Optimal. Leaf size=28 \[ b n \text{PolyLog}(3,-e x)-\text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0275801, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2374, 6589} \[ b n \text{PolyLog}(3,-e x)-\text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx &=-\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)+(b n) \int \frac{\text{Li}_2(-e x)}{x} \, dx\\ &=-\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)+b n \text{Li}_3(-e x)\\ \end{align*}
Mathematica [A] time = 0.0089549, size = 34, normalized size = 1.21 \[ -a \text{PolyLog}(2,-e x)-b \text{PolyLog}(2,-e x) \log \left (c x^n\right )+b n \text{PolyLog}(3,-e x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.187, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( ex+1 \right ) }{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left (e x + 1\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a \log \left (e x + 1\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left (e x + 1\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]