Integrand size = 18, antiderivative size = 519 \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\frac {x^4}{4 a^2}+\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^3 x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {b^2 \log \left (b+a \sinh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )} \]
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Time = 0.73 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5545, 4276, 3405, 3403, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=-\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}+\frac {b^2 \log \left (a \sinh \left (c+d x^2\right )+b\right )}{2 a^2 d^2 \left (a^2+b^2\right )}-\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d \sqrt {a^2+b^2}}+\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d \sqrt {a^2+b^2}}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a d \left (a^2+b^2\right ) \left (a \sinh \left (c+d x^2\right )+b\right )}+\frac {b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b-\sqrt {a^2+b^2}}\right )}{2 a^2 d^2 \left (a^2+b^2\right )^{3/2}}-\frac {b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b+\sqrt {a^2+b^2}}\right )}{2 a^2 d^2 \left (a^2+b^2\right )^{3/2}}+\frac {b^3 x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}+1\right )}{2 a^2 d \left (a^2+b^2\right )^{3/2}}-\frac {b^3 x^2 \log \left (\frac {a e^{c+d x^2}}{\sqrt {a^2+b^2}+b}+1\right )}{2 a^2 d \left (a^2+b^2\right )^{3/2}}+\frac {x^4}{4 a^2} \]
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3403
Rule 3405
Rule 4276
Rule 5545
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b \text {csch}(c+d x))^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {x}{a^2}+\frac {b^2 x}{a^2 (b+a \sinh (c+d x))^2}-\frac {2 b x}{a^2 (b+a \sinh (c+d x))}\right ) \, dx,x,x^2\right ) \\ & = \frac {x^4}{4 a^2}-\frac {b \text {Subst}\left (\int \frac {x}{b+a \sinh (c+d x)} \, dx,x,x^2\right )}{a^2}+\frac {b^2 \text {Subst}\left (\int \frac {x}{(b+a \sinh (c+d x))^2} \, dx,x,x^2\right )}{2 a^2} \\ & = \frac {x^4}{4 a^2}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )}-\frac {(2 b) \text {Subst}\left (\int \frac {e^{c+d x} x}{-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}{b+a \sinh (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 {\cosh (c+d x)}{b+a \sinh (c+d x)} \, dx,x,x^2\right )}{2 a \left (a^2+b^2\right ) d} \\ & = \frac {x^4}{4 a^2}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )}+\frac {b^3 \text {Subst}\left (\int \frac {e^{c+d x} x}{-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 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 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 {1}{b+x} \, dx,x,a \sinh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2+b^2\right ) d^2} \\ & = \frac {x^4}{4 a^2}-\frac {b x^2 \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 x^2 \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 \log \left (b+a \sinh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )}+\frac {b^3 \text {Subst}\left (\int \frac {e^{c+d x} x}{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 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 \text {Subst}\left (\int \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 \text {Subst}\left (\int \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 {x^4}{4 a^2}+\frac {b^3 x^2 \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^2 \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^3 x^2 \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^2 \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 \log \left (b+a \sinh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b^2 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )}+\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^3 \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 \text {Subst}\left (\int \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,x^2\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d} \\ & = \frac {x^4}{4 a^2}+\frac {b^3 x^2 \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^2 \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^3 x^2 \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^2 \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 \log \left (b+a \sinh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2+b^2\right ) d^2}-\frac {b \operatorname {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 \operatorname {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 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )}-\frac {b^3 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {b^3 \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^2}\right )}{2 a^2 \left (a^2+b^2\right )^{3/2} d^2} \\ & = \frac {x^4}{4 a^2}+\frac {b^3 x^2 \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^2 \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^3 x^2 \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^2 \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 \log \left (b+a \sinh \left (c+d x^2\right )\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {b^3 \operatorname {PolyLog}\left (2,-\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^2}-\frac {b \operatorname {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 \operatorname {PolyLog}\left (2,-\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^2}+\frac {b \operatorname {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 x^2 \cosh \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d x^2\right )\right )} \\ \end{align*}
Time = 3.91 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.42 \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\frac {\text {csch}^2\left (c+d x^2\right ) \left (b+a \sinh \left (c+d x^2\right )\right ) \left (-\frac {2 a b^2 d x^2 \cosh \left (c+d x^2\right )}{a^2+b^2}+\left (-c+d x^2\right ) \left (c+d x^2\right ) \left (b+a \sinh \left (c+d x^2\right )\right )-\frac {2 b \left (a^2+b^2\right ) \left (-b \sqrt {-\left (a^2+b^2\right )^2} \left (c+d x^2\right )+2 b^2 \sqrt {a^2+b^2} \arctan \left (\frac {b+a e^{c+d x^2}}{\sqrt {-a^2-b^2}}\right )+2 b^2 \sqrt {-a^2-b^2} \text {arctanh}\left (\frac {b+a e^{c+d x^2}}{\sqrt {a^2+b^2}}\right )-4 a^2 \sqrt {-a^2-b^2} c \text {arctanh}\left (\frac {b+a e^{c+d x^2}}{\sqrt {a^2+b^2}}\right )-2 b^2 \sqrt {-a^2-b^2} c \text {arctanh}\left (\frac {b+a e^{c+d x^2}}{\sqrt {a^2+b^2}}\right )-2 a^2 \sqrt {-a^2-b^2} \left (c+d x^2\right ) \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )-b^2 \sqrt {-a^2-b^2} \left (c+d x^2\right ) \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {a^2+b^2}}\right )+2 a^2 \sqrt {-a^2-b^2} \left (c+d x^2\right ) \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )+b^2 \sqrt {-a^2-b^2} \left (c+d x^2\right ) \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )+b \sqrt {-\left (a^2+b^2\right )^2} \log \left (2 b e^{c+d x^2}+a \left (-1+e^{2 \left (c+d x^2\right )}\right )\right )-\sqrt {-a^2-b^2} \left (2 a^2+b^2\right ) \operatorname {PolyLog}\left (2,\frac {a e^{c+d x^2}}{-b+\sqrt {a^2+b^2}}\right )+\sqrt {-a^2-b^2} \left (2 a^2+b^2\right ) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {a^2+b^2}}\right )\right ) \left (b+a \sinh \left (c+d x^2\right )\right )}{\left (-\left (a^2+b^2\right )^2\right )^{3/2}}\right )}{4 a^2 d^2 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \]
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\[\int \frac {x^{3}}{{\left (a +b \,\operatorname {csch}\left (d \,x^{2}+c \right )\right )}^{2}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 2383 vs. \(2 (461) = 922\).
Time = 0.31 (sec) , antiderivative size = 2383, normalized size of antiderivative = 4.59 \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{3}}{\left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )^{2}}\, dx \]
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\[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^3}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^3}{{\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right )}^2} \,d x \]
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