\[ \int \left (-1+\sec ^2(2 x)\right )^3 \sin (2 x) \, dx \]
Optimal antiderivative \[ \frac {\cos \left (2 x \right )}{2}+\frac {3 \sec \left (2 x \right )}{2}-\frac {\left (\sec ^{3}\left (2 x \right )\right )}{2}+\frac {\left (\sec ^{5}\left (2 x \right )\right )}{10} \]
command
integrate((-1+sec(2*x)**2)**3*sin(2*x),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {\cos {\left (2 x \right )}}{2} - \frac {1}{2 \left (2 \cos ^{2}{\left (x \right )} - 1\right )^{3}} + \frac {1}{10 \left (2 \cos ^{2}{\left (x \right )} - 1\right )^{5}} + \frac {3}{4 \left (\cos ^{2}{\left (x \right )} - \frac {1}{2}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________