\[ \int \frac {\text {sech}(c+d x)}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2} \, dx \]
Optimal antiderivative \[ \frac {a^{2} \left (a^{4}-3 b^{4}\right ) \arctan \left (\sinh \left (d x +c \right )\right )}{\left (a^{4}+b^{4}\right )^{2} d}+\frac {b^{2} \left (3 a^{4}-b^{4}\right ) \ln \left (\cosh \left (d x +c \right )\right )}{\left (a^{4}+b^{4}\right )^{2} d}-\frac {2 b^{2} \left (3 a^{4}-b^{4}\right ) \ln \left (a +b \left (\sqrt {\sinh }\left (d x +c \right )\right )\right )}{\left (a^{4}+b^{4}\right )^{2} d}-\frac {a b \left (a^{4}+2 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\sinh \left (d x +c \right )-\sqrt {2}\, \left (\sqrt {\sinh }\left (d x +c \right )\right )\right ) \sqrt {2}}{2 \left (a^{4}+b^{4}\right )^{2} d}+\frac {a b \left (a^{4}+2 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\sinh \left (d x +c \right )+\sqrt {2}\, \left (\sqrt {\sinh }\left (d x +c \right )\right )\right ) \sqrt {2}}{2 \left (a^{4}+b^{4}\right )^{2} d}-\frac {a b \left (a^{4}-2 a^{2} b^{2}-b^{4}\right ) \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\sinh }\left (d x +c \right )\right )\right ) \sqrt {2}}{\left (a^{4}+b^{4}\right )^{2} d}-\frac {a b \left (a^{4}-2 a^{2} b^{2}-b^{4}\right ) \arctan \left (1+\sqrt {2}\, \left (\sqrt {\sinh }\left (d x +c \right )\right )\right ) \sqrt {2}}{\left (a^{4}+b^{4}\right )^{2} d}+\frac {2 a \,b^{2}}{\left (a^{4}+b^{4}\right ) d \left (a +b \left (\sqrt {\sinh }\left (d x +c \right )\right )\right )} \]
command
integrate(sech(d*x+c)/(a+b*sinh(d*x+c)^(1/2))^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]