Optimal. Leaf size=52 \[ \frac{1}{4} x^4 \sqrt{\frac{1}{a^2 x^4}+1}+\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^4}+1}\right )}{4 a^2}+\frac{x^2}{2 a} \]
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Rubi [A] time = 0.0383185, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6336, 30, 266, 47, 63, 208} \[ \frac{1}{4} x^4 \sqrt{\frac{1}{a^2 x^4}+1}+\frac{\tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^4}+1}\right )}{4 a^2}+\frac{x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{\text{csch}^{-1}\left (a x^2\right )} x^3 \, dx &=\frac{\int x \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^4}} x^3 \, dx\\ &=\frac{x^2}{2 a}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a^2}}}{x^2} \, dx,x,\frac{1}{x^4}\right )\\ &=\frac{x^2}{2 a}+\frac{1}{4} \sqrt{1+\frac{1}{a^2 x^4}} x^4-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a^2}}} \, dx,x,\frac{1}{x^4}\right )}{8 a^2}\\ &=\frac{x^2}{2 a}+\frac{1}{4} \sqrt{1+\frac{1}{a^2 x^4}} x^4-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-a^2+a^2 x^2} \, dx,x,\sqrt{1+\frac{1}{a^2 x^4}}\right )\\ &=\frac{x^2}{2 a}+\frac{1}{4} \sqrt{1+\frac{1}{a^2 x^4}} x^4+\frac{\tanh ^{-1}\left (\sqrt{1+\frac{1}{a^2 x^4}}\right )}{4 a^2}\\ \end{align*}
Mathematica [A] time = 0.0535126, size = 53, normalized size = 1.02 \[ \frac{a x^2 \left (a x^2 \sqrt{\frac{1}{a^2 x^4}+1}+2\right )+\log \left (x^2 \left (\sqrt{\frac{1}{a^2 x^4}+1}+1\right )\right )}{4 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.29, size = 94, normalized size = 1.8 \begin{align*}{\frac{{x}^{2}}{4\,{a}^{2}}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ({x}^{2}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}{a}^{2}+\ln \left ({x}^{2}+\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}}}}+{\frac{{x}^{2}}{2\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03212, size = 109, normalized size = 2.1 \begin{align*} \frac{x^{2}}{2 \, a} + \frac{\sqrt{\frac{1}{a^{2} x^{4}} + 1}}{4 \,{\left (a^{2}{\left (\frac{1}{a^{2} x^{4}} + 1\right )} - a^{2}\right )}} + \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{4}} + 1} + 1\right )}{8 \, a^{2}} - \frac{\log \left (\sqrt{\frac{1}{a^{2} x^{4}} + 1} - 1\right )}{8 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.59534, size = 149, normalized size = 2.87 \begin{align*} \frac{a^{2} x^{4} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + 2 \, a x^{2} - \log \left (a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} - a x^{2}\right )}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.29354, size = 36, normalized size = 0.69 \begin{align*} \frac{x^{2} \sqrt{a^{2} x^{4} + 1}}{4 a} + \frac{x^{2}}{2 a} + \frac{\operatorname{asinh}{\left (a x^{2} \right )}}{4 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13288, size = 74, normalized size = 1.42 \begin{align*} \frac{\sqrt{a^{2} x^{4} + 1} x^{2}{\left | a \right |}}{4 \, a^{2}} + \frac{x^{2}}{2 \, a} - \frac{\log \left (-x^{2}{\left | a \right |} + \sqrt{a^{2} x^{4} + 1}\right )}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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