Optimal. Leaf size=68 \[ -\frac{b \text{PolyLog}\left (2,-e^{c+d x^2}\right )}{2 d^2}+\frac{b \text{PolyLog}\left (2,e^{c+d x^2}\right )}{2 d^2}+\frac{a x^4}{4}-\frac{b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d} \]
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Rubi [A] time = 0.0781825, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {14, 5437, 4182, 2279, 2391} \[ -\frac{b \text{PolyLog}\left (2,-e^{c+d x^2}\right )}{2 d^2}+\frac{b \text{PolyLog}\left (2,e^{c+d x^2}\right )}{2 d^2}+\frac{a x^4}{4}-\frac{b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 5437
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int x^3 \left (a+b \text{csch}\left (c+d x^2\right )\right ) \, dx &=\int \left (a x^3+b x^3 \text{csch}\left (c+d x^2\right )\right ) \, dx\\ &=\frac{a x^4}{4}+b \int x^3 \text{csch}\left (c+d x^2\right ) \, dx\\ &=\frac{a x^4}{4}+\frac{1}{2} b \operatorname{Subst}\left (\int x \text{csch}(c+d x) \, dx,x,x^2\right )\\ &=\frac{a x^4}{4}-\frac{b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac{b \operatorname{Subst}\left (\int \log \left (1-e^{c+d x}\right ) \, dx,x,x^2\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \log \left (1+e^{c+d x}\right ) \, dx,x,x^2\right )}{2 d}\\ &=\frac{a x^4}{4}-\frac{b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac{b \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{c+d x^2}\right )}{2 d^2}+\frac{b \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{c+d x^2}\right )}{2 d^2}\\ &=\frac{a x^4}{4}-\frac{b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac{b \text{Li}_2\left (-e^{c+d x^2}\right )}{2 d^2}+\frac{b \text{Li}_2\left (e^{c+d x^2}\right )}{2 d^2}\\ \end{align*}
Mathematica [A] time = 0.135797, size = 108, normalized size = 1.59 \[ \frac{1}{4} \left (\frac{2 b \left (\text{PolyLog}\left (2,-e^{-c-d x^2}\right )-\text{PolyLog}\left (2,e^{-c-d x^2}\right )+\left (c+d x^2\right ) \left (\log \left (1-e^{-c-d x^2}\right )-\log \left (e^{-c-d x^2}+1\right )\right )-c \log \left (\tanh \left (\frac{1}{2} \left (c+d x^2\right )\right )\right )\right )}{d^2}+a x^4\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b{\rm csch} \left (d{x}^{2}+c\right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, a x^{4} + 2 \, b \int \frac{x^{3}}{e^{\left (d x^{2} + c\right )} - e^{\left (-d x^{2} - c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.62218, size = 386, normalized size = 5.68 \begin{align*} \frac{a d^{2} x^{4} - 2 \, b d x^{2} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + 1\right ) - 2 \, b c \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - 1\right ) + 2 \, b{\rm Li}_2\left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) - 2 \, b{\rm Li}_2\left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right ) + 2 \,{\left (b d x^{2} + b c\right )} \log \left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right ) + 1\right )}{4 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \left (a + b \operatorname{csch}{\left (c + d x^{2} \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{csch}\left (d x^{2} + c\right ) + a\right )} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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