\[ \int \frac {1}{(5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {-3 \cos \! \left (e x +d \right )+4 \sin \! \left (e x +d \right )}{10 e \left (5+4 \cos \! \left (e x +d \right )+3 \sin \! \left (e x +d \right )\right )^{\frac {3}{2}}}+\frac {\arctanh \! \left (\frac {\sin \left (d +e x -\arctan \left (\frac {3}{4}\right )\right ) \sqrt {2}}{2 \sqrt {1+\cos \left (d +e x -\arctan \left (\frac {3}{4}\right )\right )}}\right ) \sqrt {10}}{100 e} \]
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
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {1}{100} \, {\left (\frac {\sqrt {10} \log \left (\frac {{\left | -2 \, \sqrt {10} + 2 \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - 2 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 6 \right |}}{{\left | 2 \, \sqrt {10} + 2 \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - 2 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 6 \right |}}\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 3\right )} - \frac {20 \, {\left (19 \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{3} - 51 \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{2} - 17 \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} + 17 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 3\right )}}{{\left ({\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{2} - 6 \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} + 6 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 1\right )}^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 3\right )}\right )} e^{\left (-1\right )} \]