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

Optimal. Leaf size=16 \[ \frac{\text{PolyLog}\left (2,1-\frac{x^n}{d}\right )}{n} \]

[Out]

PolyLog[2, 1 - x^n/d]/n

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Rubi [A]  time = 0.0631588, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2336, 2315} \[ \frac{\text{PolyLog}\left (2,1-\frac{x^n}{d}\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

PolyLog[2, 1 - x^n/d]/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 (\frac{x^n}{d}\right )}{d-x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log \left (\frac{x}{d}\right )}{d-x} \, dx,x,x^n\right )}{n}\\ &=\frac{\text{Li}_2\left (1-\frac{x^n}{d}\right )}{n}\\ \end{align*}

Mathematica [A]  time = 0.0092833, size = 17, normalized size = 1.06 \[ \frac{\text{PolyLog}\left (2,\frac{d-x^n}{d}\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

PolyLog[2, (d - x^n)/d]/n

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Maple [A]  time = 0.043, size = 13, normalized size = 0.8 \begin{align*}{\frac{1}{n}{\it dilog} \left ({\frac{{x}^{n}}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.7504, size = 61, normalized size = 3.81 \begin{align*} \frac{\log \left (d\right ) \log \left (-d + x^{n}\right )}{n} - \frac{\log \left (x^{n}\right ) \log \left (-\frac{x^{n}}{d} + 1\right ) +{\rm Li}_2\left (\frac{x^{n}}{d}\right )}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.28396, size = 112, normalized size = 7. \begin{align*} -\frac{n \log \left (x\right ) \log \left (\frac{d - x^{n}}{d}\right ) + \log \left (-d + x^{n}\right ) \log \left (\frac{1}{d}\right ) +{\rm Li}_2\left (-\frac{d - x^{n}}{d} + 1\right )}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(n*log(x)*log((d - x^n)/d) + log(-d + x^n)*log(1/d) + dilog(-(d - x^n)/d + 1))/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(x**n/d)/(d-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 (\frac{x^{n}}{d}\right )}{d - x^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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