\(\int \frac {\log (\frac {a+x}{x})}{x} \, dx\) [390]

Optimal result

Integrand size = 12, antiderivative size = 8 \[ \int \frac {\log \left (\frac {a+x}{x}\right )}{x} \, dx=\operatorname {PolyLog}\left (2,-\frac {a}{x}\right ) \]

[Out]

polylog(2,-a/x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.50, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2497} \[ \int \frac {\log \left (\frac {a+x}{x}\right )}{x} \, dx=\operatorname {PolyLog}\left (2,1-\frac {a+x}{x}\right ) \]

[In]

Int[Log[(a + x)/x]/x,x]

[Out]

PolyLog[2, 1 - (a + x)/x]

Rule 2497

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*} \text {integral}& = \text {Li}_2\left (1-\frac {a+x}{x}\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(34\) vs. \(2(8)=16\).

Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 4.25 \[ \int \frac {\log \left (\frac {a+x}{x}\right )}{x} \, dx=-\log \left (-\frac {a}{x}\right ) \log \left (\frac {a+x}{x}\right )-\operatorname {PolyLog}\left (2,-\frac {-a-x}{x}\right ) \]

[In]

Integrate[Log[(a + x)/x]/x,x]

[Out]

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

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.12

method	result	size
derivativedivides	\(\operatorname {dilog}\left (1+\frac {a}{x}\right )\)	\(9\)
default	\(\operatorname {dilog}\left (1+\frac {a}{x}\right )\)	\(9\)
risch	\(\operatorname {dilog}\left (1+\frac {a}{x}\right )\)	\(9\)
parts	\(\ln \left (\frac {a +x}{x}\right ) \ln \left (x \right )+\frac {\ln \left (x \right )^{2}}{2}-\operatorname {dilog}\left (\frac {a +x}{a}\right )-\ln \left (x \right ) \ln \left (\frac {a +x}{a}\right )\)	\(41\)

[In]

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

[Out]

dilog(1+a/x)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.38 \[ \int \frac {\log \left (\frac {a+x}{x}\right )}{x} \, dx={\rm Li}_2\left (-\frac {a + x}{x} + 1\right ) \]

[In]

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

[Out]

dilog(-(a + x)/x + 1)

Sympy [F]

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

[In]

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

[Out]

Integral(log(a/x + 1)/x, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (7) = 14\).

Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 7.38 \[ \int \frac {\log \left (\frac {a+x}{x}\right )}{x} \, dx=-{\left (\log \left (a + x\right ) - \log \left (x\right )\right )} \log \left (x\right ) + \log \left (a + x\right ) \log \left (x\right ) - \frac {1}{2} \, \log \left (x\right )^{2} + \log \left (x\right ) \log \left (\frac {a + x}{x}\right ) - \log \left (x\right ) \log \left (\frac {x}{a} + 1\right ) - {\rm Li}_2\left (-\frac {x}{a}\right ) \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (7) = 14\).

Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 8.50 \[ \int \frac {\log \left (\frac {a+x}{x}\right )}{x} \, dx=-\frac {a^{3} {\left (\frac {1}{\frac {a + x}{x} - 1} - \log \left (\frac {{\left | a + x \right |}}{{\left | x \right |}}\right ) + \log \left ({\left | \frac {a + x}{x} - 1 \right |}\right )\right )} + \frac {a^{3} \log \left (\frac {a + x}{x}\right )}{{\left (\frac {a + x}{x} - 1\right )}^{2}}}{2 \, a^{2}} \]

[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

Mupad [B] (verification not implemented)

Time = 1.45 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (\frac {a+x}{x}\right )}{x} \, dx=\mathrm {polylog}\left (2,-\frac {a}{x}\right ) \]

[In]

int(log((a + x)/x)/x,x)

[Out]

polylog(2, -a/x)