Optimal. Leaf size=21 \[ \log (6) \log (x)-\frac {\text {Li}_2\left (-\frac {e x^n}{3}\right )}{n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2454, 2392, 2391} \[ \log (6) \log (x)-\frac {\text {PolyLog}\left (2,-\frac {e x^n}{3}\right )}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2391
Rule 2392
Rule 2454
Rubi steps
\begin {align*} \int \frac {\log \left (2 \left (3+e x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\log (2 (3+e x))}{x} \, dx,x,x^n\right )}{n}\\ &=\log (6) \log (x)+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {e x}{3}\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\log (6) \log (x)-\frac {\text {Li}_2\left (-\frac {e x^n}{3}\right )}{n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \[ \log (6) \log (x)-\frac {\text {Li}_2\left (-\frac {e x^n}{3}\right )}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.43, size = 41, normalized size = 1.95 \[ \frac {n \log \left (2 \, e x^{n} + 6\right ) \log \relax (x) - n \log \left (\frac {1}{3} \, e x^{n} + 1\right ) \log \relax (x) - {\rm Li}_2\left (-\frac {1}{3} \, e x^{n}\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (2 \, e x^{n} + 6\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 57, normalized size = 2.71 \[ \frac {\ln \left (-\frac {e \,x^{n}}{3}\right ) \ln \left (2 e \,x^{n}+6\right )}{n}-\frac {\ln \left (-\frac {e \,x^{n}}{3}\right ) \ln \left (\frac {e \,x^{n}}{3}+1\right )}{n}-\frac {\dilog \left (\frac {e \,x^{n}}{3}+1\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, n \log \relax (x)^{2} + 3 \, n \int \frac {\log \relax (x)}{e x x^{n} + 3 \, x}\,{d x} + \log \relax (2) \log \relax (x) + \log \left (e x^{n} + 3\right ) \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\ln \left (2\,e\,x^n+6\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 3.87, size = 78, normalized size = 3.71 \[ \begin {cases} \log {\relax (6 )} \log {\relax (x )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{n} e^{i \pi }}{3}\right )}{n} & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (6 )} \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{n} e^{i \pi }}{3}\right )}{n} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (6 )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (6 )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{n} e^{i \pi }}{3}\right )}{n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________