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

Optimal. Leaf size=14 \[ \frac {\text {Li}_2\left (-a x^{-n}\right )}{n} \]

[Out]

polylog(2,-a/(x^n))/n

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Rubi [A]  time = 0.02, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ \frac {\text {Li}_2\left (-a x^{-n}\right )}{n} \]

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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fricas [B]  time = 0.43, size = 61, normalized size = 4.36 \[ \frac {n^{2} \log \relax (x)^{2} - 2 \, n \log \relax (x) \log \left (\frac {a + x^{n}}{a}\right ) + 2 \, n \log \relax (x) \log \left (\frac {a + x^{n}}{x^{n}}\right ) - 2 \, {\rm Li}_2\left (-\frac {a + x^{n}}{a} + 1\right )}{2 \, n} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {a + x^{n}}{x^{n}}\right )}{x}\,{d x} \]

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)

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maple [A]  time = 0.08, size = 15, normalized size = 1.07 \[ \frac {\dilog \left (a \,x^{-n}+1\right )}{n} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ a n \int \frac {\log \relax (x)}{a x + x x^{n}}\,{d x} + \log \left (a + x^{n}\right ) \log \relax (x) - \log \relax (x) \log \left (x^{n}\right ) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {\ln \left (\frac {a+x^n}{x^n}\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log((a + x^n)/x^n)/x,x)

[Out]

int(log((a + x^n)/x^n)/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (a x^{- n} + 1 \right )}}{x}\, dx \]

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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