Optimal. Leaf size=60 \[ -\frac{2 \sqrt{\pi } \left (a x^n\right )^{\frac{1}{n}} \text{Erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{n^{3/2} x}-\frac{2}{n x \sqrt{\log \left (a x^n\right )}} \]
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Rubi [A] time = 0.0501952, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2306, 2310, 2180, 2205} \[ -\frac{2 \sqrt{\pi } \left (a x^n\right )^{\frac{1}{n}} \text{Erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{n^{3/2} x}-\frac{2}{n x \sqrt{\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
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Rule 2306
Rule 2310
Rule 2180
Rule 2205
Rubi steps
\begin{align*} \int \frac{1}{x^2 \log ^{\frac{3}{2}}\left (a x^n\right )} \, dx &=-\frac{2}{n x \sqrt{\log \left (a x^n\right )}}-\frac{2 \int \frac{1}{x^2 \sqrt{\log \left (a x^n\right )}} \, dx}{n}\\ &=-\frac{2}{n x \sqrt{\log \left (a x^n\right )}}-\frac{\left (2 \left (a x^n\right )^{\frac{1}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{-\frac{x}{n}}}{\sqrt{x}} \, dx,x,\log \left (a x^n\right )\right )}{n^2 x}\\ &=-\frac{2}{n x \sqrt{\log \left (a x^n\right )}}-\frac{\left (4 \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^2 x}\\ &=-\frac{2 \sqrt{\pi } \left (a x^n\right )^{\frac{1}{n}} \text{erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{n^{3/2} x}-\frac{2}{n x \sqrt{\log \left (a x^n\right )}}\\ \end{align*}
Mathematica [A] time = 0.0490753, size = 58, normalized size = 0.97 \[ \frac{2 \left (\left (a x^n\right )^{\frac{1}{n}} \sqrt{\frac{\log \left (a x^n\right )}{n}} \text{Gamma}\left (\frac{1}{2},\frac{\log \left (a x^n\right )}{n}\right )-1\right )}{n x \sqrt{\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.173, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{-{\frac{3}{2}}}}\, 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} \log \left (a x^{n}\right )^{\frac{3}{2}}}\,{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} \log{\left (a x^{n} \right )}^{\frac{3}{2}}}\, 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} \log \left (a x^{n}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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