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

Optimal. Leaf size=8 \[ \text {Li}_2\left (-\frac {a}{x}\right ) \]

[Out]

polylog(2,-a/x)

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.50, 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*}

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Mathematica [B]  time = 0.00, size = 34, normalized size = 4.25 \[ -\text {Li}_2\left (-\frac {-a-x}{x}\right )-\log \left (-\frac {a}{x}\right ) \log \left (\frac {a+x}{x}\right ) \]

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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fricas [A]  time = 0.42, size = 11, normalized size = 1.38 \[ {\rm Li}_2\left (-\frac {a + x}{x} + 1\right ) \]

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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giac [B]  time = 0.27, size = 68, normalized size = 8.50 \[ -\frac {a^{3} {\left (\frac {1}{\frac {a + x}{x} - 1} - \log \left (\frac {{\left | a + x \right |}}{{\left | x \right |}}\right ) + \log \left ({\left | \frac {a + x}{x} - 1 \right |}\right )\right )} + \frac {a^{3} \log \left (\frac {a + x}{x}\right )}{{\left (\frac {a + x}{x} - 1\right )}^{2}}}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^3*(1/((a + x)/x - 1) - log(abs(a + x)/abs(x)) + log(abs((a + x)/x - 1))) + a^3*log((a + x)/x)/((a + x)
/x - 1)^2)/a^2

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maple [A]  time = 0.07, size = 9, normalized size = 1.12 \[ \dilog \left (\frac {a}{x}+1\right ) \]

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 = 0.46, size = 59, normalized size = 7.38 \[ -{\left (\log \left (a + x\right ) - \log \relax (x)\right )} \log \relax (x) + \log \left (a + x\right ) \log \relax (x) - \frac {1}{2} \, \log \relax (x)^{2} + \log \relax (x) \log \left (\frac {a + x}{x}\right ) - \log \relax (x) \log \left (\frac {x}{a} + 1\right ) - {\rm Li}_2\left (-\frac {x}{a}\right ) \]

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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mupad [B]  time = 0.31, size = 8, normalized size = 1.00 \[ \mathrm {polylog}\left (2,-\frac {a}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log((a + x)/x)/x,x)

[Out]

polylog(2, -a/x)

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

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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