Optimal. Leaf size=93 \[ \frac {8 \sqrt {2 \pi } x^2 \left (a x^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {8 x^2}{3 n^2 \sqrt {\log \left (a x^n\right )}}-\frac {2 x^2}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2306, 2310, 2180, 2204} \[ \frac {8 \sqrt {2 \pi } x^2 \left (a x^n\right )^{-2/n} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {8 x^2}{3 n^2 \sqrt {\log \left (a x^n\right )}}-\frac {2 x^2}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2180
Rule 2204
Rule 2306
Rule 2310
Rubi steps
\begin {align*} \int \frac {x}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx &=-\frac {2 x^2}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {4 \int \frac {x}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx}{3 n}\\ &=-\frac {2 x^2}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {8 x^2}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {16 \int \frac {x}{\sqrt {\log \left (a x^n\right )}} \, dx}{3 n^2}\\ &=-\frac {2 x^2}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {8 x^2}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (16 x^2 \left (a x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{3 n^3}\\ &=-\frac {2 x^2}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {8 x^2}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (32 x^2 \left (a x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int e^{\frac {2 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{3 n^3}\\ &=\frac {8 \sqrt {2 \pi } x^2 \left (a x^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {2 x^2}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {8 x^2}{3 n^2 \sqrt {\log \left (a x^n\right )}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 92, normalized size = 0.99 \[ -\frac {2 x^2 \left (a x^n\right )^{-2/n} \left (\left (a x^n\right )^{2/n} \left (4 \log \left (a x^n\right )+n\right )+4 \sqrt {2} n \left (-\frac {\log \left (a x^n\right )}{n}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {2 \log \left (a x^n\right )}{n}\right )\right )}{3 n^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\log \left (a x^{n}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {x}{\ln \left (a \,x^{n}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\log \left (a x^{n}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\ln \left (a\,x^n\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\log {\left (a x^{n} \right )}^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________