98.6 Problem number 595

\[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^{5/2}} \, dx \]

Optimal antiderivative \[ \frac {\frac {2 b \cosh \left (x \right )}{3}+\frac {2 a \sinh \left (x \right )}{3}}{\left (a^{2}-b^{2}\right ) \left (a \cosh \left (x \right )+b \sinh \left (x \right )\right )^{\frac {3}{2}}}-\frac {2 i \sqrt {\frac {\cos \left (i x -\arctan \left (a , -i b \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {i x}{2}-\frac {\arctan \left (a , -i b \right )}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {a \cosh \left (x \right )+b \sinh \left (x \right )}{\sqrt {a^{2}-b^{2}}}}}{3 \cos \left (\frac {i x}{2}-\frac {\arctan \left (a , -i b \right )}{2}\right ) \left (a^{2}-b^{2}\right ) \sqrt {a \cosh \left (x \right )+b \sinh \left (x \right )}} \]

command

integrate(1/(a*cosh(x)+b*sinh(x))^(5/2),x, algorithm="fricas")

Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output

\[ \frac {2 \, {\left ({\left (\sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, \sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{4} + 2 \, \sqrt {2} {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + \sqrt {2} {\left (a^{2} - b^{2}\right )}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (\sqrt {2} {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{3} + \sqrt {2} {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2} {\left (a^{2} - 2 \, a b + b^{2}\right )}\right )} \sqrt {a + b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{3} + 3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{3} - {\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) + {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} - a^{2} + b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )}\right )}}{3 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \sinh \left (x\right )^{4} + 2 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5} + 3 \, {\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{5} + 3 \, a^{4} b + 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5}\right )} \cosh \left (x\right )^{3} + {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (\frac {\sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )}}{a^{3} \cosh \left (x\right )^{3} + 3 \, a^{2} b \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, a b^{2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + b^{3} \sinh \left (x\right )^{3}}, x\right ) \]