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 = 0.05, antiderivative size = 60, 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 = {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 2180
Rule 2205
Rule 2306
Rule 2310
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*}
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Mathematica [A] time = 0.05, size = 58, normalized size = 0.97 \[ \frac {2 \left (\left (a x^n\right )^{\frac {1}{n}} \sqrt {\frac {\log \left (a x^n\right )}{n}} \Gamma \left (\frac {1}{2},\frac {\log \left (a x^n\right )}{n}\right )-1\right )}{n x \sqrt {\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
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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 \[ \int \frac {1}{x^{2} \log \left (a x^{n}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \ln \left (a \,x^{n}\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {1}{x^{2} \log \left (a x^{n}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{x^2\,{\ln \left (a\,x^n\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{2} \log {\left (a x^{n} \right )}^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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