\(\int \frac {x^{-1+n} \log (e x^n)}{1-e x^n} \, dx\) [345]

Optimal result

Integrand size = 22, antiderivative size = 17 \[ \int \frac {x^{-1+n} \log \left (e x^n\right )}{1-e x^n} \, dx=\frac {\operatorname {PolyLog}\left (2,1-e x^n\right )}{e n} \]

[Out]

polylog(2,1-e*x^n)/e/n

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2374, 2352} \[ \int \frac {x^{-1+n} \log \left (e x^n\right )}{1-e x^n} \, dx=\frac {\operatorname {PolyLog}\left (2,1-e x^n\right )}{e n} \]

[In]

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

[Out]

PolyLog[2, 1 - e*x^n]/(e*n)

Rule 2352

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

Rule 2374

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-1+n} \log \left (e x^n\right )}{1-e x^n} \, dx=\frac {\operatorname {PolyLog}\left (2,1-e x^n\right )}{e n} \]

[In]

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

[Out]

PolyLog[2, 1 - e*x^n]/(e*n)

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {\operatorname {dilog}\left (e \,x^{n}\right )}{e n}\)	\(14\)
risch	\(-\frac {\ln \left (1-e \,x^{n}\right ) \ln \left (x^{n}\right )}{n e}+\frac {\ln \left (1-e \,x^{n}\right ) \ln \left (e \,x^{n}\right )}{n e}+\frac {\operatorname {dilog}\left (e \,x^{n}\right )}{e n}+\frac {\left (-\frac {i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e \,x^{n}\right )^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e \,x^{n}\right ) \operatorname {csgn}\left (i x^{n}\right )}{2}+\frac {i \pi \operatorname {csgn}\left (i e \,x^{n}\right )^{3}}{2}-\frac {i \pi \operatorname {csgn}\left (i e \,x^{n}\right )^{2} \operatorname {csgn}\left (i x^{n}\right )}{2}-\ln \left (e \right )\right ) \ln \left (e \,x^{n}-1\right )}{n e}\)	\(156\)
meijerg	\(\frac {i \left (-1\right )^{\frac {\operatorname {csgn}\left (i e \right )}{2}-\frac {\operatorname {csgn}\left (i x^{n}\right )}{2}-\frac {\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i e \right )}{2}-\frac {1}{n}-\frac {n -1}{n}} \ln \left (e \right ) \ln \left (1+i x^{n} e \left (-1\right )^{-\frac {\operatorname {csgn}\left (i e \right )}{2}+\frac {\operatorname {csgn}\left (i x^{n}\right )}{2}+\frac {\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i e \right )}{2}}\right )}{e n}-\frac {i \left (-1\right )^{\frac {\operatorname {csgn}\left (i e \right )}{2}-\frac {\operatorname {csgn}\left (i x^{n}\right )}{2}-\frac {\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i e \right )}{2}} \ln \left (x \right ) \ln \left (1+i x^{n} e \left (-1\right )^{-\frac {\operatorname {csgn}\left (i e \right )}{2}+\frac {\operatorname {csgn}\left (i x^{n}\right )}{2}+\frac {\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i e \right )}{2}}\right )}{e}-\frac {i \left (-1\right )^{\frac {\operatorname {csgn}\left (i e \right )}{2}-\frac {\operatorname {csgn}\left (i x^{n}\right )}{2}-\frac {\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i e \right )}{2}} \operatorname {Li}_{2}\left (-i x^{n} e \left (-1\right )^{-\frac {\operatorname {csgn}\left (i e \right )}{2}+\frac {\operatorname {csgn}\left (i x^{n}\right )}{2}+\frac {\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i e \right )}{2}}\right )}{n e}\)	\(270\)

[In]

int(x^(n-1)*ln(e*x^n)/(1-e*x^n),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (16) = 32\).

Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.29 \[ \int \frac {x^{-1+n} \log \left (e x^n\right )}{1-e x^n} \, dx=-\frac {n \log \left (-e x^{n} + 1\right ) \log \left (x\right ) + \log \left (e x^{n} - 1\right ) \log \left (e\right ) + {\rm Li}_2\left (e x^{n}\right )}{e n} \]

[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)

Sympy [F(-2)]

Exception generated. \[ \int \frac {x^{-1+n} \log \left (e x^n\right )}{1-e x^n} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (16) = 32\).

Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 3.06 \[ \int \frac {x^{-1+n} \log \left (e x^n\right )}{1-e x^n} \, dx=-\frac {\log \left (e\right ) \log \left (\frac {e x^{n} - 1}{e}\right )}{e n} - \frac {\log \left (-e x^{n} + 1\right ) \log \left (x^{n}\right ) + {\rm Li}_2\left (e x^{n}\right )}{e n} \]

[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)

Giac [F]

\[ \int \frac {x^{-1+n} \log \left (e x^n\right )}{1-e x^n} \, dx=\int { -\frac {x^{n - 1} \log \left (e x^{n}\right )}{e x^{n} - 1} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {x^{-1+n} \log \left (e x^n\right )}{1-e x^n} \, dx=\frac {{\mathrm {Li}}_{\mathrm {2}}\left (e\,x^n\right )}{e\,n} \]

[In]

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

[Out]

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