\[ \int \frac {a+b \csc ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (a +b \,\mathrm {arccsc}\! \left (c x \right )\right ) \ln \! \left (1-\frac {\mathrm {I} c \left (\frac {\mathrm {I}}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) \sqrt {-d}}{\sqrt {e}-\sqrt {c^{2} d +e}}\right )}{4 \left (-d \right )^{\frac {3}{2}} \sqrt {e}}-\frac {\left (a +b \,\mathrm {arccsc}\! \left (c x \right )\right ) \ln \! \left (1+\frac {\mathrm {I} c \left (\frac {\mathrm {I}}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) \sqrt {-d}}{\sqrt {e}-\sqrt {c^{2} d +e}}\right )}{4 \left (-d \right )^{\frac {3}{2}} \sqrt {e}}+\frac {\left (a +b \,\mathrm {arccsc}\! \left (c x \right )\right ) \ln \! \left (1-\frac {\mathrm {I} c \left (\frac {\mathrm {I}}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) \sqrt {-d}}{\sqrt {e}+\sqrt {c^{2} d +e}}\right )}{4 \left (-d \right )^{\frac {3}{2}} \sqrt {e}}-\frac {\left (a +b \,\mathrm {arccsc}\! \left (c x \right )\right ) \ln \! \left (1+\frac {\mathrm {I} c \left (\frac {\mathrm {I}}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) \sqrt {-d}}{\sqrt {e}+\sqrt {c^{2} d +e}}\right )}{4 \left (-d \right )^{\frac {3}{2}} \sqrt {e}}+\frac {\mathrm {I} b \polylog \! \left (2, \frac {\mathrm {-I} c \left (\frac {\mathrm {I}}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) \sqrt {-d}}{\sqrt {e}-\sqrt {c^{2} d +e}}\right )}{4 \left (-d \right )^{\frac {3}{2}} \sqrt {e}}-\frac {\mathrm {I} b \polylog \! \left (2, \frac {\mathrm {I} c \left (\frac {\mathrm {I}}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) \sqrt {-d}}{\sqrt {e}-\sqrt {c^{2} d +e}}\right )}{4 \left (-d \right )^{\frac {3}{2}} \sqrt {e}}+\frac {\mathrm {I} b \polylog \! \left (2, \frac {\mathrm {-I} c \left (\frac {\mathrm {I}}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) \sqrt {-d}}{\sqrt {e}+\sqrt {c^{2} d +e}}\right )}{4 \left (-d \right )^{\frac {3}{2}} \sqrt {e}}-\frac {\mathrm {I} b \polylog \! \left (2, \frac {\mathrm {I} c \left (\frac {\mathrm {I}}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) \sqrt {-d}}{\sqrt {e}+\sqrt {c^{2} d +e}}\right )}{4 \left (-d \right )^{\frac {3}{2}} \sqrt {e}}+\frac {-a -b \,\mathrm {arccsc}\! \left (c x \right )}{4 d \left (-\frac {d}{x}+\sqrt {-d}\, \sqrt {e}\right )}+\frac {a +b \,\mathrm {arccsc}\! \left (c x \right )}{4 d \left (\frac {d}{x}+\sqrt {-d}\, \sqrt {e}\right )}+\frac {b \arctanh \! \left (\frac {c^{2} d -\frac {\sqrt {-d}\, \sqrt {e}}{x}}{c \sqrt {d}\, \sqrt {c^{2} d +e}\, \sqrt {1-\frac {1}{c^{2} x^{2}}}}\right )}{4 d^{\frac {3}{2}} \sqrt {c^{2} d +e}}+\frac {b \arctanh \! \left (\frac {c^{2} d +\frac {\sqrt {-d}\, \sqrt {e}}{x}}{c \sqrt {d}\, \sqrt {c^{2} d +e}\, \sqrt {1-\frac {1}{c^{2} x^{2}}}}\right )}{4 d^{\frac {3}{2}} \sqrt {c^{2} d +e}} \]
command
int((a+b*arccsc(c*x))/(e*x^2+d)^2,x)
Maple 2022.1 output
hanged
Maple 2021.1 output
\[ \text {output too large to display} \]