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

Optimal result

Integrand size = 16, antiderivative size = 14 \[ \int \frac {\log \left (x^{-n} \left (a+x^n\right )\right )}{x} \, dx=\frac {\operatorname {PolyLog}\left (2,-a x^{-n}\right )}{n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2511, 2438} \[ \int \frac {\log \left (x^{-n} \left (a+x^n\right )\right )}{x} \, dx=\frac {\operatorname {PolyLog}\left (2,-a x^{-n}\right )}{n} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (x^{-n} \left (a+x^n\right )\right )}{x} \, dx=\frac {\operatorname {PolyLog}\left (2,-a x^{-n}\right )}{n} \]

[In]

Integrate[Log[(a + x^n)/x^n]/x,x]

[Out]

PolyLog[2, -(a/x^n)]/n

Maple [A] (verified)

Time = 2.16 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (13) = 26\).

Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 4.36 \[ \int \frac {\log \left (x^{-n} \left (a+x^n\right )\right )}{x} \, dx=\frac {n^{2} \log \left (x\right )^{2} - 2 \, n \log \left (x\right ) \log \left (\frac {a + x^{n}}{a}\right ) + 2 \, n \log \left (x\right ) \log \left (\frac {a + x^{n}}{x^{n}}\right ) - 2 \, {\rm Li}_2\left (-\frac {a + x^{n}}{a} + 1\right )}{2 \, n} \]

[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

Sympy [F]

\[ \int \frac {\log \left (x^{-n} \left (a+x^n\right )\right )}{x} \, dx=\int \frac {\log {\left (a x^{- n} + 1 \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(log(a/x**n + 1)/x, x)

Maxima [F]

\[ \int \frac {\log \left (x^{-n} \left (a+x^n\right )\right )}{x} \, dx=\int { \frac {\log \left (\frac {a + x^{n}}{x^{n}}\right )}{x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\log \left (x^{-n} \left (a+x^n\right )\right )}{x} \, dx=\int { \frac {\log \left (\frac {a + x^{n}}{x^{n}}\right )}{x} \,d x } \]

[In]

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

[Out]

integrate(log((a + x^n)/x^n)/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (x^{-n} \left (a+x^n\right )\right )}{x} \, dx=\int \frac {\ln \left (\frac {a+x^n}{x^n}\right )}{x} \,d x \]

[In]

int(log((a + x^n)/x^n)/x,x)

[Out]

int(log((a + x^n)/x^n)/x, x)