Optimal. Leaf size=40 \[ -\frac{\sqrt{\frac{\pi }{2}} x \text{Erf}\left (\frac{\sqrt{-\log \left (a x^2\right )}}{\sqrt{2}}\right )}{\sqrt{a x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0288386, antiderivative size = 40, 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, 2205} \[ -\frac{\sqrt{\frac{\pi }{2}} x \text{Erf}\left (\frac{\sqrt{-\log \left (a x^2\right )}}{\sqrt{2}}\right )}{\sqrt{a x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2300
Rule 2180
Rule 2205
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-\log \left (a x^2\right )}} \, dx &=\frac{x \operatorname{Subst}\left (\int \frac{e^{x/2}}{\sqrt{-x}} \, dx,x,\log \left (a x^2\right )\right )}{2 \sqrt{a x^2}}\\ &=-\frac{x \operatorname{Subst}\left (\int e^{-\frac{x^2}{2}} \, dx,x,\sqrt{-\log \left (a x^2\right )}\right )}{\sqrt{a x^2}}\\ &=-\frac{\sqrt{\frac{\pi }{2}} x \text{erf}\left (\frac{\sqrt{-\log \left (a x^2\right )}}{\sqrt{2}}\right )}{\sqrt{a x^2}}\\ \end{align*}
Mathematica [A] time = 0.0106863, size = 59, normalized size = 1.48 \[ \frac{\sqrt{\frac{\pi }{2}} x \sqrt{\log \left (a x^2\right )} \text{Erfi}\left (\frac{\sqrt{\log \left (a x^2\right )}}{\sqrt{2}}\right )}{\sqrt{a x^2} \sqrt{-\log \left (a x^2\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{-\ln \left ( a{x}^{2} \right ) }}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\log \left (a x^{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \log{\left (a x^{2} \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-\log \left (a x^{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]