Optimal. Leaf size=165 \[ -\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\sqrt{a} x\right ),\frac{1}{2}\right )}{a^{3/2} \sqrt{\frac{1}{a^2 x^4}+1}}+x \sqrt{\frac{1}{a^2 x^4}+1}-\frac{2 \sqrt{\frac{1}{a^2 x^4}+1}}{x \left (a+\frac{1}{x^2}\right )}+\frac{2 \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{a^{3/2} \sqrt{\frac{1}{a^2 x^4}+1}}-\frac{1}{a x} \]
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Rubi [A] time = 0.0803542, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {6331, 30, 242, 277, 305, 220, 1196} \[ x \sqrt{\frac{1}{a^2 x^4}+1}-\frac{2 \sqrt{\frac{1}{a^2 x^4}+1}}{x \left (a+\frac{1}{x^2}\right )}-\frac{\sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{a^{3/2} \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{2 \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{a^{3/2} \sqrt{\frac{1}{a^2 x^4}+1}}-\frac{1}{a x} \]
Antiderivative was successfully verified.
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Rule 6331
Rule 30
Rule 242
Rule 277
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int e^{\text{csch}^{-1}\left (a x^2\right )} \, dx &=\frac{\int \frac{1}{x^2} \, dx}{a}+\int \sqrt{1+\frac{1}{a^2 x^4}} \, dx\\ &=-\frac{1}{a x}-\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^4}{a^2}}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{a x}+\sqrt{1+\frac{1}{a^2 x^4}} x-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}\\ &=-\frac{1}{a x}+\sqrt{1+\frac{1}{a^2 x^4}} x-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}+\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 )}{a}\\ &=-\frac{1}{a x}-\frac{2 \sqrt{1+\frac{1}{a^2 x^4}}}{\left (a+\frac{1}{x^2}\right ) x}+\sqrt{1+\frac{1}{a^2 x^4}} x+\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 )}{a^{3/2} \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 )}{a^{3/2} \sqrt{1+\frac{1}{a^2 x^4}}}\\ \end{align*}
Mathematica [C] time = 0.141479, size = 96, normalized size = 0.58 \[ \frac{\sqrt{2} x e^{\text{csch}^{-1}\left (a x^2\right )} \sqrt{\frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{e^{2 \text{csch}^{-1}\left (a x^2\right )}-1}} \left (4 \sqrt{1-e^{2 \text{csch}^{-1}\left (a x^2\right )}} \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{3}{2},\frac{7}{4},e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )-3\right )}{3 \sqrt{a x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.18, size = 146, normalized size = 0.9 \begin{align*}{\frac{x}{{a}^{2}{x}^{4}+1}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( -\sqrt{ia}{x}^{4}{a}^{2}+2\,i\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}x{\it EllipticF} \left ( x\sqrt{ia},i \right ) a-2\,i\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}x{\it EllipticE} \left ( x\sqrt{ia},i \right ) a-\sqrt{ia} \right ){\frac{1}{\sqrt{ia}}}}-{\frac{1}{ax}} \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{1}{4}\right ) \,_2F_1\left (\begin{matrix} -\frac{1}{2},-\frac{1}{4} \\ \frac{3}{4} \end{matrix} ; -a^{2} x^{4} \right )}{4 \, x \Gamma \left (\frac{3}{4}\right )}}{a} - \frac{1}{a x} \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^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.908202, size = 42, normalized size = 0.25 \begin{align*} - \frac{x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{e^{i \pi }}{a^{2} x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{4}\right )} - \frac{1}{a x} \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 \sqrt{\frac{1}{a^{2} x^{4}} + 1} + \frac{1}{a x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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