Optimal. Leaf size=86 \[ -\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\sqrt{a} x\right ),\frac{1}{2}\right )}{3 a^{5/2} \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{1}{3} x^3 \sqrt{\frac{1}{a^2 x^4}+1}+\frac{x}{a} \]
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Rubi [A] time = 0.0532904, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6336, 8, 335, 277, 220} \[ \frac{1}{3} x^3 \sqrt{\frac{1}{a^2 x^4}+1}-\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{3 a^{5/2} \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{x}{a} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 8
Rule 335
Rule 277
Rule 220
Rubi steps
\begin{align*} \int e^{\text{csch}^{-1}\left (a x^2\right )} x^2 \, dx &=\frac{\int 1 \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^4}} x^2 \, dx\\ &=\frac{x}{a}-\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^4}{a^2}}}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{x}{a}+\frac{1}{3} \sqrt{1+\frac{1}{a^2 x^4}} x^3-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 a^2}\\ &=\frac{x}{a}+\frac{1}{3} \sqrt{1+\frac{1}{a^2 x^4}} x^3-\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{3 a^{5/2} \sqrt{1+\frac{1}{a^2 x^4}}}\\ \end{align*}
Mathematica [C] time = 0.214752, size = 113, normalized size = 1.31 \[ -\frac{2 \sqrt{2} x e^{-\text{csch}^{-1}\left (a x^2\right )} \left (\frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{e^{2 \text{csch}^{-1}\left (a x^2\right )}-1}\right )^{3/2} \left (\left (1-e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )^{3/2} \left (-\text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )\right )-2 e^{2 \text{csch}^{-1}\left (a x^2\right )}+1\right )}{3 a \sqrt{a x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.187, size = 104, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}}{3\,{a}^{2}{x}^{4}+3}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( \sqrt{ia}{x}^{5}{a}^{2}+2\,\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}{\it EllipticF} \left ( x\sqrt{ia},i \right ) +x\sqrt{ia} \right ){\frac{1}{\sqrt{ia}}}}+{\frac{x}{a}} \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{x}{a} + \frac{\frac{x \Gamma \left (\frac{1}{4}\right ) \,_2F_1\left (\begin{matrix} -\frac{1}{2},\frac{1}{4} \\ \frac{5}{4} \end{matrix} ; -a^{2} x^{4} \right )}{4 \, \Gamma \left (\frac{5}{4}\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1}{a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.27897, size = 41, normalized size = 0.48 \begin{align*} - \frac{x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{e^{i \pi }}{a^{2} x^{4}}} \right )}}{4 \Gamma \left (\frac{1}{4}\right )} + \frac{x}{a} \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 x^{2}{\left (\sqrt{\frac{1}{a^{2} x^{4}} + 1} + \frac{1}{a x^{2}}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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