Optimal. Leaf size=40 \[ \frac{1}{2} x^2 \sqrt{\frac{1}{a^2 x^4}+1}-\frac{\text{csch}^{-1}\left (a x^2\right )}{2 a}+\frac{\log (x)}{a} \]
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Rubi [A] time = 0.0362239, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6336, 29, 335, 275, 277, 215} \[ \frac{1}{2} x^2 \sqrt{\frac{1}{a^2 x^4}+1}-\frac{\text{csch}^{-1}\left (a x^2\right )}{2 a}+\frac{\log (x)}{a} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 29
Rule 335
Rule 275
Rule 277
Rule 215
Rubi steps
\begin{align*} \int e^{\text{csch}^{-1}\left (a x^2\right )} x \, dx &=\frac{\int \frac{1}{x} \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^4}} x \, dx\\ &=\frac{\log (x)}{a}-\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^4}{a^2}}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\log (x)}{a}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^2}{a^2}}}{x^2} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^4}} x^2+\frac{\log (x)}{a}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a^2}\\ &=\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^4}} x^2-\frac{\text{csch}^{-1}\left (a x^2\right )}{2 a}+\frac{\log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.031044, size = 42, normalized size = 1.05 \[ \frac{a x^2 \sqrt{\frac{1}{a^2 x^4}+1}+\log \left (a x^2\right )-\sinh ^{-1}\left (\frac{1}{a x^2}\right )}{2 a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.285, size = 116, normalized size = 2.9 \begin{align*}{\frac{{x}^{2}}{2\,{a}^{2}}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( \sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}{a}^{2}-\ln \left ( 2\,{\frac{1}{{a}^{2}{x}^{2}} \left ( \sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}{a}^{2}+1 \right ) } \right ) \right ){\frac{1}{\sqrt{{a}^{-2}}}}{\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}}}}+{\frac{\ln \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.988743, size = 96, normalized size = 2.4 \begin{align*} \frac{1}{2} \, x^{2} \sqrt{\frac{1}{a^{2} x^{4}} + 1} - \frac{\log \left (a x^{2} \sqrt{\frac{1}{a^{2} x^{4}} + 1} + 1\right )}{4 \, a} + \frac{\log \left (a x^{2} \sqrt{\frac{1}{a^{2} x^{4}} + 1} - 1\right )}{4 \, a} + \frac{\log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.54821, size = 205, normalized size = 5.12 \begin{align*} \frac{2 \, a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} - \log \left (a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1\right ) + \log \left (a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} - 1\right ) + 4 \, \log \left (x\right )}{4 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.24026, size = 58, normalized size = 1.45 \begin{align*} \frac{x^{2}}{2 \sqrt{1 + \frac{1}{a^{2} x^{4}}}} + \frac{\log{\left (x \right )}}{a} - \frac{\operatorname{asinh}{\left (\frac{1}{a x^{2}} \right )}}{2 a} + \frac{1}{2 a^{2} x^{2} \sqrt{1 + \frac{1}{a^{2} x^{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13163, size = 82, normalized size = 2.05 \begin{align*} \frac{{\left (a -{\left | a \right |}\right )} \log \left (\sqrt{a^{2} x^{4} + 1} + 1\right ) +{\left (a +{\left | a \right |}\right )} \log \left (\sqrt{a^{2} x^{4} + 1} - 1\right ) + 2 \, \sqrt{a^{2} x^{4} + 1}{\left | a \right |}}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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