Optimal. Leaf size=25 \[ \frac {\log \left (a x^n\right ) \sqrt {\log ^2\left (a x^n\right )}}{2 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {15, 30} \[ \frac {\log \left (a x^n\right ) \sqrt {\log ^2\left (a x^n\right )}}{2 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 30
Rubi steps
\begin {align*} \int \frac {\sqrt {\log ^2\left (a x^n\right )}}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {x^2} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac {\sqrt {\log ^2\left (a x^n\right )} \operatorname {Subst}\left (\int x \, dx,x,\log \left (a x^n\right )\right )}{n \log \left (a x^n\right )}\\ &=\frac {\log \left (a x^n\right ) \sqrt {\log ^2\left (a x^n\right )}}{2 n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {\log \left (a x^n\right ) \sqrt {\log ^2\left (a x^n\right )}}{2 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.55, size = 13, normalized size = 0.52 \[ \frac {1}{2} \, n \log \relax (x)^{2} + \log \relax (a) \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 27, normalized size = 1.08 \[ \frac {1}{2} \, n \log \relax (x)^{2} \mathrm {sgn}\left (\log \left (a x^{n}\right )\right ) + \log \relax (a) \log \relax (x) \mathrm {sgn}\left (\log \left (a x^{n}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.35, size = 21, normalized size = 0.84 \[ \frac {\mathrm {csgn}\left (\ln \left (a \,x^{n}\right )\right ) \ln \left (a \,x^{n}\right )^{2}}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.63, size = 20, normalized size = 0.80 \[ -\frac {1}{2} \, n \log \relax (x)^{2} + \log \relax (a) \log \relax (x) + \log \relax (x) \log \left (x^{n}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.40, size = 21, normalized size = 0.84 \[ \frac {\ln \left (a\,x^n\right )\,\sqrt {{\ln \left (a\,x^n\right )}^2}}{2\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\log {\left (a x^{n} \right )}^{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________