Integrand size = 18, antiderivative size = 241 \[ \int \frac {x^3}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}+\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{2 a \sqrt {-a^2+b^2} d^2} \]
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Time = 0.36 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5544, 4276, 3401, 2296, 2221, 2317, 2438} \[ \int \frac {x^3}{a+b \text {sech}\left (c+d x^2\right )} \, dx=-\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b-\sqrt {b^2-a^2}}\right )}{2 a d^2 \sqrt {b^2-a^2}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^2+c}}{b+\sqrt {b^2-a^2}}\right )}{2 a d^2 \sqrt {b^2-a^2}}-\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{b-\sqrt {b^2-a^2}}+1\right )}{2 a d \sqrt {b^2-a^2}}+\frac {b x^2 \log \left (\frac {a e^{c+d x^2}}{\sqrt {b^2-a^2}+b}+1\right )}{2 a d \sqrt {b^2-a^2}}+\frac {x^4}{4 a} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3401
Rule 4276
Rule 5544
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{a+b \text {sech}(c+d x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {x}{a}-\frac {b x}{a (b+a \cosh (c+d x))}\right ) \, dx,x,x^2\right ) \\ & = \frac {x^4}{4 a}-\frac {b \text {Subst}\left (\int \frac {x}{b+a \cosh (c+d x)} \, dx,x,x^2\right )}{2 a} \\ & = \frac {x^4}{4 a}-\frac {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} \\ & = \frac {x^4}{4 a}-\frac {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 )}{\sqrt {-a^2+b^2}}+\frac {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 )}{\sqrt {-a^2+b^2}} \\ & = \frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \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 )}{2 a \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 )}{2 a \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 )}{2 a \sqrt {-a^2+b^2} d} \\ & = \frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \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 )}{2 a \sqrt {-a^2+b^2} d}+\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 )}{2 a \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 )}{2 a \sqrt {-a^2+b^2} d^2} \\ & = \frac {x^4}{4 a}-\frac {b x^2 \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \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 )}{2 a \sqrt {-a^2+b^2} d}-\frac {b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )}{2 a \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 )}{2 a \sqrt {-a^2+b^2} d^2} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\frac {d x^2 \left (\sqrt {-a^2+b^2} d x^2-2 b \log \left (1+\frac {a e^{c+d x^2}}{b-\sqrt {-a^2+b^2}}\right )+2 b \log \left (1+\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )\right )-2 b \operatorname {PolyLog}\left (2,\frac {a e^{c+d x^2}}{-b+\sqrt {-a^2+b^2}}\right )+2 b \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^2}}{b+\sqrt {-a^2+b^2}}\right )}{4 a \sqrt {-a^2+b^2} d^2} \]
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\[\int \frac {x^{3}}{a +b \,\operatorname {sech}\left (d \,x^{2}+c \right )}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (209) = 418\).
Time = 0.26 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.22 \[ \int \frac {x^3}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\frac {{\left (a^{2} - b^{2}\right )} d^{2} x^{4} + 2 \, a b c \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d x^{2} + c\right ) + 2 \, a \sinh \left (d x^{2} + c\right ) + 2 \, a \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} + 2 \, b\right ) - 2 \, a b c \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} \log \left (2 \, a \cosh \left (d x^{2} + c\right ) + 2 \, a \sinh \left (d x^{2} + c\right ) - 2 \, a \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} + 2 \, b\right ) + 2 \, a b \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} {\rm Li}_2\left (-\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) + {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} + a}{a} + 1\right ) - 2 \, a b \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} {\rm Li}_2\left (-\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) - {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} + a}{a} + 1\right ) + 2 \, {\left (a b d x^{2} + a b c\right )} \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} \log \left (\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) + {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} + a}{a}\right ) - 2 \, {\left (a b d x^{2} + a b c\right )} \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} \log \left (\frac {b \cosh \left (d x^{2} + c\right ) + b \sinh \left (d x^{2} + c\right ) - {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{a^{2}}} + a}{a}\right )}{4 \, {\left (a^{3} - a b^{2}\right )} d^{2}} \]
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\[ \int \frac {x^3}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\int \frac {x^{3}}{a + b \operatorname {sech}{\left (c + d x^{2} \right )}}\, dx \]
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Exception generated. \[ \int \frac {x^3}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^3}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\int { \frac {x^{3}}{b \operatorname {sech}\left (d x^{2} + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {x^3}{a+b \text {sech}\left (c+d x^2\right )} \, dx=\int \frac {x^3}{a+\frac {b}{\mathrm {cosh}\left (d\,x^2+c\right )}} \,d x \]
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