Optimal. Leaf size=15 \[ \frac {2 \sqrt {\log \left (a x^n\right )}}{n} \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2302, 30} \[ \frac {2 \sqrt {\log \left (a x^n\right )}}{n} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {\log \left (a x^n\right )}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac {2 \sqrt {\log \left (a x^n\right )}}{n}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \[ \frac {2 \sqrt {\log \left (a x^n\right )}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 14, normalized size = 0.93 \[ \frac {2 \, \sqrt {n \log \relax (x) + \log \relax (a)}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 14, normalized size = 0.93 \[ \frac {2 \, \sqrt {n \log \relax (x) + \log \relax (a)}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 14, normalized size = 0.93 \[ \frac {2 \sqrt {\ln \left (a \,x^{n}\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 13, normalized size = 0.87 \[ \frac {2 \, \sqrt {\log \left (a x^{n}\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.58, size = 13, normalized size = 0.87 \[ \frac {2\,\sqrt {\ln \left (a\,x^n\right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.08, size = 24, normalized size = 1.60 \[ \begin {cases} \frac {2 \sqrt {n \log {\relax (x )} + \log {\relax (a )}}}{n} & \text {for}\: n \neq 0 \\\frac {\log {\relax (x )}}{\sqrt {\log {\relax (a )}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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