Optimal. Leaf size=40 \[ \frac{\sqrt{\pi } \left (a x^n\right )^{\frac{1}{n}} \text{Erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{n} x} \]
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Rubi [A] time = 0.0327074, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2310, 2180, 2205} \[ \frac{\sqrt{\pi } \left (a x^n\right )^{\frac{1}{n}} \text{Erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{n} x} \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2180
Rule 2205
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{\log \left (a x^n\right )}} \, dx &=\frac{\left (a x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int \frac{e^{-\frac{x}{n}}}{\sqrt{x}} \, dx,x,\log \left (a x^n\right )\right )}{n x}\\ &=\frac{\left (2 \left (a x^n\right )^{\frac{1}{n}}\right ) \operatorname{Subst}\left (\int e^{-\frac{x^2}{n}} \, dx,x,\sqrt{\log \left (a x^n\right )}\right )}{n x}\\ &=\frac{\sqrt{\pi } \left (a x^n\right )^{\frac{1}{n}} \text{erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{\sqrt{n} x}\\ \end{align*}
Mathematica [A] time = 0.0342237, size = 52, normalized size = 1.3 \[ -\frac{\left (a x^n\right )^{\frac{1}{n}} \sqrt{\frac{\log \left (a x^n\right )}{n}} \text{Gamma}\left (\frac{1}{2},\frac{\log \left (a x^n\right )}{n}\right )}{x \sqrt{\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.171, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt{\ln \left ( a{x}^{n} \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}{x^{2} \sqrt{\log \left (a x^{n}\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}{x^{2} \sqrt{\log{\left (a x^{n} \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}{x^{2} \sqrt{\log \left (a x^{n}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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