Optimal. Leaf size=51 \[ \frac {\sqrt {\frac {\pi }{3}} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}} \]
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Rubi [A] time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, 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, 2204} \[ \frac {\sqrt {\frac {\pi }{3}} x^3 \left (a x^n\right )^{-3/n} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2310
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx &=\frac {\left (x^3 \left (a x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac {\left (2 x^3 \left (a x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int e^{\frac {3 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{n}\\ &=\frac {\sqrt {\frac {\pi }{3}} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 1.00 \[ \frac {\sqrt {\frac {\pi }{3}} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{\sqrt {n}} \]
Antiderivative was successfully verified.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {\log \left (a x^{n}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {\ln \left (a \,x^{n}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {\log \left (a x^{n}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{\sqrt {\ln \left (a\,x^n\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {\log {\left (a x^{n} \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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