47.9 Problem number 700

\[ \int \frac {(c+d \sin (e+f x))^2}{a+b \sin (e+f x)} \, dx \]

Optimal antiderivative \[ \frac {d \left (-a d +2 b c \right ) x}{b^{2}}-\frac {d^{2} \cos \left (f x +e \right )}{b f}+\frac {2 \left (-a d +b c \right )^{2} \arctan \left (\frac {b +a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{b^{2} f \sqrt {a^{2}-b^{2}}} \]

command

integrate((c+d*sin(f*x+e))**2/(a+b*sin(f*x+e)),x)

Sympy 1.10.1 under Python 3.10.4 output

\[ \text {output too large to display} \]

Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________