\(\int \frac {\log (2+e x)}{x} \, dx\) [74]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [B] (verified)
   Fricas [F]
   Sympy [C] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [F]
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 10, antiderivative size = 16 \[ \int \frac {\log (2+e x)}{x} \, dx=\log (2) \log (x)-\operatorname {PolyLog}\left (2,-\frac {e x}{2}\right ) \]

[Out]

ln(2)*ln(x)-polylog(2,-1/2*e*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2439, 2438} \[ \int \frac {\log (2+e x)}{x} \, dx=\log (2) \log (x)-\operatorname {PolyLog}\left (2,-\frac {e x}{2}\right ) \]

[In]

Int[Log[2 + e*x]/x,x]

[Out]

Log[2]*Log[x] - PolyLog[2, -1/2*(e*x)]

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 2439

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*d])*Log[x], x] + Dist[
b, Int[Log[1 + e*(x/d)]/x, x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 0]

Rubi steps \begin{align*} \text {integral}& = \log (2) \log (x)+\int \frac {\log \left (1+\frac {e x}{2}\right )}{x} \, dx \\ & = \log (2) \log (x)-\text {Li}_2\left (-\frac {e x}{2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\log (2+e x)}{x} \, dx=\log (2) \log (x)-\operatorname {PolyLog}\left (2,-\frac {e x}{2}\right ) \]

[In]

Integrate[Log[2 + e*x]/x,x]

[Out]

Log[2]*Log[x] - PolyLog[2, -1/2*(e*x)]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(32\) vs. \(2(14)=28\).

Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06

method result size
derivativedivides \(\left (\ln \left (e x +2\right )-\ln \left (\frac {e x}{2}+1\right )\right ) \ln \left (-\frac {e x}{2}\right )-\operatorname {dilog}\left (\frac {e x}{2}+1\right )\) \(33\)
default \(\left (\ln \left (e x +2\right )-\ln \left (\frac {e x}{2}+1\right )\right ) \ln \left (-\frac {e x}{2}\right )-\operatorname {dilog}\left (\frac {e x}{2}+1\right )\) \(33\)
risch \(\left (\ln \left (e x +2\right )-\ln \left (\frac {e x}{2}+1\right )\right ) \ln \left (-\frac {e x}{2}\right )-\operatorname {dilog}\left (\frac {e x}{2}+1\right )\) \(33\)
parts \(\ln \left (e x +2\right ) \ln \left (x \right )-e \left (\frac {\operatorname {dilog}\left (\frac {e x}{2}+1\right )}{e}+\frac {\ln \left (x \right ) \ln \left (\frac {e x}{2}+1\right )}{e}\right )\) \(39\)

[In]

int(ln(e*x+2)/x,x,method=_RETURNVERBOSE)

[Out]

(ln(e*x+2)-ln(1/2*e*x+1))*ln(-1/2*e*x)-dilog(1/2*e*x+1)

Fricas [F]

\[ \int \frac {\log (2+e x)}{x} \, dx=\int { \frac {\log \left (e x + 2\right )}{x} \,d x } \]

[In]

integrate(log(e*x+2)/x,x, algorithm="fricas")

[Out]

integral(log(e*x + 2)/x, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.47 (sec) , antiderivative size = 87, normalized size of antiderivative = 5.44 \[ \int \frac {\log (2+e x)}{x} \, dx=\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{2}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (2 \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{2}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (2 \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{2}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (2 \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (2 \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{2}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate(ln(e*x+2)/x,x)

[Out]

Piecewise((-polylog(2, e*x*exp_polar(I*pi)/2), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(2)*log(x) - polylog(2, e*x
*exp_polar(I*pi)/2), Abs(x) < 1), (-log(2)*log(1/x) - polylog(2, e*x*exp_polar(I*pi)/2), 1/Abs(x) < 1), (-meij
erg(((), (1, 1)), ((0, 0), ()), x)*log(2) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(2) - polylog(2, e*x*exp
_polar(I*pi)/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {\log (2+e x)}{x} \, dx=\log \left (e x + 2\right ) \log \left (-\frac {1}{2} \, e x\right ) + {\rm Li}_2\left (\frac {1}{2} \, e x + 1\right ) \]

[In]

integrate(log(e*x+2)/x,x, algorithm="maxima")

[Out]

log(e*x + 2)*log(-1/2*e*x) + dilog(1/2*e*x + 1)

Giac [F]

\[ \int \frac {\log (2+e x)}{x} \, dx=\int { \frac {\log \left (e x + 2\right )}{x} \,d x } \]

[In]

integrate(log(e*x+2)/x,x, algorithm="giac")

[Out]

integrate(log(e*x + 2)/x, x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\log (2+e x)}{x} \, dx={\mathrm {Li}}_{\mathrm {2}}\left (-\frac {e\,x}{2}\right )+\ln \left (e\,x+2\right )\,\ln \left (-\frac {e\,x}{2}\right ) \]

[In]

int(log(e*x + 2)/x,x)

[Out]

dilog(-(e*x)/2) + log(e*x + 2)*log(-(e*x)/2)