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