Optimal. Leaf size=115 \[ -\frac{2}{3} \sqrt{a} \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{a} x\right ),-1\right )+\frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{3 a x^3}+\frac{2}{3 a x^3}-\frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{x} \]
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Rubi [A] time = 0.0510283, antiderivative size = 115, 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 = {6335, 30, 259, 325, 221} \[ \frac{2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{3 a x^3}+\frac{2}{3 a x^3}-\frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{x}-\frac{2}{3} \sqrt{a} \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} F\left (\left .\sin ^{-1}\left (\sqrt{a} x\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 259
Rule 325
Rule 221
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{x^2} \, dx &=-\frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{x}-\frac{2 \int \frac{1}{x^4} \, dx}{a}-\frac{\left (2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{x^4 \sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{a}\\ &=\frac{2}{3 a x^3}-\frac{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^4 \sqrt{1-a^2 x^4}} \, dx}{a}\\ &=\frac{2}{3 a x^3}-\frac{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}}{3 a x^3}-\frac{1}{3} \left (2 a \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{\sqrt{1-a^2 x^4}} \, dx\\ &=\frac{2}{3 a x^3}-\frac{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}}{3 a x^3}-\frac{2}{3} \sqrt{a} \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 )\\ \end{align*}
Mathematica [C] time = 0.191668, size = 123, normalized size = 1.07 \[ \frac{2 i \sqrt{-a} \sqrt{\frac{1-a x^2}{a x^2+1}} \sqrt{1-a^2 x^4} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-a} x\right ),-1\right )}{3 a x^2-3}-\frac{\sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^2+1\right )}{3 a x^3}-\frac{1}{3 a x^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.208, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{3\,x \left ({a}^{2}{x}^{4}-1 \right ) }\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ( 2\,\sqrt{-a{x}^{2}+1}\sqrt{a{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{a},i \right ){x}^{3}{a}^{3/2}-{a}^{2}{x}^{4}+1 \right ) }-{\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{\int \frac{\sqrt{a x^{2} + 1} \sqrt{-a x^{2} + 1}}{x^{4}}\,{d x}}{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 x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} + 1}{a x^{4}}, 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^{4}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x^{2}}} \sqrt{1 + \frac{1}{a x^{2}}}}{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 \frac{\sqrt{\frac{1}{a x^{2}} + 1} \sqrt{\frac{1}{a x^{2}} - 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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