3.345 \(\int \frac{x^{-1+n} \log (e x^n)}{1-e x^n} \, dx\)

Optimal. Leaf size=17 \[ \frac{\text{PolyLog}\left (2,1-e x^n\right )}{e n} \]

[Out]

PolyLog[2, 1 - e*x^n]/(e*n)

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Rubi [A]  time = 0.0651177, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2336, 2315} \[ \frac{\text{PolyLog}\left (2,1-e x^n\right )}{e n} \]

Antiderivative was successfully verified.

[In]

Int[(x^(-1 + n)*Log[e*x^n])/(1 - e*x^n),x]

[Out]

PolyLog[2, 1 - e*x^n]/(e*n)

Rule 2336

Int[((a_.) + Log[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :>
Dist[f^m/n, Subst[Int[(d + e*x)^q*(a + b*Log[c*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}
, x] && EqQ[m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && EqQ[r, n]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int \frac{x^{-1+n} \log \left (e x^n\right )}{1-e x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log (e x)}{1-e x} \, dx,x,x^n\right )}{n}\\ &=\frac{\text{Li}_2\left (1-e x^n\right )}{e n}\\ \end{align*}

Mathematica [A]  time = 0.0106647, size = 17, normalized size = 1. \[ \frac{\text{PolyLog}\left (2,1-e x^n\right )}{e n} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(-1 + n)*Log[e*x^n])/(1 - e*x^n),x]

[Out]

PolyLog[2, 1 - e*x^n]/(e*n)

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Maple [A]  time = 0.041, size = 14, normalized size = 0.8 \begin{align*}{\frac{{\it dilog} \left ( e{x}^{n} \right ) }{en}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+n)*ln(e*x^n)/(1-e*x^n),x)

[Out]

1/e/n*dilog(e*x^n)

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Maxima [B]  time = 1.84436, size = 70, normalized size = 4.12 \begin{align*} -\frac{\log \left (e\right ) \log \left (\frac{e x^{n} - 1}{e}\right )}{e n} - \frac{\log \left (-e x^{n} + 1\right ) \log \left (x^{n}\right ) +{\rm Li}_2\left (e x^{n}\right )}{e n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*log(e*x^n)/(1-e*x^n),x, algorithm="maxima")

[Out]

-log(e)*log((e*x^n - 1)/e)/(e*n) - (log(-e*x^n + 1)*log(x^n) + dilog(e*x^n))/(e*n)

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Fricas [B]  time = 1.29888, size = 100, normalized size = 5.88 \begin{align*} -\frac{n \log \left (-e x^{n} + 1\right ) \log \left (x\right ) + \log \left (e x^{n} - 1\right ) \log \left (e\right ) +{\rm Li}_2\left (e x^{n}\right )}{e n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*log(e*x^n)/(1-e*x^n),x, algorithm="fricas")

[Out]

-(n*log(-e*x^n + 1)*log(x) + log(e*x^n - 1)*log(e) + dilog(e*x^n))/(e*n)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+n)*ln(e*x**n)/(1-e*x**n),x)

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{n - 1} \log \left (e x^{n}\right )}{e x^{n} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*log(e*x^n)/(1-e*x^n),x, algorithm="giac")

[Out]

integrate(-x^(n - 1)*log(e*x^n)/(e*x^n - 1), x)