\[ \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {3 \cos \! \left (e x +d \right )-4 \sin \! \left (e x +d \right )}{10 e \left (-5+4 \cos \! \left (e x +d \right )+3 \sin \! \left (e x +d \right )\right )^{\frac {3}{2}}}+\frac {\arctan \! \left (\frac {\sin \left (d +e x -\arctan \left (\frac {3}{4}\right )\right ) \sqrt {2}}{2 \sqrt {-1+\cos \left (d +e x -\arctan \left (\frac {3}{4}\right )\right )}}\right ) \sqrt {10}}{100 e} \]
command
integrate(1/(-5+4*cos(e*x+d)+3*sin(e*x+d))^(3/2),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
\[ -\frac {1}{450} \, {\left (\frac {9 \, \sqrt {10} \arctan \left (\frac {1}{10} \, \sqrt {10} {\left (-3 i \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} + 3 i \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - i\right )}\right )}{\mathrm {sgn}\left (-3 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 1\right )} + \frac {10 \, {\left (33 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{3} - 7 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{2} + 21 i \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - 21 i \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 9 i\right )}}{{\left (-3 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{2} - 2 i \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} + 2 i \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 3 i\right )}^{2} \mathrm {sgn}\left (-3 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 1\right )}\right )} e^{\left (-1\right )} \]