right ) \left (\sinh ^{2}\left (d x +c \right )\right )\right )^{2}}+\frac {3 \sinh \! \left (d x +c \right )}{8 a^{2} d \left (a +\left (a +b \right ) \left (\sinh ^{2}\left (d x +c \right )\right )\right )}+\frac {3 \arctan \! \left (\frac {\sinh \left (d x +c \right ) \sqrt {a +b}}{\sqrt {a}}\right )}{8 a^{\frac {5}{2}} d \sqrt {a +b}} \]
command
integrate(sech(d*x+c)^5/(a+b*tanh(d*x+c)^2)^3,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 {3 \, {\left (a^{3} - 10 \, a^{2} b + 5 \, a b^{2} + {\left (5 \, a^{2} - 10 \, a b + b^{2}\right )} \sqrt {-a b}\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 )} + \sqrt {{\left (a^{3} e^{\left (2 \, c\right )} - a^{2} b e^{\left (2 \, c\right )}\right )}^{2} - {\left (a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}\right )} {\left (a^{3} + a^{2} b\right )}}}{a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{a^{8} - 13 \, a^{7} b - 14 \, a^{6} b^{2} + 14 \, a^{5} b^{3} + 13 \, a^{4} b^{4} - a^{3} b^{5} + 2 \, {\left (3 \, a^{7} - 4 \, a^{6} b - 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + 3 \, a^{3} b^{4}\right )} \sqrt {-a b}} + \frac {3 \, {\left (a^{3} - 10 \, a^{2} b + 5 \, a b^{2} - {\left (5 \, a^{2} - 10 \, a b + b^{2}\right )} \sqrt {-a b}\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 )} - \sqrt {{\left (a^{3} e^{\left (2 \, c\right )} - a^{2} b e^{\left (2 \, c\right )}\right )}^{2} - {\left (a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}\right )} {\left (a^{3} + a^{2} b\right )}}}{a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{a^{8} - 13 \, a^{7} b - 14 \, a^{6} b^{2} + 14 \, a^{5} b^{3} + 13 \, a^{4} b^{4} - a^{3} b^{5} - 2 \, {\left (3 \, a^{7} - 4 \, a^{6} b - 14 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + 3 \, a^{3} b^{4}\right )} \sqrt {-a b}} + \frac {2 \, {\left (3 \, a e^{\left (7 \, d x + 7 \, c\right )} + 3 \, b e^{\left (7 \, d x + 7 \, c\right )} + 11 \, a e^{\left (5 \, d x + 5 \, c\right )} - 9 \, b e^{\left (5 \, d x + 5 \, c\right )} - 11 \, a e^{\left (3 \, d x + 3 \, c\right )} + 9 \, b e^{\left (3 \, d x + 3 \, c\right )} - 3 \, a e^{\left (d x + c\right )} - 3 \, b e^{\left (d x + c\right )}\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2} a^{2}}}{8 \, d} \]