\[ \int \frac {\cos (x) \left (-\cos ^2(x)+2 \sqrt [4]{1+2 \sin (x)}\right )}{(1+2 \sin (x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {4}{\left (1+2 \sin \left (x \right )\right )^{\frac {1}{4}}}+\frac {\left (1+2 \sin \left (x \right )\right )^{\frac {3}{2}}}{12}+\frac {3}{4 \sqrt {1+2 \sin \left (x \right )}}-\frac {\sqrt {1+2 \sin \left (x \right )}}{2} \]
command
integrate(cos(x)*(-cos(x)**2+2*(1+2*sin(x))**(1/4))/(1+2*sin(x))**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {4 \left (2 \sin {\left (x \right )} + 1\right )^{\frac {3}{4}} \sin ^{2}{\left (x \right )}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} - \frac {2 \left (2 \sin {\left (x \right )} + 1\right )^{\frac {3}{4}} \sin {\left (x \right )}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} + \frac {3 \left (2 \sin {\left (x \right )} + 1\right )^{\frac {3}{4}} \cos ^{2}{\left (x \right )}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} - \frac {2 \left (2 \sin {\left (x \right )} + 1\right )^{\frac {3}{4}}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} - \frac {24 \sin {\left (x \right )}}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} - \frac {12}{6 \sqrt [4]{2 \sin {\left (x \right )} + 1} \sin {\left (x \right )} + 3 \sqrt [4]{2 \sin {\left (x \right )} + 1}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________