3.390 \(\int \frac{\log (\frac{a+x}{x})}{x} \, dx\)

Optimal. Leaf size=8 \[ \text{PolyLog}\left (2,-\frac{a}{x}\right ) \]

[Out]

PolyLog[2, -(a/x)]

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Rubi [A]  time = 0.0095414, antiderivative size = 12, normalized size of antiderivative = 1.5, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2447} \[ \text{PolyLog}\left (2,1-\frac{a+x}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[(a + x)/x]/x,x]

[Out]

PolyLog[2, 1 - (a + x)/x]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

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

Mathematica [B]  time = 0.0028248, size = 34, normalized size = 4.25 \[ -\text{PolyLog}\left (2,-\frac{-a-x}{x}\right )-\log \left (-\frac{a}{x}\right ) \log \left (\frac{a+x}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[(a + x)/x]/x,x]

[Out]

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

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Maple [A]  time = 0.081, size = 9, normalized size = 1.1 \begin{align*}{\it dilog} \left ( 1+{\frac{a}{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

dilog(1+a/x)

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Maxima [B]  time = 1.03128, size = 80, normalized size = 10. \begin{align*} -{\left (\log \left (a + x\right ) - \log \left (x\right )\right )} \log \left (x\right ) + \log \left (a + x\right ) \log \left (x\right ) - \frac{1}{2} \, \log \left (x\right )^{2} + \log \left (x\right ) \log \left (\frac{a + x}{x}\right ) - \log \left (x\right ) \log \left (\frac{x}{a} + 1\right ) -{\rm Li}_2\left (-\frac{x}{a}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.43906, size = 31, normalized size = 3.88 \begin{align*}{\rm Li}_2\left (-\frac{a + x}{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

dilog(-(a + x)/x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(log(a/x + 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}{x}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((a + x)/x)/x, x)