Optimal. Leaf size=84 \[ \frac{4 \sqrt{\pi } \left (a x^n\right )^{\frac{1}{n}} \text{Erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{3 n^{5/2} x}+\frac{4}{3 n^2 x \sqrt{\log \left (a x^n\right )}}-\frac{2}{3 n x \log ^{\frac{3}{2}}\left (a x^n\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0688181, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2306, 2310, 2180, 2205} \[ \frac{4 \sqrt{\pi } \left (a x^n\right )^{\frac{1}{n}} \text{Erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{3 n^{5/2} x}+\frac{4}{3 n^2 x \sqrt{\log \left (a x^n\right )}}-\frac{2}{3 n x \log ^{\frac{3}{2}}\left (a x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2306
Rule 2310
Rule 2180
Rule 2205
Rubi steps
\begin{align*} \int \frac{1}{x^2 \log ^{\frac{5}{2}}\left (a x^n\right )} \, dx &=-\frac{2}{3 n x \log ^{\frac{3}{2}}\left (a x^n\right )}-\frac{2 \int \frac{1}{x^2 \log ^{\frac{3}{2}}\left (a x^n\right )} \, dx}{3 n}\\ &=-\frac{2}{3 n x \log ^{\frac{3}{2}}\left (a x^n\right )}+\frac{4}{3 n^2 x \sqrt{\log \left (a x^n\right )}}+\frac{4 \int \frac{1}{x^2 \sqrt{\log \left (a x^n\right )}} \, dx}{3 n^2}\\ &=-\frac{2}{3 n x \log ^{\frac{3}{2}}\left (a x^n\right )}+\frac{4}{3 n^2 x \sqrt{\log \left (a x^n\right )}}+\frac{\left (4 \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 )}{3 n^3 x}\\ &=-\frac{2}{3 n x \log ^{\frac{3}{2}}\left (a x^n\right )}+\frac{4}{3 n^2 x \sqrt{\log \left (a x^n\right )}}+\frac{\left (8 \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 )}{3 n^3 x}\\ &=\frac{4 \sqrt{\pi } \left (a x^n\right )^{\frac{1}{n}} \text{erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{3 n^{5/2} x}-\frac{2}{3 n x \log ^{\frac{3}{2}}\left (a x^n\right )}+\frac{4}{3 n^2 x \sqrt{\log \left (a x^n\right )}}\\ \end{align*}
Mathematica [A] time = 0.0542735, size = 70, normalized size = 0.83 \[ -\frac{2 \left (2 n \left (a x^n\right )^{\frac{1}{n}} \left (\frac{\log \left (a x^n\right )}{n}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},\frac{\log \left (a x^n\right )}{n}\right )-2 \log \left (a x^n\right )+n\right )}{3 n^2 x \log ^{\frac{3}{2}}\left (a x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.182, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{-{\frac{5}{2}}}}\, 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}{x^{2} \log \left (a x^{n}\right )^{\frac{5}{2}}}\,{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}{x^{2} \log \left (a x^{n}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]