Optimal. Leaf size=75 \[ \frac{\sqrt{a} \left (a x^2+1\right ) \sqrt{\frac{a^2 x^4+1}{\left (a x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\sqrt{a} x\right ),\frac{1}{2}\right )}{\sqrt{a^2 x^4+1}}-\frac{\sinh ^{-1}\left (a x^2\right )}{x} \]
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Rubi [A] time = 0.0225441, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5902, 12, 220} \[ \frac{\sqrt{a} \left (a x^2+1\right ) \sqrt{\frac{a^2 x^4+1}{\left (a x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{\sqrt{a^2 x^4+1}}-\frac{\sinh ^{-1}\left (a x^2\right )}{x} \]
Antiderivative was successfully verified.
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Rule 5902
Rule 12
Rule 220
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}\left (a x^2\right )}{x^2} \, dx &=-\frac{\sinh ^{-1}\left (a x^2\right )}{x}+\int \frac{2 a}{\sqrt{1+a^2 x^4}} \, dx\\ &=-\frac{\sinh ^{-1}\left (a x^2\right )}{x}+(2 a) \int \frac{1}{\sqrt{1+a^2 x^4}} \, dx\\ &=-\frac{\sinh ^{-1}\left (a x^2\right )}{x}+\frac{\sqrt{a} \left (1+a x^2\right ) \sqrt{\frac{1+a^2 x^4}{\left (1+a x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt{a} x\right )|\frac{1}{2}\right )}{\sqrt{1+a^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0369385, size = 42, normalized size = 0.56 \[ -\frac{\sinh ^{-1}\left (a x^2\right )+2 \sqrt{i a} x \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{i a} x\right ),-1\right )}{x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.006, size = 66, normalized size = 0.9 \begin{align*} -{\frac{{\it Arcsinh} \left ( a{x}^{2} \right ) }{x}}+2\,{\frac{a\sqrt{1-ia{x}^{2}}\sqrt{1+ia{x}^{2}}{\it EllipticF} \left ( x\sqrt{ia},i \right ) }{\sqrt{ia}\sqrt{{a}^{2}{x}^{4}+1}}} \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{1}{4} \, a^{2}{\left (\frac{i \, \sqrt{2}{\left (\log \left (\frac{i \, \sqrt{2}{\left (2 \, \sqrt{a^{2}} x + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (a^{2}\right )}^{\frac{1}{4}}} + 1\right ) - \log \left (-\frac{i \, \sqrt{2}{\left (2 \, \sqrt{a^{2}} x + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (a^{2}\right )}^{\frac{1}{4}}} + 1\right )\right )}}{{\left (a^{2}\right )}^{\frac{3}{4}}} + \frac{i \, \sqrt{2}{\left (\log \left (\frac{i \, \sqrt{2}{\left (2 \, \sqrt{a^{2}} x - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (a^{2}\right )}^{\frac{1}{4}}} + 1\right ) - \log \left (-\frac{i \, \sqrt{2}{\left (2 \, \sqrt{a^{2}} x - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}\right )}}{2 \,{\left (a^{2}\right )}^{\frac{1}{4}}} + 1\right )\right )}}{{\left (a^{2}\right )}^{\frac{3}{4}}} + \frac{\sqrt{2} \log \left (\sqrt{a^{2}} x^{2} + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (a^{2}\right )}^{\frac{3}{4}}} - \frac{\sqrt{2} \log \left (\sqrt{a^{2}} x^{2} - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (a^{2}\right )}^{\frac{3}{4}}}\right )} + 2 \, a \int \frac{1}{a^{3} x^{6} + a x^{2} +{\left (a^{2} x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} - \frac{\log \left (a x^{2} + \sqrt{a^{2} x^{4} + 1}\right )}{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{\operatorname{arsinh}\left (a x^{2}\right )}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asinh}{\left (a x^{2} \right )}}{x^{2}}\, dx \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{\operatorname{arsinh}\left (a x^{2}\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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