Optimal. Leaf size=994 \[ \frac {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^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^2 \text {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^3 \text {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {2 b \text {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {b^3 \text {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac {2 b \text {PolyLog}\left (3,-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.49, antiderivative size = 994, 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 = {5544, 4276,
3405, 3401, 2296, 2221, 2611, 2320, 6724, 5681, 2317, 2438} \begin {gather*} \frac {x^6}{6 a^2}-\frac {b \log \left (\frac {e^{d x^2+c} a}{b-\sqrt {b^2-a^2}}+1\right ) x^4}{a^2 \sqrt {b^2-a^2} d}+\frac {b^3 \log \left (\frac {e^{d x^2+c} a}{b-\sqrt {b^2-a^2}}+1\right ) x^4}{2 a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {b \log \left (\frac {e^{d x^2+c} a}{b+\sqrt {b^2-a^2}}+1\right ) x^4}{a^2 \sqrt {b^2-a^2} d}-\frac {b^3 \log \left (\frac {e^{d x^2+c} a}{b+\sqrt {b^2-a^2}}+1\right ) x^4}{2 a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {b^2 \sinh \left (d x^2+c\right ) x^4}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (d x^2+c\right )\right )}+\frac {b^2 x^4}{2 a^2 \left (a^2-b^2\right ) d}-\frac {b^2 \log \left (\frac {e^{d x^2+c} a}{b-\sqrt {b^2-a^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac {b^2 \log \left (\frac {e^{d x^2+c} a}{b+\sqrt {b^2-a^2}}+1\right ) x^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac {2 b \text {Li}_2\left (-\frac {a e^{d x^2+c}}{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 {a e^{d x^2+c}}{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 {a e^{d x^2+c}}{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 {a e^{d x^2+c}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {b^2 \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {b^2 \text {Li}_2\left (-\frac {a e^{d x^2+c}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {2 b \text {Li}_3\left (-\frac {a e^{d x^2+c}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^3}-\frac {b^3 \text {Li}_3\left (-\frac {a e^{d x^2+c}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac {2 b \text {Li}_3\left (-\frac {a e^{d x^2+c}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^3}+\frac {b^3 \text {Li}_3\left (-\frac {a e^{d x^2+c}}{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 3401
Rule 3405
Rule 4276
Rule 5544
Rule 5681
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b \text {sech}(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 \cosh (c+d x))^2}-\frac {2 b x^2}{a^2 (b+a \cosh (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 \cosh (c+d x)} \, dx,x,x^2\right )}{a^2}+\frac {b^2 \text {Subst}\left (\int \frac {x^2}{(b+a \cosh (c+d x))^2} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac {x^6}{6 a^2}+\frac {b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}-\frac {(2 b) \text {Subst}\left (\int \frac {e^{c+d x} x^2}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^2\right )}{a^2}-\frac {b^3 \text {Subst}\left (\int \frac {x^2}{b+a \cosh (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 \sinh (c+d x)}{b+a \cosh (c+d x)} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}\\ &=\frac {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 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}-\frac {b^3 \text {Subst}\left (\int \frac {e^{c+d x} x^2}{a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right )}-\frac {(2 b) \text {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2}}+\frac {(2 b) \text {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{a \sqrt {-a^2+b^2}}-\frac {b^2 \text {Subst}\left (\int \frac {e^{c+d x} x}{b-\sqrt {-a^2+b^2}+a e^{c+d x}} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}-\frac {b^2 \text {Subst}\left (\int \frac {e^{c+d x} x}{b+\sqrt {-a^2+b^2}+a e^{c+d x}} \, dx,x,x^2\right )}{a \left (a^2-b^2\right ) d}\\ &=\frac {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^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}+\frac {b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}+\frac {b^3 \text {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^2\right )}{a \left (-a^2+b^2\right )^{3/2}}-\frac {b^3 \text {Subst}\left (\int \frac {e^{c+d x} x^2}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{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^{c+d x}}{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^{c+d x}}{b+\sqrt {-a^2+b^2}}\right ) \, dx,x,x^2\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {(2 b) \text {Subst}\left (\int x \log \left (1+\frac {2 a e^{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 b) \text {Subst}\left (\int x \log \left (1+\frac {2 a e^{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 {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^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}+\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{b-\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {(2 b) \text {Subst}\left (\int \text {Li}_2\left (-\frac {2 a e^{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 a e^{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 {b^3 \text {Subst}\left (\int x \log \left (1+\frac {2 a e^{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 {b^3 \text {Subst}\left (\int x \log \left (1+\frac {2 a e^{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 {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^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 \text {Li}_2\left (-\frac {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^2 \text {Li}_2\left (-\frac {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}+\frac {(2 b) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {(2 b) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {b^3 \text {Subst}\left (\int \text {Li}_2\left (-\frac {2 a e^{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 a e^{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^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^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 \text {Li}_2\left (-\frac {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^2 \text {Li}_2\left (-\frac {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}+\frac {2 b \text {Li}_3\left (-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}-\frac {2 b \text {Li}_3\left (-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}-\frac {b^3 \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {b^3 \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}\\ &=\frac {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^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}+\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}-\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2}-\frac {b^3 x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a^2 \left (-a^2+b^2\right )^{3/2} d}+\frac {b x^4 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d}-\frac {b^2 \text {Li}_2\left (-\frac {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^2 \text {Li}_2\left (-\frac {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{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 {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2}-\frac {b^3 \text {Li}_3\left (-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}+\frac {2 b \text {Li}_3\left (-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {b^3 \text {Li}_3\left (-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3}-\frac {2 b \text {Li}_3\left (-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3}+\frac {b^2 x^4 \sinh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (b+a \cosh \left (c+d x^2\right )\right )}\\ \end {align*}
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Mathematica [A]
time = 8.65, size = 1565, normalized size = 1.57 \begin {gather*} \frac {\left (b+a \cosh \left (c+d x^2\right )\right ) \text {sech}^2\left (c+d x^2\right ) \left (x^6 \left (b+a \cosh \left (c+d x^2\right )\right )-\frac {3 b e^{2 c} \left (b+a \cosh \left (c+d x^2\right )\right ) \left (2 b d^2 e^{2 c} \sqrt {\left (-a^2+b^2\right ) e^{2 c}} x^4-2 b d \sqrt {\left (-a^2+b^2\right ) e^{2 c}} x^2 \log \left (1+\frac {a e^{2 c+d x^2}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-2 b d e^{2 c} \sqrt {\left (-a^2+b^2\right ) e^{2 c}} x^2 \log \left (1+\frac {a e^{2 c+d x^2}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-2 a^2 d^2 e^c x^4 \log \left (1+\frac {a e^{2 c+d x^2}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+b^2 d^2 e^c x^4 \log \left (1+\frac {a e^{2 c+d x^2}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-2 a^2 d^2 e^{3 c} x^4 \log \left (1+\frac {a e^{2 c+d x^2}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+b^2 d^2 e^{3 c} x^4 \log \left (1+\frac {a e^{2 c+d x^2}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-2 b d \sqrt {\left (-a^2+b^2\right ) e^{2 c}} x^2 \log \left (1+\frac {a e^{2 c+d x^2}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-2 b d e^{2 c} \sqrt {\left (-a^2+b^2\right ) e^{2 c}} x^2 \log \left (1+\frac {a e^{2 c+d x^2}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+2 a^2 d^2 e^c x^4 \log \left (1+\frac {a e^{2 c+d x^2}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-b^2 d^2 e^c x^4 \log \left (1+\frac {a e^{2 c+d x^2}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+2 a^2 d^2 e^{3 c} x^4 \log \left (1+\frac {a e^{2 c+d x^2}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-b^2 d^2 e^{3 c} x^4 \log \left (1+\frac {a e^{2 c+d x^2}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+2 \left (1+e^{2 c}\right ) \left (-b \sqrt {\left (-a^2+b^2\right ) e^{2 c}}-2 a^2 d e^c x^2+b^2 d e^c x^2\right ) \text {PolyLog}\left (2,-\frac {a e^{2 c+d x^2}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-2 \left (1+e^{2 c}\right ) \left (b \sqrt {\left (-a^2+b^2\right ) e^{2 c}}-2 a^2 d e^c x^2+b^2 d e^c x^2\right ) \text {PolyLog}\left (2,-\frac {a e^{2 c+d x^2}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+4 a^2 e^c \text {PolyLog}\left (3,-\frac {a e^{2 c+d x^2}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-2 b^2 e^c \text {PolyLog}\left (3,-\frac {a e^{2 c+d x^2}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+4 a^2 e^{3 c} \text {PolyLog}\left (3,-\frac {a e^{2 c+d x^2}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-2 b^2 e^{3 c} \text {PolyLog}\left (3,-\frac {a e^{2 c+d x^2}}{b e^c-\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-4 a^2 e^c \text {PolyLog}\left (3,-\frac {a e^{2 c+d x^2}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+2 b^2 e^c \text {PolyLog}\left (3,-\frac {a e^{2 c+d x^2}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )-4 a^2 e^{3 c} \text {PolyLog}\left (3,-\frac {a e^{2 c+d x^2}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )+2 b^2 e^{3 c} \text {PolyLog}\left (3,-\frac {a e^{2 c+d x^2}}{b e^c+\sqrt {\left (-a^2+b^2\right ) e^{2 c}}}\right )\right )}{d^3 \left (\left (-a^2+b^2\right ) e^{2 c}\right )^{3/2} \left (1+e^{2 c}\right )}+\frac {3 b^2 x^4 \text {sech}(c) \left (-b \sinh (c)+a \sinh \left (d x^2\right )\right )}{(a-b) (a+b) d}\right )}{6 a^2 \left (a+b \text {sech}\left (c+d x^2\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.33, size = 0, normalized size = 0.00 \[\int \frac {x^{5}}{\left (a +b \,\mathrm {sech}\left (d \,x^{2}+c \right )\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3918 vs.
\(2 (906) = 1812\).
time = 0.42, size = 3918, normalized size = 3.94 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\left (a + b \operatorname {sech}{\left (c + d x^{2} \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5}{{\left (a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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