Optimal. Leaf size=29 \[ \frac {\log \left (a x^n\right ) \left (b \log ^m\left (a x^n\right )\right )^p}{n (1+m p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 29, 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}
\begin {gather*} \frac {\log \left (a x^n\right ) \left (b \log ^m\left (a x^n\right )\right )^p}{n (m p+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 30
Rubi steps
\begin {align*} \int \frac {\left (b \log ^m\left (a x^n\right )\right )^p}{x} \, dx &=\frac {\text {Subst}\left (\int \left (b x^m\right )^p \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac {\left (\log ^{-m p}\left (a x^n\right ) \left (b \log ^m\left (a x^n\right )\right )^p\right ) \text {Subst}\left (\int x^{m p} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac {\log \left (a x^n\right ) \left (b \log ^m\left (a x^n\right )\right )^p}{n (1+m p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} \frac {\log \left (a x^n\right ) \left (b \log ^m\left (a x^n\right )\right )^p}{n (1+m p)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (b \ln \left (a \,x^{n}\right )^{m}\right )^{p}}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 33, normalized size = 1.14 \begin {gather*} \frac {{\left (n \log \left (x\right ) + \log \left (a\right )\right )} e^{\left (m p \log \left (n \log \left (x\right ) + \log \left (a\right )\right ) + p \log \left (b\right )\right )}}{m n p + n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b \log {\left (a x^{n} \right )}^{m}\right )^{p}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.72, size = 35, normalized size = 1.21 \begin {gather*} \frac {{\left (n \log \left (x\right ) + \log \left (a\right )\right )} e^{\left (m p \log \left (n \log \left (x\right ) + \log \left (a\right )\right ) + p \log \left (b\right )\right )}}{{\left (m p + 1\right )} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.36, size = 29, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a\,x^n\right )\,{\left (b\,{\ln \left (a\,x^n\right )}^m\right )}^p}{n\,\left (m\,p+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________