Optimal. Leaf size=23 \[ -\frac {\log \left (c \left (b x^n\right )^p\right )}{x}-\frac {n p}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 23, 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 = {2304, 2445} \[ -\frac {\log \left (c \left (b x^n\right )^p\right )}{x}-\frac {n p}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2304
Rule 2445
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (b x^n\right )^p\right )}{x^2} \, dx &=\operatorname {Subst}\left (\int \frac {\log \left (b^p c x^{n p}\right )}{x^2} \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=-\frac {n p}{x}-\frac {\log \left (c \left (b x^n\right )^p\right )}{x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 23, normalized size = 1.00 \[ -\frac {\log \left (c \left (b x^n\right )^p\right )}{x}-\frac {n p}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 20, normalized size = 0.87 \[ -\frac {n p \log \relax (x) + n p + p \log \relax (b) + \log \relax (c)}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 25, normalized size = 1.09 \[ -\frac {n p \log \relax (x)}{x} - \frac {n p + p \log \relax (b) + \log \relax (c)}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.05, size = 23, normalized size = 1.00 \[ -\frac {n p}{x} - \frac {\log \left (\left (b x^{n}\right )^{p} c\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.88, size = 19, normalized size = 0.83 \[ -\frac {\ln \left (c\,{\left (b\,x^n\right )}^p\right )+n\,p}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.00, size = 26, normalized size = 1.13 \[ - \frac {n p \log {\relax (x )}}{x} - \frac {n p}{x} - \frac {p \log {\relax (b )}}{x} - \frac {\log {\relax (c )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________