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

Optimal. Leaf size=20 \[ -\frac{\text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{e n} \]

[Out]

-(PolyLog[2, 1 + (e*x^n)/d]/(e*n))

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Rubi [A]  time = 0.0685046, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2336, 2315} \[ -\frac{\text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{e n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0101116, size = 21, normalized size = 1.05 \[ -\frac{\text{PolyLog}\left (2,\frac{d+e x^n}{d}\right )}{e n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(PolyLog[2, (d + e*x^n)/d]/(e*n))

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Maple [A]  time = 0.042, size = 19, normalized size = 1. \begin{align*} -{\frac{1}{en}{\it dilog} \left ( -{\frac{e{x}^{n}}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.822, size = 86, normalized size = 4.3 \begin{align*} -\frac{{\left (\log \left (d\right ) - \log \left (e\right )\right )} \log \left (\frac{e x^{n} + d}{e}\right )}{e n} + \frac{\log \left (\frac{e x^{n}}{d} + 1\right ) \log \left (-x^{n}\right ) +{\rm Li}_2\left (-\frac{e x^{n}}{d}\right )}{e n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.26075, size = 124, normalized size = 6.2 \begin{align*} \frac{n \log \left (x\right ) \log \left (\frac{e x^{n} + d}{d}\right ) + \log \left (e x^{n} + d\right ) \log \left (-\frac{e}{d}\right ) +{\rm Li}_2\left (-\frac{e x^{n} + d}{d} + 1\right )}{e n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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