Optimal. Leaf size=39 \[ \sqrt{\frac{\pi }{2}} x \sqrt{\frac{a}{x^2}} \text{Erfi}\left (\frac{\sqrt{-\log \left (\frac{a}{x^2}\right )}}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0246292, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2300, 2180, 2204} \[ \sqrt{\frac{\pi }{2}} x \sqrt{\frac{a}{x^2}} \text{Erfi}\left (\frac{\sqrt{-\log \left (\frac{a}{x^2}\right )}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 2300
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-\log \left (\frac{a}{x^2}\right )}} \, dx &=-\left (\frac{1}{2} \left (\sqrt{\frac{a}{x^2}} x\right ) \operatorname{Subst}\left (\int \frac{e^{-x/2}}{\sqrt{-x}} \, dx,x,\log \left (\frac{a}{x^2}\right )\right )\right )\\ &=\left (\sqrt{\frac{a}{x^2}} x\right ) \operatorname{Subst}\left (\int e^{\frac{x^2}{2}} \, dx,x,\sqrt{-\log \left (\frac{a}{x^2}\right )}\right )\\ &=\sqrt{\frac{\pi }{2}} \sqrt{\frac{a}{x^2}} x \text{erfi}\left (\frac{\sqrt{-\log \left (\frac{a}{x^2}\right )}}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0124686, size = 60, normalized size = 1.54 \[ -\frac{\sqrt{\frac{\pi }{2}} x \sqrt{\frac{a}{x^2}} \sqrt{\log \left (\frac{a}{x^2}\right )} \text{Erf}\left (\frac{\sqrt{\log \left (\frac{a}{x^2}\right )}}{\sqrt{2}}\right )}{\sqrt{-\log \left (\frac{a}{x^2}\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{-\ln \left ({\frac{a}{{x}^{2}}} \right ) }}}\, 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}{\sqrt{-\log \left (\frac{a}{x^{2}}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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}{\sqrt{- \log{\left (\frac{a}{x^{2}} \right )}}}\, 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}{\sqrt{-\log \left (\frac{a}{x^{2}}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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