\[ \int \frac {\sec ^2(x)}{a+b \sin (x)+c \sin ^2(x)} \, dx \]
Optimal antiderivative \[ \frac {\cos \left (x \right )}{2 \left (a +b +c \right ) \left (1-\sin \left (x \right )\right )}-\frac {\cos \left (x \right )}{2 \left (a -b +c \right ) \left (1+\sin \left (x \right )\right )}-\frac {b c \arctan \left (\frac {\left (2 c +\left (b -\sqrt {-4 a c +b^{2}}\right ) \tan \left (\frac {x}{2}\right )\right ) \sqrt {2}}{2 \sqrt {b^{2}-2 c \left (a +c \right )-b \sqrt {-4 a c +b^{2}}}}\right ) \sqrt {2}\, \left (1+\frac {b^{2}-2 c \left (a +c \right )}{b \sqrt {-4 a c +b^{2}}}\right )}{\left (a -b +c \right ) \left (a +b +c \right ) \sqrt {b^{2}-2 c \left (a +c \right )-b \sqrt {-4 a c +b^{2}}}}-\frac {b c \arctan \left (\frac {\left (2 c +\left (b +\sqrt {-4 a c +b^{2}}\right ) \tan \left (\frac {x}{2}\right )\right ) \sqrt {2}}{2 \sqrt {b^{2}-2 c \left (a +c \right )+b \sqrt {-4 a c +b^{2}}}}\right ) \sqrt {2}\, \left (1+\frac {-b^{2}+2 c \left (a +c \right )}{b \sqrt {-4 a c +b^{2}}}\right )}{\left (a -b +c \right ) \left (a +b +c \right ) \sqrt {b^{2}-2 c \left (a +c \right )+b \sqrt {-4 a c +b^{2}}}} \]
command
integrate(sec(x)^2/(a+b*sin(x)+c*sin(x)^2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]