3.4.92 \(\int \frac {\log (x^{-n} (a+x^n))}{x} \, dx\) [392]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, 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 = {2511, 2438} \begin {gather*} \frac {\text {PolyLog}\left (2,-a x^{-n}\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[(a + x^n)/x^n]/x,x]

[Out]

PolyLog[2, -(a/x^n)]/n

Rule 2438

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 2511

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*}

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Integrate[Log[(a + x^n)/x^n]/x,x]

[Out]

PolyLog[2, -(a/x^n)]/n

________________________________________________________________________________________

Maple [A]
time = 1.83, size = 15, normalized size = 1.07

method	result	size
derivativedivides	\(\frac {\dilog \left (1+a \,x^{-n}\right )}{n}\)	\(15\)
default	\(\frac {\dilog \left (1+a \,x^{-n}\right )}{n}\)	\(15\)
risch	\(-\ln \left (x \right ) \ln \left (x^{n}\right )+\frac {n \ln \left (x \right )^{2}}{2}+\frac {i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x^{-n}\right ) \mathrm {csgn}\left (i x^{-n} \left (a +x^{n}\right )\right )^{2}}{2}+\frac {i \pi \ln \left (x \right ) \mathrm {csgn}\left (i \left (a +x^{n}\right )\right ) \mathrm {csgn}\left (i x^{-n} \left (a +x^{n}\right )\right )^{2}}{2}-\frac {i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x^{-n} \left (a +x^{n}\right )\right )^{3}}{2}-\frac {i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x^{-n}\right ) \mathrm {csgn}\left (i \left (a +x^{n}\right )\right ) \mathrm {csgn}\left (i x^{-n} \left (a +x^{n}\right )\right )}{2}+\frac {\ln \left (a +x^{n}\right ) \ln \left (-\frac {x^{n}}{a}\right )}{n}+\frac {\dilog \left (-\frac {x^{n}}{a}\right )}{n}\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln((a+x^n)/(x^n))/x,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.42, size = 19, normalized size = 1.36 \begin {gather*} \frac {{\rm Li}_2\left (-\frac {a + x^{n}}{x^{n}} + 1\right )}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

dilog(-(a + x^n)/x^n + 1)/n

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

[Out]

integrate(log((a + x^n)/x^n)/x, x)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________