\[ \int \left (a \cosh ^3(x)\right )^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {26 i a^{2} \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\right ) \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}{77 \cosh \left (\frac {x}{2}\right ) \cosh \left (x \right )^{\frac {3}{2}}}+\frac {78 a^{2} \cosh \left (x \right ) \sinh \left (x \right ) \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}{385}+\frac {26 a^{2} \left (\cosh ^{3}\left (x \right )\right ) \sinh \left (x \right ) \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}{165}+\frac {2 a^{2} \left (\cosh ^{5}\left (x \right )\right ) \sinh \left (x \right ) \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}}{15}+\frac {26 a^{2} \sqrt {a \left (\cosh ^{3}\left (x \right )\right )}\, \tanh \left (x \right )}{77} \]
command
integrate((a*cosh(x)^3)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {24960 \, {\left (\sqrt {2} a^{2} \cosh \left (x\right )^{7} + 7 \, \sqrt {2} a^{2} \cosh \left (x\right )^{6} \sinh \left (x\right ) + 21 \, \sqrt {2} a^{2} \cosh \left (x\right )^{5} \sinh \left (x\right )^{2} + 35 \, \sqrt {2} a^{2} \cosh \left (x\right )^{4} \sinh \left (x\right )^{3} + 35 \, \sqrt {2} a^{2} \cosh \left (x\right )^{3} \sinh \left (x\right )^{4} + 21 \, \sqrt {2} a^{2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{5} + 7 \, \sqrt {2} a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{6} + \sqrt {2} a^{2} \sinh \left (x\right )^{7}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (77 \, a^{2} \cosh \left (x\right )^{14} + 1078 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{13} + 77 \, a^{2} \sinh \left (x\right )^{14} + 749 \, a^{2} \cosh \left (x\right )^{12} + 7 \, {\left (1001 \, a^{2} \cosh \left (x\right )^{2} + 107 \, a^{2}\right )} \sinh \left (x\right )^{12} + 3657 \, a^{2} \cosh \left (x\right )^{10} + 28 \, {\left (1001 \, a^{2} \cosh \left (x\right )^{3} + 321 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{11} + {\left (77077 \, a^{2} \cosh \left (x\right )^{4} + 49434 \, a^{2} \cosh \left (x\right )^{2} + 3657 \, a^{2}\right )} \sinh \left (x\right )^{10} + 15465 \, a^{2} \cosh \left (x\right )^{8} + 2 \, {\left (77077 \, a^{2} \cosh \left (x\right )^{5} + 82390 \, a^{2} \cosh \left (x\right )^{3} + 18285 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{9} + 3 \, {\left (77077 \, a^{2} \cosh \left (x\right )^{6} + 123585 \, a^{2} \cosh \left (x\right )^{4} + 54855 \, a^{2} \cosh \left (x\right )^{2} + 5155 \, a^{2}\right )} \sinh \left (x\right )^{8} - 15465 \, a^{2} \cosh \left (x\right )^{6} + 24 \, {\left (11011 \, a^{2} \cosh \left (x\right )^{7} + 24717 \, a^{2} \cosh \left (x\right )^{5} + 18285 \, a^{2} \cosh \left (x\right )^{3} + 5155 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{7} + 3 \, {\left (77077 \, a^{2} \cosh \left (x\right )^{8} + 230692 \, a^{2} \cosh \left (x\right )^{6} + 255990 \, a^{2} \cosh \left (x\right )^{4} + 144340 \, a^{2} \cosh \left (x\right )^{2} - 5155 \, a^{2}\right )} \sinh \left (x\right )^{6} - 3657 \, a^{2} \cosh \left (x\right )^{4} + 2 \, {\left (77077 \, a^{2} \cosh \left (x\right )^{9} + 296604 \, a^{2} \cosh \left (x\right )^{7} + 460782 \, a^{2} \cosh \left (x\right )^{5} + 433020 \, a^{2} \cosh \left (x\right )^{3} - 46395 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + {\left (77077 \, a^{2} \cosh \left (x\right )^{10} + 370755 \, a^{2} \cosh \left (x\right )^{8} + 767970 \, a^{2} \cosh \left (x\right )^{6} + 1082550 \, a^{2} \cosh \left (x\right )^{4} - 231975 \, a^{2} \cosh \left (x\right )^{2} - 3657 \, a^{2}\right )} \sinh \left (x\right )^{4} - 749 \, a^{2} \cosh \left (x\right )^{2} + 4 \, {\left (7007 \, a^{2} \cosh \left (x\right )^{11} + 41195 \, a^{2} \cosh \left (x\right )^{9} + 109710 \, a^{2} \cosh \left (x\right )^{7} + 216510 \, a^{2} \cosh \left (x\right )^{5} - 77325 \, a^{2} \cosh \left (x\right )^{3} - 3657 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (7007 \, a^{2} \cosh \left (x\right )^{12} + 49434 \, a^{2} \cosh \left (x\right )^{10} + 164565 \, a^{2} \cosh \left (x\right )^{8} + 433020 \, a^{2} \cosh \left (x\right )^{6} - 231975 \, a^{2} \cosh \left (x\right )^{4} - 21942 \, a^{2} \cosh \left (x\right )^{2} - 749 \, a^{2}\right )} \sinh \left (x\right )^{2} - 77 \, a^{2} + 2 \, {\left (539 \, a^{2} \cosh \left (x\right )^{13} + 4494 \, a^{2} \cosh \left (x\right )^{11} + 18285 \, a^{2} \cosh \left (x\right )^{9} + 61860 \, a^{2} \cosh \left (x\right )^{7} - 46395 \, a^{2} \cosh \left (x\right )^{5} - 7314 \, a^{2} \cosh \left (x\right )^{3} - 749 \, a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right )}}{73920 \, {\left (\cosh \left (x\right )^{7} + 7 \, \cosh \left (x\right )^{6} \sinh \left (x\right ) + 21 \, \cosh \left (x\right )^{5} \sinh \left (x\right )^{2} + 35 \, \cosh \left (x\right )^{4} \sinh \left (x\right )^{3} + 35 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{4} + 21 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{5} + 7 \, \cosh \left (x\right ) \sinh \left (x\right )^{6} + \sinh \left (x\right )^{7}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {a \cosh \left (x\right )^{3}} a^{2} \cosh \left (x\right )^{6}, x\right ) \]