\[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {2 a +b \cot \left (e x +d \right )}{2 \sqrt {a}\, \sqrt {a +b \cot \left (e x +d \right )+c \left (\cot ^{2}\left (e x +d \right )\right )}}\right )}{a^{\frac {3}{2}} e}+\frac {3 \left (-4 a c +5 b^{2}\right ) \arctanh \left (\frac {2 a +b \cot \left (e x +d \right )}{2 \sqrt {a}\, \sqrt {a +b \cot \left (e x +d \right )+c \left (\cot ^{2}\left (e x +d \right )\right )}}\right )}{8 a^{\frac {7}{2}} e}+\frac {2 b^{2}-4 a c +2 b c \cot \left (e x +d \right )}{a \left (-4 a c +b^{2}\right ) e \sqrt {a +b \cot \left (e x +d \right )+c \left (\cot ^{2}\left (e x +d \right )\right )}}-\frac {2 \left (a \left (b^{2}-2 \left (a -c \right ) c \right )+b c \left (a +c \right ) \cot \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 \cot \left (e x +d \right )+c \left (\cot ^{2}\left (e x +d \right )\right )}}+\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 \cot \left (e x +d \right ) \left (2 a -2 c +\sqrt {a^{2}-2 a c +b^{2}+c^{2}}\right )\right ) \sqrt {2}}{2 \sqrt {a +b \cot \left (e x +d \right )+c \left (\cot ^{2}\left (e x +d \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}}}}\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}-b \cot \left (e x +d \right ) \left (2 a -2 c -\sqrt {a^{2}-2 a c +b^{2}+c^{2}}\right )-\left (a -c \right ) \left (a -c +\sqrt {a^{2}-2 a c +b^{2}+c^{2}}\right )\right ) \sqrt {2}}{2 \sqrt {a +b \cot \left (e x +d \right )+c \left (\cot ^{2}\left (e x +d \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}}}}\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 {b \left (-52 a c +15 b^{2}\right ) \sqrt {a +b \cot \left (e x +d \right )+c \left (\cot ^{2}\left (e x +d \right )\right )}\, \tan \left (e x +d \right )}{4 a^{3} \left (-4 a c +b^{2}\right ) e}-\frac {2 \left (b^{2}-2 a c +b c \cot \left (e x +d \right )\right ) \left (\tan ^{2}\left (e x +d \right )\right )}{a \left (-4 a c +b^{2}\right ) e \sqrt {a +b \cot \left (e x +d \right )+c \left (\cot ^{2}\left (e x +d \right )\right )}}+\frac {\left (-12 a c +5 b^{2}\right ) \sqrt {a +b \cot \left (e x +d \right )+c \left (\cot ^{2}\left (e x +d \right )\right )}\, \left (\tan ^{2}\left (e x +d \right )\right )}{2 a^{2} \left (-4 a c +b^{2}\right ) e} \]
command
int(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(3/2),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(29682798\) |
Maple 2021.1 output
\[ \int \frac {\tan ^{3}\left (e x +d \right )}{\left (a +b \cot \left (e x +d \right )+c \left (\cot ^{2}\left (e x +d \right )\right )\right )^{\frac {3}{2}}}\, dx \]________________________________________________________________________________________