Optimal. Leaf size=80 \[ \frac {4 \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {4 x}{3 n^2 \sqrt {\log \left (a x^n\right )}}-\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2297, 2300, 2180, 2204} \[ \frac {4 \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {Erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {4 x}{3 n^2 \sqrt {\log \left (a x^n\right )}}-\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2297
Rule 2300
Rubi steps
\begin {align*} \int \frac {1}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx &=-\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {2 \int \frac {1}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx}{3 n}\\ &=-\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {4 \int \frac {1}{\sqrt {\log \left (a x^n\right )}} \, dx}{3 n^2}\\ &=-\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (4 x \left (a x^n\right )^{-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}\\ &=-\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x}{3 n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (8 x \left (a x^n\right )^{-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}\\ &=\frac {4 \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2}}-\frac {2 x}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x}{3 n^2 \sqrt {\log \left (a x^n\right )}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 83, normalized size = 1.04 \[ -\frac {2 x \left (a x^n\right )^{-1/n} \left (\left (a x^n\right )^{\frac {1}{n}} \left (2 \log \left (a x^n\right )+n\right )+2 n \left (-\frac {\log \left (a x^n\right )}{n}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {\log \left (a x^n\right )}{n}\right )\right )}{3 n^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \]
Antiderivative was successfully verified.
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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 \[ \int \frac {1}{\log \left (a x^{n}\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{\ln \left (a \,x^{n}\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {1}{\log \left (a x^{n}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\ln \left (a\,x^n\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\log {\left (a x^{n} \right )}^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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