Integrand size = 14, antiderivative size = 17 \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x} \, dx=\frac {2 \log ^{\frac {3}{2}}\left (a x^n\right )}{3 n} \]
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Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2339, 30} \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x} \, dx=\frac {2 \log ^{\frac {3}{2}}\left (a x^n\right )}{3 n} \]
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Rule 30
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {x} \, dx,x,\log \left (a x^n\right )\right )}{n} \\ & = \frac {2 \log ^{\frac {3}{2}}\left (a x^n\right )}{3 n} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x} \, dx=\frac {2 \log ^{\frac {3}{2}}\left (a x^n\right )}{3 n} \]
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Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {2 \ln \left (a \,x^{n}\right )^{\frac {3}{2}}}{3 n}\) | \(14\) |
default | \(\frac {2 \ln \left (a \,x^{n}\right )^{\frac {3}{2}}}{3 n}\) | \(14\) |
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none
Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x} \, dx=\frac {2 \, {\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{\frac {3}{2}}}{3 \, n} \]
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Time = 0.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x} \, dx=- \begin {cases} - \sqrt {\log {\left (a \right )}} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {2 \log {\left (a x^{n} \right )}^{\frac {3}{2}}}{3 n} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x} \, dx=\frac {2 \, \log \left (a x^{n}\right )^{\frac {3}{2}}}{3 \, n} \]
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none
Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x} \, dx=\frac {2 \, {\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{\frac {3}{2}}}{3 \, n} \]
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Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {\log \left (a x^n\right )}}{x} \, dx=\frac {2\,{\ln \left (a\,x^n\right )}^{3/2}}{3\,n} \]
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