Optimal. Leaf size=147 \[ \frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{a} x\right ),-1\right )}{\sqrt{a}}-\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{a x}+x e^{\text{sech}^{-1}\left (a x^2\right )}-\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} E\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{\sqrt{a}}-\frac{2}{a x} \]
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Rubi [A] time = 0.0680163, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6330, 30, 259, 325, 307, 221, 1199, 424} \[ -\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{a x}+x e^{\text{sech}^{-1}\left (a x^2\right )}+\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{\sqrt{a}}-\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} E\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{\sqrt{a}}-\frac{2}{a x} \]
Antiderivative was successfully verified.
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Rule 6330
Rule 30
Rule 259
Rule 325
Rule 307
Rule 221
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int e^{\text{sech}^{-1}\left (a x^2\right )} \, dx &=e^{\text{sech}^{-1}\left (a x^2\right )} x+\frac{2 \int \frac{1}{x^2} \, dx}{a}+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{x^2 \sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{a}\\ &=-\frac{2}{a x}+e^{\text{sech}^{-1}\left (a x^2\right )} x+\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{x^2 \sqrt{1-a^2 x^4}} \, dx}{a}\\ &=-\frac{2}{a x}+e^{\text{sech}^{-1}\left (a x^2\right )} x-\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{a x}-\left (2 a \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^2}{\sqrt{1-a^2 x^4}} \, dx\\ &=-\frac{2}{a x}+e^{\text{sech}^{-1}\left (a x^2\right )} x-\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{a x}+\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{\sqrt{1-a^2 x^4}} \, dx-\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1+a x^2}{\sqrt{1-a^2 x^4}} \, dx\\ &=-\frac{2}{a x}+e^{\text{sech}^{-1}\left (a x^2\right )} x-\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{a x}+\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{\sqrt{a}}-\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{\sqrt{1+a x^2}}{\sqrt{1-a x^2}} \, dx\\ &=-\frac{2}{a x}+e^{\text{sech}^{-1}\left (a x^2\right )} x-\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{a x}-\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} E\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{\sqrt{a}}+\frac{2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [C] time = 0.281666, size = 135, normalized size = 0.92 \[ -\frac{2 i \sqrt{\frac{1-a x^2}{a x^2+1}} \sqrt{1-a^2 x^4} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-a} x\right )\right |-1\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-a} x\right ),-1\right )\right )}{\sqrt{-a} \left (a x^2-1\right )}+\sqrt{\frac{1-a x^2}{a x^2+1}} \left (-\frac{1}{a x}-x\right )-\frac{1}{a x} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.187, size = 132, normalized size = 0.9 \begin{align*} -{\frac{1}{ax}}-{\frac{x}{{a}^{2}{x}^{4}-1}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ({a}^{2}{x}^{4}+2\,\sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}x{\it EllipticF} \left ( x\sqrt{a},i \right ) \sqrt{a}-2\,\sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}x{\it EllipticE} \left ( x\sqrt{a},i \right ) \sqrt{a}-1 \right ) } \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{\int \frac{\sqrt{a x^{2} + 1} \sqrt{-a x^{2} + 1}}{x^{2}}\,{d x}}{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 x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} + 1}{a 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*} \frac{\int \frac{1}{x^{2}}\, dx + \int a \sqrt{-1 + \frac{1}{a x^{2}}} \sqrt{1 + \frac{1}{a x^{2}}}\, dx}{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 \sqrt{\frac{1}{a x^{2}} + 1} \sqrt{\frac{1}{a x^{2}} - 1} + \frac{1}{a x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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