Optimal. Leaf size=1124 \[ -\frac {i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac {x^6}{6 a^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {i b^3 x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {i b^3 x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {i b^2 \text {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {i b^2 \text {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {b^3 x^2 \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {2 b x^2 \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^3 x^2 \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {2 b x^2 \text {PolyLog}\left (2,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {i b^3 \text {PolyLog}\left (3,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {2 i b \text {PolyLog}\left (3,\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {i b^3 \text {PolyLog}\left (3,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac {2 i b \text {PolyLog}\left (3,\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {b^2 x^4 \cos \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d x^2\right )\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.69, antiderivative size = 1124, normalized size of antiderivative = 1.00, number of steps
used = 31, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4290, 4276,
3405, 3404, 2296, 2221, 2611, 2320, 6724, 4617, 2317, 2438} \begin {gather*} \frac {x^6}{6 a^2}+\frac {i b \log \left (1-\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^4}{a^2 \sqrt {b^2-a^2} d}-\frac {i b^3 \log \left (1-\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^4}{2 a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {i b \log \left (1-\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^4}{a^2 \sqrt {b^2-a^2} d}+\frac {i b^3 \log \left (1-\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^4}{2 a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}-\frac {b^2 \cos \left (d x^2+c\right ) x^4}{2 a \left (a^2-b^2\right ) d \left (b+a \sin \left (d x^2+c\right )\right )}+\frac {b^2 \log \left (\frac {e^{i \left (d x^2+c\right )} a}{i b-\sqrt {a^2-b^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^2 \log \left (\frac {e^{i \left (d x^2+c\right )} a}{i b+\sqrt {a^2-b^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac {2 b \text {Li}_2\left (\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a^2 \sqrt {b^2-a^2} d^2}-\frac {b^3 \text {Li}_2\left (\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {2 b \text {Li}_2\left (\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \sqrt {b^2-a^2} d^2}+\frac {b^3 \text {Li}_2\left (\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {i b^2 \text {Li}_2\left (-\frac {a e^{i \left (d x^2+c\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {i b^2 \text {Li}_2\left (-\frac {a e^{i \left (d x^2+c\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {2 i b \text {Li}_3\left (\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^3}-\frac {i b^3 \text {Li}_3\left (\frac {i a e^{i \left (d x^2+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac {2 i b \text {Li}_3\left (\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^3}+\frac {i b^3 \text {Li}_3\left (\frac {i a e^{i \left (d x^2+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3404
Rule 3405
Rule 4276
Rule 4290
Rule 4617
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b \csc \left (c+d x^2\right )\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b \csc (c+d x))^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {x^2}{a^2}+\frac {b^2 x^2}{a^2 (b+a \sin (c+d x))^2}-\frac {2 b x^2}{a^2 (b+a \sin (c+d x))}\right ) \, dx,x,x^2\right )\\ &=\frac {x^6}{6 a^2}-\frac {b \text {Subst}\left (\int \frac {x^2}{b+a \sin (c+d x)} \, dx,x,x^2\right )}{a^2}+\frac {b^2 \text {Subst}\left (\int \frac {x^2}{(b+a \sin (c+d x))^2} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac {x^6}{6 a^2}-\frac {b^2 x^4 \cos \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d x^2\right )\right )}-\frac {(2 b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,x^2\right )}{a^2}-\frac {b^3 \text {Subst}\left (\int \frac {x^2}{b+a \sin (c+d x)} \, dx,x,x^2\right )}{2 a^2 \left (a^2-b^2\right )}+\frac {b^2 \text {Subst}\left (\int \frac {x \cos (c+d x)}{b+a \sin (c+d x)} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}\\ &=-\frac {i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac {x^6}{6 a^2}-\frac {b^2 x^4 \cos \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d x^2\right )\right )}-\frac {b^3 \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right )}+\frac {(2 i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2}}-\frac {(2 i b) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2}}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{i b-\sqrt {a^2-b^2}+a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{i b+\sqrt {a^2-b^2}+a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}\\ &=-\frac {i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac {x^6}{6 a^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 x^4 \cos \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d x^2\right )\right )}-\frac {\left (i b^3\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b-2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \left (-a^2+b^2\right )^{3/2}}+\frac {\left (i b^3\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b+2 \sqrt {-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,x^2\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac {b^2 \text {Subst}\left (\int \log \left (1+\frac {a e^{i (c+d x)}}{i b-\sqrt {a^2-b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {b^2 \text {Subst}\left (\int \log \left (1+\frac {a e^{i (c+d x)}}{i b+\sqrt {a^2-b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {(2 i b) \text {Subst}\left (\int x \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {(2 i b) \text {Subst}\left (\int x \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt {-a^2+b^2} d}\\ &=-\frac {i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac {x^6}{6 a^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {i b^3 x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {i b^3 x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {2 b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {2 b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^2 x^4 \cos \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d x^2\right )\right )}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{i b-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{i b+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {(2 b) \text {Subst}\left (\int \text {Li}_2\left (\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {(2 b) \text {Subst}\left (\int \text {Li}_2\left (\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {\left (i b^3\right ) \text {Subst}\left (\int x \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {\left (i b^3\right ) \text {Subst}\left (\int x \log \left (1-\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d}\\ &=-\frac {i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac {x^6}{6 a^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {i b^3 x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {i b^3 x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {i b^2 \text {Li}_2\left (-\frac {a e^{i \left (c+d x^2\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {i b^2 \text {Li}_2\left (-\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {b^3 x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {2 b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^3 x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {2 b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^2 x^4 \cos \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d x^2\right )\right )}+\frac {(2 i b) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i a x}{b-\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {(2 i b) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {b^3 \text {Subst}\left (\int \text {Li}_2\left (\frac {2 i a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {b^3 \text {Subst}\left (\int \text {Li}_2\left (\frac {2 i a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}\\ &=-\frac {i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac {x^6}{6 a^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {i b^3 x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {i b^3 x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {i b^2 \text {Li}_2\left (-\frac {a e^{i \left (c+d x^2\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {i b^2 \text {Li}_2\left (-\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {b^3 x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {2 b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^3 x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {2 b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {2 i b \text {Li}_3\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {2 i b \text {Li}_3\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {b^2 x^4 \cos \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d x^2\right )\right )}-\frac {\left (i b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i a x}{b-\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {\left (i b^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^2\right )}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}\\ &=-\frac {i b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}+\frac {x^6}{6 a^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^2 x^2 \log \left (1+\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {i b^3 x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {i b^3 x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {i b x^4 \log \left (1-\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {i b^2 \text {Li}_2\left (-\frac {a e^{i \left (c+d x^2\right )}}{i b-\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {i b^2 \text {Li}_2\left (-\frac {a e^{i \left (c+d x^2\right )}}{i b+\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {b^3 x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}+\frac {2 b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^3 x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2}-\frac {2 b x^2 \text {Li}_2\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {i b^3 \text {Li}_3\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {2 i b \text {Li}_3\left (\frac {i a e^{i \left (c+d x^2\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {i b^3 \text {Li}_3\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac {2 i b \text {Li}_3\left (\frac {i a e^{i \left (c+d x^2\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {b^2 x^4 \cos \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d x^2\right )\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 9.66, size = 2033, normalized size = 1.81 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {x^{5}}{\left (a +b \csc \left (d \,x^{2}+c \right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 3032 vs. \(2 (966) = 1932\).
time = 3.00, size = 3032, normalized size = 2.70 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5}{{\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________