Optimal. Leaf size=202 \[ -\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\sqrt{a} x\right ),\frac{1}{2}\right )}{5 a^{7/2} \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{1}{5} x^5 \sqrt{\frac{1}{a^2 x^4}+1}+\frac{2 x \sqrt{\frac{1}{a^2 x^4}+1}}{5 a^2}-\frac{2 \sqrt{\frac{1}{a^2 x^4}+1}}{5 a^2 x \left (a+\frac{1}{x^2}\right )}+\frac{2 \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 a^{7/2} \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{x^3}{3 a} \]
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Rubi [A] time = 0.124744, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6336, 30, 335, 277, 325, 305, 220, 1196} \[ \frac{1}{5} x^5 \sqrt{\frac{1}{a^2 x^4}+1}+\frac{2 x \sqrt{\frac{1}{a^2 x^4}+1}}{5 a^2}-\frac{2 \sqrt{\frac{1}{a^2 x^4}+1}}{5 a^2 x \left (a+\frac{1}{x^2}\right )}-\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 a^{7/2} \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{2 \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 a^{7/2} \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{x^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 335
Rule 277
Rule 325
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int e^{\text{csch}^{-1}\left (a x^2\right )} x^4 \, dx &=\frac{\int x^2 \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^4}} x^4 \, dx\\ &=\frac{x^3}{3 a}-\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^4}{a^2}}}{x^6} \, dx,x,\frac{1}{x}\right )\\ &=\frac{x^3}{3 a}+\frac{1}{5} \sqrt{1+\frac{1}{a^2 x^4}} x^5-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{5 a^2}\\ &=\frac{2 \sqrt{1+\frac{1}{a^2 x^4}} x}{5 a^2}+\frac{x^3}{3 a}+\frac{1}{5} \sqrt{1+\frac{1}{a^2 x^4}} x^5-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{5 a^4}\\ &=\frac{2 \sqrt{1+\frac{1}{a^2 x^4}} x}{5 a^2}+\frac{x^3}{3 a}+\frac{1}{5} \sqrt{1+\frac{1}{a^2 x^4}} x^5-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{5 a^3}+\frac{2 \operatorname{Subst}\left (\int \frac{1-\frac{x^2}{a}}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{5 a^3}\\ &=-\frac{2 \sqrt{1+\frac{1}{a^2 x^4}}}{5 a^2 \left (a+\frac{1}{x^2}\right ) x}+\frac{2 \sqrt{1+\frac{1}{a^2 x^4}} x}{5 a^2}+\frac{x^3}{3 a}+\frac{1}{5} \sqrt{1+\frac{1}{a^2 x^4}} x^5+\frac{2 \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 a^{7/2} \sqrt{1+\frac{1}{a^2 x^4}}}-\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 )}{5 a^{7/2} \sqrt{1+\frac{1}{a^2 x^4}}}\\ \end{align*}
Mathematica [C] time = 0.248762, size = 112, normalized size = 0.55 \[ \frac{4 \sqrt{2} x^5 e^{-\text{csch}^{-1}\left (a x^2\right )} \left (\frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{e^{2 \text{csch}^{-1}\left (a x^2\right )}-1}\right )^{5/2} \left (4 \left (1-e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )^{5/2} \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{7}{2},\frac{7}{4},e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )+7 e^{2 \text{csch}^{-1}\left (a x^2\right )}-4\right )}{21 \left (a x^2\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.211, size = 150, normalized size = 0.7 \begin{align*}{\frac{{x}^{2}}{5\,a \left ({a}^{2}{x}^{4}+1 \right ) }\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( \sqrt{ia}{x}^{7}{a}^{3}+{x}^{3}a\sqrt{ia}+2\,i\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}{\it EllipticF} \left ( x\sqrt{ia},i \right ) -2\,i\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}{\it EllipticE} \left ( x\sqrt{ia},i \right ) \right ){\frac{1}{\sqrt{ia}}}}+{\frac{{x}^{3}}{3\,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^{3}}{3 \, a} + \frac{\frac{x^{3} \Gamma \left (\frac{3}{4}\right ) \,_2F_1\left (\begin{matrix} -\frac{1}{2},\frac{3}{4} \\ \frac{7}{4} \end{matrix} ; -a^{2} x^{4} \right )}{4 \, \Gamma \left (\frac{7}{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^{4} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + x^{2}}{a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.35218, size = 48, normalized size = 0.24 \begin{align*} - \frac{x^{5} \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{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^{3}}{3 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^{4}{\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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