3.392 \(\int \frac{\log (x^{-n} (a+x^n))}{x} \, dx\)

Optimal. Leaf size=14 \[ \frac{\text{PolyLog}\left (2,-a x^{-n}\right )}{n} \]

[Out]

PolyLog[2, -(a/x^n)]/n

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Rubi [A]  time = 0.0188247, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2461, 2391} \[ \frac{\text{PolyLog}\left (2,-a x^{-n}\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[Log[(a + x^n)/x^n]/x,x]

[Out]

PolyLog[2, -(a/x^n)]/n

Rule 2461

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.)*((f_.)*(x_))^(m_.), x_Symbol] :> Int[(f*x)^m*(a + b*Log[c*Expa
ndToSum[v, x]^p])^q, x] /; FreeQ[{a, b, c, f, m, p, q}, x] && BinomialQ[v, x] &&  !BinomialMatchQ[v, x]

Rule 2391

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

Rubi steps

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

Mathematica [A]  time = 0.0041753, size = 14, normalized size = 1. \[ \frac{\text{PolyLog}\left (2,-a x^{-n}\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(a + x^n)/x^n]/x,x]

[Out]

PolyLog[2, -(a/x^n)]/n

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Maple [A]  time = 0.086, size = 15, normalized size = 1.1 \begin{align*}{\frac{1}{n}{\it dilog} \left ( 1+{\frac{a}{{x}^{n}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln((a+x^n)/(x^n))/x,x)

[Out]

1/n*dilog(1+a/(x^n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((a+x^n)/(x^n))/x,x, algorithm="maxima")

[Out]

a*n*integrate(log(x)/(a*x + x*x^n), x) + log(a + x^n)*log(x) - log(x)*log(x^n)

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Fricas [B]  time = 1.64604, size = 151, normalized size = 10.79 \begin{align*} \frac{n^{2} \log \left (x\right )^{2} - 2 \, n \log \left (x\right ) \log \left (\frac{a + x^{n}}{a}\right ) + 2 \, n \log \left (x\right ) \log \left (\frac{a + x^{n}}{x^{n}}\right ) - 2 \,{\rm Li}_2\left (-\frac{a + x^{n}}{a} + 1\right )}{2 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((a+x^n)/(x^n))/x,x, algorithm="fricas")

[Out]

1/2*(n^2*log(x)^2 - 2*n*log(x)*log((a + x^n)/a) + 2*n*log(x)*log((a + x^n)/x^n) - 2*dilog(-(a + x^n)/a + 1))/n

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln((a+x**n)/(x**n))/x,x)

[Out]

Integral(log(a*x**(-n) + 1)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((a+x^n)/(x^n))/x,x, algorithm="giac")

[Out]

integrate(log((a + x^n)/x^n)/x, x)