Optimal. Leaf size=72 \[ -\frac{\sqrt [4]{1-a x} (a x+1)^{3/4}}{x}-a \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+a \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right ) \]
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Rubi [A] time = 0.0363453, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6126, 94, 93, 298, 203, 206} \[ -\frac{\sqrt [4]{1-a x} (a x+1)^{3/4}}{x}-a \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+a \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right ) \]
Antiderivative was successfully verified.
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Rule 6126
Rule 94
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-\frac{1}{2} \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac{\sqrt [4]{1-a x}}{x^2 \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{x}-\frac{1}{2} a \int \frac{1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{x}-(2 a) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{x}+a \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-a \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac{\sqrt [4]{1-a x} (1+a x)^{3/4}}{x}-a \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+a \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0136235, size = 55, normalized size = 0.76 \[ \frac{\sqrt [4]{1-a x} \left (2 a x \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{1-a x}{a x+1}\right )-a x-1\right )}{x \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}}}}}\, dx \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*} \int \frac{1}{x^{2} \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74347, size = 298, normalized size = 4.14 \begin{align*} -\frac{2 \, a x \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) - a x \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + a x \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right ) + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{2 \, x} \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{1}{x^{2} \sqrt{\frac{a x + 1}{\sqrt{- a^{2} x^{2} + 1}}}}\, 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{1}{x^{2} \sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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