Optimal. Leaf size=91 \[ -\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\sqrt{a} x\right ),\frac{1}{2}\right )}{3 \sqrt{a} \sqrt{\frac{1}{a^2 x^4}+1}}-\frac{\sqrt{\frac{1}{a^2 x^4}+1}}{3 x}-\frac{1}{3 a x^3} \]
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Rubi [A] time = 0.0472524, antiderivative size = 91, 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, 30, 335, 195, 220} \[ -\frac{\sqrt{\frac{1}{a^2 x^4}+1}}{3 x}-\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{3 \sqrt{a} \sqrt{\frac{1}{a^2 x^4}+1}}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 335
Rule 195
Rule 220
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{x^2} \, dx &=\frac{\int \frac{1}{x^4} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^4}}}{x^2} \, dx\\ &=-\frac{1}{3 a x^3}-\operatorname{Subst}\left (\int \sqrt{1+\frac{x^4}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{3 a x^3}-\frac{\sqrt{1+\frac{1}{a^2 x^4}}}{3 x}-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{3 a x^3}-\frac{\sqrt{1+\frac{1}{a^2 x^4}}}{3 x}-\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 \sqrt{a} \sqrt{1+\frac{1}{a^2 x^4}}}\\ \end{align*}
Mathematica [C] time = 0.160232, size = 96, normalized size = 1.05 \[ -\frac{a x \sqrt{\frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{2 e^{2 \text{csch}^{-1}\left (a x^2\right )}-2}} \left (4 \sqrt{1-e^{2 \text{csch}^{-1}\left (a x^2\right )}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )+e^{2 \text{csch}^{-1}\left (a x^2\right )}-1\right )}{3 \sqrt{a x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.18, size = 111, normalized size = 1.2 \begin{align*} -{\frac{1}{3\,x \left ({a}^{2}{x}^{4}+1 \right ) }\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( -2\,\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}{\it EllipticF} \left ( x\sqrt{ia},i \right ){x}^{3}{a}^{2}+\sqrt{ia}{x}^{4}{a}^{2}+\sqrt{ia} \right ){\frac{1}{\sqrt{ia}}}}-{\frac{1}{3\,{x}^{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{\frac{\Gamma \left (-\frac{3}{4}\right ) \,_2F_1\left (\begin{matrix} -\frac{3}{4},-\frac{1}{2} \\ \frac{1}{4} \end{matrix} ; -a^{2} x^{4} \right )}{4 \, x^{3} \Gamma \left (\frac{1}{4}\right )}}{a} - \frac{1}{3 \, a x^{3}} \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^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.60444, size = 42, normalized size = 0.46 \begin{align*} - \frac{\Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{e^{i \pi }}{a^{2} x^{4}}} \right )}}{4 x \Gamma \left (\frac{5}{4}\right )} - \frac{1}{3 a x^{3}} \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 \frac{\sqrt{\frac{1}{a^{2} x^{4}} + 1} + \frac{1}{a x^{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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