\[ \int \frac {\tan ^7(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {3 b \arctanh \! \left (\frac {b +2 c \tan \left (e x +d \right )}{2 \sqrt {c}\, \sqrt {a +b \tan \left (e x +d \right )+c \left (\tan ^{2}\left (e x +d \right )\right )}}\right )}{2 c^{\frac {5}{2}} e}-\frac {5 b \left (-12 a c +7 b^{2}\right ) \arctanh \! \left (\frac {b +2 c \tan \left (e x +d \right )}{2 \sqrt {c}\, \sqrt {a +b \tan \left (e x +d \right )+c \left (\tan ^{2}\left (e x +d \right )\right )}}\right )}{16 c^{\frac {9}{2}} e}+\frac {\arctanh \! \left (\frac {\left (b^{2}-\left (a -c \right ) \left (a -c -\sqrt {a^{2}-2 a c +b^{2}+c^{2}}\right )-b \left (2 a -2 c +\sqrt {a^{2}-2 a c +b^{2}+c^{2}}\right ) \tan \left (e x +d \right )\right ) \sqrt {2}}{2 \sqrt {2 a -2 c +\sqrt {a^{2}-2 a c +b^{2}+c^{2}}}\, \sqrt {a^{2}-b^{2}-2 a c +c^{2}-\left (a -c \right ) \sqrt {a^{2}-2 a c +b^{2}+c^{2}}}\, \sqrt {a +b \tan \left (e x +d \right )+c \left (\tan ^{2}\left (e x +d \right )\right )}}\right ) \sqrt {2 a -2 c +\sqrt {a^{2}-2 a c +b^{2}+c^{2}}}\, \sqrt {a^{2}-b^{2}-2 a c +c^{2}-\left (a -c \right ) \sqrt {a^{2}-2 a c +b^{2}+c^{2}}}\, \sqrt {2}}{2 \left (a^{2}-2 a c +b^{2}+c^{2}\right )^{\frac {3}{2}} e}-\frac {\arctanh \! \left (\frac {\left (b^{2}-\left (a -c \right ) \left (a -c +\sqrt {a^{2}-2 a c +b^{2}+c^{2}}\right )-b \left (2 a -2 c -\sqrt {a^{2}-2 a c +b^{2}+c^{2}}\right ) \tan \left (e x +d \right )\right ) \sqrt {2}}{2 \sqrt {2 a -2 c -\sqrt {a^{2}-2 a c +b^{2}+c^{2}}}\, \sqrt {a^{2}-b^{2}-2 a c +c^{2}+\left (a -c \right ) \sqrt {a^{2}-2 a c +b^{2}+c^{2}}}\, \sqrt {a +b \tan \left (e x +d \right )+c \left (\tan ^{2}\left (e x +d \right )\right )}}\right ) \sqrt {2 a -2 c -\sqrt {a^{2}-2 a c +b^{2}+c^{2}}}\, \sqrt {a^{2}-b^{2}-2 a c +c^{2}+\left (a -c \right ) \sqrt {a^{2}-2 a c +b^{2}+c^{2}}}\, \sqrt {2}}{2 \left (a^{2}-2 a c +b^{2}+c^{2}\right )^{\frac {3}{2}} e}+\frac {\left (-16 a c +7 b^{2}\right ) \sqrt {a +b \tan \! \left (e x +d \right )+c \left (\tan ^{2}\left (e x +d \right )\right )}\, \left (\tan ^{2}\left (e x +d \right )\right )}{3 c^{2} \left (-4 a c +b^{2}\right ) e}-\frac {2 b \sqrt {a +b \tan \! \left (e x +d \right )+c \left (\tan ^{2}\left (e x +d \right )\right )}\, \left (\tan ^{3}\left (e x +d \right )\right )}{c \left (-4 a c +b^{2}\right ) e}+\frac {4 a +2 b \tan \! \left (e x +d \right )}{\left (-4 a c +b^{2}\right ) e \sqrt {a +b \tan \! \left (e x +d \right )+c \left (\tan ^{2}\left (e x +d \right )\right )}}-\frac {2 \left (\tan ^{2}\left (e x +d \right )\right ) \left (2 a +b \tan \! \left (e x +d \right )\right )}{\left (-4 a c +b^{2}\right ) e \sqrt {a +b \tan \! \left (e x +d \right )+c \left (\tan ^{2}\left (e x +d \right )\right )}}+\frac {2 \left (\tan ^{4}\left (e x +d \right )\right ) \left (2 a +b \tan \! \left (e x +d \right )\right )}{\left (-4 a c +b^{2}\right ) e \sqrt {a +b \tan \! \left (e x +d \right )+c \left (\tan ^{2}\left (e x +d \right )\right )}}-\frac {\sqrt {a +b \tan \! \left (e x +d \right )+c \left (\tan ^{2}\left (e x +d \right )\right )}\, \left (3 b^{2}-8 a c -2 b c \tan \! \left (e x +d \right )\right )}{c^{2} \left (-4 a c +b^{2}\right ) e}-\frac {2 \left (a \left (b^{2}-2 \left (a -c \right ) c \right )+b c \left (a +c \right ) \tan \! \left (e x +d \right )\right )}{\left (b^{2}+\left (a -c \right )^{2}\right ) \left (-4 a c +b^{2}\right ) e \sqrt {a +b \tan \! \left (e x +d \right )+c \left (\tan ^{2}\left (e x +d \right )\right )}}+\frac {\sqrt {a +b \tan \! \left (e x +d \right )+c \left (\tan ^{2}\left (e x +d \right )\right )}\, \left (105 b^{4}-460 a \,b^{2} c +256 a^{2} c^{2}-2 b c \left (-116 a c +35 b^{2}\right ) \tan \! \left (e x +d \right )\right )}{24 c^{4} \left (-4 a c +b^{2}\right ) e} \]
command
int(tan(e*x+d)^7/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x)
Maple 2022.1 output
hanged
Maple 2021.1 output
\[ \text {output too large to display} \]