Optimal. Leaf size=181 \[ -\frac{\sqrt{a} \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) \text{EllipticF}\left (2 \cot ^{-1}\left (\sqrt{a} x\right ),\frac{1}{2}\right )}{5 \sqrt{\frac{1}{a^2 x^4}+1}}-\frac{2 a^2 \sqrt{\frac{1}{a^2 x^4}+1}}{5 x \left (a+\frac{1}{x^2}\right )}-\frac{\sqrt{\frac{1}{a^2 x^4}+1}}{5 x^3}+\frac{2 \sqrt{a} \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 \sqrt{\frac{1}{a^2 x^4}+1}}-\frac{1}{5 a x^5} \]
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Rubi [A] time = 0.0954163, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6336, 30, 335, 279, 305, 220, 1196} \[ -\frac{2 a^2 \sqrt{\frac{1}{a^2 x^4}+1}}{5 x \left (a+\frac{1}{x^2}\right )}-\frac{\sqrt{\frac{1}{a^2 x^4}+1}}{5 x^3}-\frac{\sqrt{a} \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) F\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 \sqrt{\frac{1}{a^2 x^4}+1}}+\frac{2 \sqrt{a} \sqrt{\frac{a^2+\frac{1}{x^4}}{\left (a+\frac{1}{x^2}\right )^2}} \left (a+\frac{1}{x^2}\right ) E\left (2 \cot ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{5 \sqrt{\frac{1}{a^2 x^4}+1}}-\frac{1}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 335
Rule 279
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{x^4} \, dx &=\frac{\int \frac{1}{x^6} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^4}}}{x^4} \, dx\\ &=-\frac{1}{5 a x^5}-\operatorname{Subst}\left (\int x^2 \sqrt{1+\frac{x^4}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{5 a x^5}-\frac{\sqrt{1+\frac{1}{a^2 x^4}}}{5 x^3}-\frac{2}{5} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{5 a x^5}-\frac{\sqrt{1+\frac{1}{a^2 x^4}}}{5 x^3}-\frac{1}{5} (2 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )+\frac{1}{5} (2 a) \operatorname{Subst}\left (\int \frac{1-\frac{x^2}{a}}{\sqrt{1+\frac{x^4}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{5 a x^5}-\frac{\sqrt{1+\frac{1}{a^2 x^4}}}{5 x^3}-\frac{2 a^2 \sqrt{1+\frac{1}{a^2 x^4}}}{5 \left (a+\frac{1}{x^2}\right ) x}+\frac{2 \sqrt{a} \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 \sqrt{1+\frac{1}{a^2 x^4}}}-\frac{\sqrt{a} \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 \sqrt{1+\frac{1}{a^2 x^4}}}\\ \end{align*}
Mathematica [C] time = 0.192641, size = 114, normalized size = 0.63 \[ \frac{\left (a x^2\right )^{3/2} \left (4 e^{2 \text{csch}^{-1}\left (a x^2\right )} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{3}{4},\frac{7}{4},e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )+3 \left (1-e^{2 \text{csch}^{-1}\left (a x^2\right )}\right )^{3/2}\right )}{6 x^3 \sqrt{2-2 e^{2 \text{csch}^{-1}\left (a x^2\right )}} \sqrt{\frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{e^{2 \text{csch}^{-1}\left (a x^2\right )}-1}}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.215, size = 171, normalized size = 0.9 \begin{align*}{\frac{1}{5\,{x}^{3} \left ({a}^{2}{x}^{4}+1 \right ) }\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( -2\,\sqrt{ia}{x}^{8}{a}^{4}+2\,i{a}^{3}\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}{x}^{5}{\it EllipticF} \left ( x\sqrt{ia},i \right ) -2\,i{a}^{3}\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}{x}^{5}{\it EllipticE} \left ( x\sqrt{ia},i \right ) -3\,\sqrt{ia}{x}^{4}{a}^{2}-\sqrt{ia} \right ){\frac{1}{\sqrt{ia}}}}-{\frac{1}{5\,a{x}^{5}}} \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{5}{4}\right ) \,_2F_1\left (\begin{matrix} -\frac{5}{4},-\frac{1}{2} \\ -\frac{1}{4} \end{matrix} ; -a^{2} x^{4} \right )}{4 \, x^{5} \Gamma \left (-\frac{1}{4}\right )}}{a} - \frac{1}{5 \, a x^{5}} \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^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.46518, size = 44, normalized size = 0.24 \begin{align*} - \frac{\Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{e^{i \pi }}{a^{2} x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac{7}{4}\right )} - \frac{1}{5 a x^{5}} \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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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