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

Optimal. Leaf size=17 \[ \frac {\text {Li}_2\left (1-e x^n\right )}{e n} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {2374, 2352} \begin {gather*} \frac {\text {PolyLog}\left (2,1-e x^n\right )}{e n} \end {gather*}

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

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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {\text {Li}_2\left (1-e x^n\right )}{e n} \end {gather*}

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 = 0.21, size = 14, normalized size = 0.82

method	result	size
default	\(\frac {\dilog \left (e \,x^{n}\right )}{e n}\)	\(14\)
risch	\(-\frac {\ln \left (1-e \,x^{n}\right ) \ln \left (x^{n}\right )}{n e}+\frac {\ln \left (1-e \,x^{n}\right ) \ln \left (e \,x^{n}\right )}{n e}+\frac {\dilog \left (e \,x^{n}\right )}{e n}-\frac {i \ln \left (e \,x^{n}-1\right ) \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e \,x^{n}\right )^{2}}{2 n e}+\frac {i \ln \left (e \,x^{n}-1\right ) \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e \,x^{n}\right ) \mathrm {csgn}\left (i x^{n}\right )}{2 n e}+\frac {i \ln \left (e \,x^{n}-1\right ) \pi \mathrm {csgn}\left (i e \,x^{n}\right )^{3}}{2 n e}-\frac {i \ln \left (e \,x^{n}-1\right ) \pi \mathrm {csgn}\left (i e \,x^{n}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )}{2 n e}-\frac {\ln \left (e \,x^{n}-1\right ) \ln \left (e \right )}{n e}\)	\(210\)
meijerg	\(\frac {i \left (-1\right )^{\frac {\mathrm {csgn}\left (i e \right )}{2}-\frac {\mathrm {csgn}\left (i x^{n}\right )}{2}-\frac {\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i e \right )}{2}-\frac {-1+n}{n}-\frac {1}{n}} \ln \left (e \right ) \ln \left (1+i x^{n} e \left (-1\right )^{-\frac {\mathrm {csgn}\left (i e \right )}{2}+\frac {\mathrm {csgn}\left (i x^{n}\right )}{2}+\frac {\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i e \right )}{2}}\right )}{e n}-\frac {i \left (-1\right )^{\frac {\mathrm {csgn}\left (i e \right )}{2}-\frac {\mathrm {csgn}\left (i x^{n}\right )}{2}-\frac {\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i e \right )}{2}} \ln \left (x \right ) \ln \left (1+i x^{n} e \left (-1\right )^{-\frac {\mathrm {csgn}\left (i e \right )}{2}+\frac {\mathrm {csgn}\left (i x^{n}\right )}{2}+\frac {\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i e \right )}{2}}\right )}{e}-\frac {i \left (-1\right )^{\frac {\mathrm {csgn}\left (i e \right )}{2}-\frac {\mathrm {csgn}\left (i x^{n}\right )}{2}-\frac {\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i e \right )}{2}} \polylog \left (2, -i x^{n} e \left (-1\right )^{-\frac {\mathrm {csgn}\left (i e \right )}{2}+\frac {\mathrm {csgn}\left (i x^{n}\right )}{2}+\frac {\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i e \right )}{2}}\right )}{n e}\)	\(270\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (16) = 32\).
time = 0.44, size = 54, normalized size = 3.18 \begin {gather*} -\frac {{\left (\log \left (x^{n}\right ) \log \left (-e^{\left (n \log \left (x\right ) + 1\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (n \log \left (x\right ) + 1\right )}\right )\right )} e^{\left (-1\right )}}{n} - \frac {e^{\left (-1\right )} \log \left ({\left (e^{\left (n \log \left (x\right ) + 1\right )} - 1\right )} e^{\left (-1\right )}\right )}{n} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (16) = 32\).
time = 0.35, size = 38, normalized size = 2.24 \begin {gather*} -\frac {{\left (n \log \left (-x^{n} e + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (x^{n} e\right ) + \log \left (x^{n} e - 1\right )\right )} e^{\left (-1\right )}}{n} \end {gather*}

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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 >> Invalid comparison of non-real zoo

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(x^n*e)/(x^n*e - 1), x)

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Mupad [B]
time = 3.53, size = 13, normalized size = 0.76 \begin {gather*} \frac {{\mathrm {Li}}_{\mathrm {2}}\left (e\,x^n\right )}{e\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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