Optimal. Leaf size=72 \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {n} \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{4 x^2}-\frac {\sqrt {\log \left (a x^n\right )}}{2 x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 = {2305, 2310, 2180, 2205} \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {n} \left (a x^n\right )^{2/n} \text {Erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{4 x^2}-\frac {\sqrt {\log \left (a x^n\right )}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2205
Rule 2305
Rule 2310
Rubi steps
\begin {align*} \int \frac {\sqrt {\log \left (a x^n\right )}}{x^3} \, dx &=-\frac {\sqrt {\log \left (a x^n\right )}}{2 x^2}+\frac {1}{4} n \int \frac {1}{x^3 \sqrt {\log \left (a x^n\right )}} \, dx\\ &=-\frac {\sqrt {\log \left (a x^n\right )}}{2 x^2}+\frac {\left (a x^n\right )^{2/n} \operatorname {Subst}\left (\int \frac {e^{-\frac {2 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{4 x^2}\\ &=-\frac {\sqrt {\log \left (a x^n\right )}}{2 x^2}+\frac {\left (a x^n\right )^{2/n} \operatorname {Subst}\left (\int e^{-\frac {2 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{2 x^2}\\ &=\frac {\sqrt {n} \sqrt {\frac {\pi }{2}} \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{4 x^2}-\frac {\sqrt {\log \left (a x^n\right )}}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 73, normalized size = 1.01 \[ -\frac {4 \log \left (a x^n\right )+\sqrt {2} n \left (a x^n\right )^{2/n} \sqrt {\frac {\log \left (a x^n\right )}{n}} \Gamma \left (\frac {1}{2},\frac {2 \log \left (a x^n\right )}{n}\right )}{8 x^2 \sqrt {\log \left (a x^n\right )}} \]
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 \[ \int \frac {\sqrt {\log \left (a x^{n}\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\ln \left (a \,x^{n}\right )}}{x^{3}}\, dx \]
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 \[ \int \frac {\sqrt {\log \left (a x^{n}\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\ln \left (a\,x^n\right )}}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {\log {\left (a x^{n} \right )}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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