\[ \int \frac {\tan ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (a -b \right ) \cot \left (d x +c \right ) \EllipticE \left (\frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sqrt {a +b}}, \sqrt {\frac {a +b}{a -b}}\right ) \sqrt {a +b}\, \sqrt {\frac {b \left (1-\sec \left (d x +c \right )\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sec \left (d x +c \right )\right )}{a -b}}}{b^{2} d}-\frac {2 \cot \left (d x +c \right ) \EllipticF \left (\frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sqrt {a +b}}, \sqrt {\frac {a +b}{a -b}}\right ) \sqrt {a +b}\, \sqrt {\frac {b \left (1-\sec \left (d x +c \right )\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sec \left (d x +c \right )\right )}{a -b}}}{b d}+\frac {2 \cot \left (d x +c \right ) \EllipticPi \left (\frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sqrt {a +b}}, \frac {a +b}{a}, \sqrt {\frac {a +b}{a -b}}\right ) \sqrt {a +b}\, \sqrt {\frac {b \left (1-\sec \left (d x +c \right )\right )}{a +b}}\, \sqrt {-\frac {b \left (1+\sec \left (d x +c \right )\right )}{a -b}}}{a d} \]
command
Integrate[Tan[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]],x]
Mathematica 13.1 output
\[ \text {Result too large to show} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________