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

Optimal. Leaf size=47 \[ -\frac{\text{PolyLog}\left (2,\frac{a x^{-n}}{b}+1\right )}{n}-\frac{\log \left (-\frac{a x^{-n}}{b}\right ) \log \left (a x^{-n}+b\right )}{n} \]

[Out]

-((Log[-(a/(b*x^n))]*Log[b + a/x^n])/n) - PolyLog[2, 1 + a/(b*x^n)]/n

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Rubi [A]  time = 0.0503694, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2461, 2454, 2394, 2315} \[ -\frac{\text{PolyLog}\left (2,\frac{a x^{-n}}{b}+1\right )}{n}-\frac{\log \left (-\frac{a x^{-n}}{b}\right ) \log \left (a x^{-n}+b\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[Log[(a + b*x^n)/x^n]/x,x]

[Out]

-((Log[-(a/(b*x^n))]*Log[b + a/x^n])/n) - PolyLog[2, 1 + a/(b*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 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

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

Mathematica [A]  time = 0.0171986, size = 44, normalized size = 0.94 \[ -\frac{\text{PolyLog}\left (2,\frac{a x^{-n}+b}{b}\right )+\log \left (-\frac{a x^{-n}}{b}\right ) \log \left (a x^{-n}+b\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(a + b*x^n)/x^n]/x,x]

[Out]

-((Log[-(a/(b*x^n))]*Log[b + a/x^n] + PolyLog[2, (b + a/x^n)/b])/n)

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Maple [A]  time = 0.085, size = 46, normalized size = 1. \begin{align*} -{\frac{1}{n}\ln \left ( -{\frac{a}{b{x}^{n}}} \right ) \ln \left ( b+{\frac{a}{{x}^{n}}} \right ) }-{\frac{1}{n}{\it dilog} \left ( -{\frac{a}{b{x}^{n}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(-a/b/(x^n))*ln(b+a/(x^n))/n-1/n*dilog(-a/b/(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 )}{b x x^{n} + a x}\,{d x} + \log \left (b x^{n} + a\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+b*x^n)/(x^n))/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.69493, size = 159, normalized size = 3.38 \begin{align*} \frac{n^{2} \log \left (x\right )^{2} - 2 \, n \log \left (x\right ) \log \left (\frac{b x^{n} + a}{a}\right ) + 2 \, n \log \left (x\right ) \log \left (\frac{b x^{n} + a}{x^{n}}\right ) - 2 \,{\rm Li}_2\left (-\frac{b x^{n} + a}{a} + 1\right )}{2 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(n^2*log(x)^2 - 2*n*log(x)*log((b*x^n + a)/a) + 2*n*log(x)*log((b*x^n + a)/x^n) - 2*dilog(-(b*x^n + a)/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} + b \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(a*x**(-n) + b)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((b*x^n + a)/x^n)/x, x)