∫ log(−2+ex)_______

Integrand size = 10, antiderivative size = 25 \[ \int \frac {\log (-2+e x)}{x} \, dx=\log \left (\frac {e x}{2}\right ) \log (-2+e x)+\operatorname {PolyLog}\left (2,1-\frac {e x}{2}\right ) \]

3.1.78.2 Mathematica [A] (veriﬁed)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\log (-2+e x)}{x} \, dx=\log \left (\frac {e x}{2}\right ) \log (-2+e x)+\operatorname {PolyLog}\left (2,\frac {1}{2} (2-e x)\right ) \]

3.1.78.3 Rubi [A] (veriﬁed)

Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2841, 25, 2752}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules deﬁnitions used are listed below.

\(\displaystyle \int \frac {\log (e x-2)}{x} \, dx\)

\(\Big \downarrow \) 2841

\(\displaystyle \log \left (\frac {e x}{2}\right ) \log (e x-2)-e \int -\frac {\log \left (\frac {e x}{2}\right )}{2-e x}dx\)

\(\Big \downarrow \) 25

\(\displaystyle e \int \frac {\log \left (\frac {e x}{2}\right )}{2-e x}dx+\log \left (\frac {e x}{2}\right ) \log (e x-2)\)

\(\Big \downarrow \) 2752

\(\displaystyle \operatorname {PolyLog}\left (2,1-\frac {e x}{2}\right )+\log \left (\frac {e x}{2}\right ) \log (e x-2)\)

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]

rule 2841

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] - Simp[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]

3.1.78.4 Maple [A] (veriﬁed)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76


method	result	size

derivativedivides	\(\operatorname {dilog}\left (\frac {e x}{2}\right )+\ln \left (\frac {e x}{2}\right ) \ln \left (e x -2\right )\)	\(19\)
default	\(\operatorname {dilog}\left (\frac {e x}{2}\right )+\ln \left (\frac {e x}{2}\right ) \ln \left (e x -2\right )\)	\(19\)
risch	\(\operatorname {dilog}\left (\frac {e x}{2}\right )+\ln \left (\frac {e x}{2}\right ) \ln \left (e x -2\right )\)	\(19\)
parts	\(\ln \left (e x -2\right ) \ln \left (x \right )-e \left (\frac {\left (\ln \left (x \right )-\ln \left (\frac {e x}{2}\right )\right ) \ln \left (1-\frac {e x}{2}\right )}{e}-\frac {\operatorname {dilog}\left (\frac {e x}{2}\right )}{e}\right )\)	\(46\)

3.1.78.5 Fricas [F]

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

3.1.78.6 Sympy [C] (veriﬁcation not implemented)

Time = 1.97 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.08 \[ \int \frac {\log (-2+e x)}{x} \, dx=\begin {cases} - \operatorname {Li}_{2}\left (\frac {e x}{2}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (2 \right )} \log {\left (x \right )} + 3 i \pi \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x}{2}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (2 \right )} \log {\left (\frac {1}{x} \right )} - 3 i \pi \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x}{2}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (2 \right )} - 3 i \pi {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (2 \right )} + 3 i \pi {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} - \operatorname {Li}_{2}\left (\frac {e x}{2}\right ) & \text {otherwise} \end {cases} \]

output

Piecewise((-polylog(2, e*x/2), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(2)*log 
(x) + 3*I*pi*log(x) - polylog(2, e*x/2), Abs(x) < 1), (-log(2)*log(1/x) - 
3*I*pi*log(1/x) - polylog(2, e*x/2), 1/Abs(x) < 1), (-meijerg(((), (1, 1)) 
, ((0, 0), ()), x)*log(2) - 3*I*pi*meijerg(((), (1, 1)), ((0, 0), ()), x) 
+ meijerg(((1, 1), ()), ((), (0, 0)), x)*log(2) + 3*I*pi*meijerg(((1, 1), 
()), ((), (0, 0)), x) - polylog(2, e*x/2), True))

3.1.78.7 Maxima [A] (veriﬁcation not implemented)

Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {\log (-2+e x)}{x} \, dx=\log \left (e x - 2\right ) \log \left (\frac {1}{2} \, e x\right ) + {\rm Li}_2\left (-\frac {1}{2} \, e x + 1\right ) \]

3.1.78.8 Giac [F]

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

3.1.78.9 Mupad [B] (veriﬁcation not implemented)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {\log (-2+e x)}{x} \, dx={\mathrm {Li}}_{\mathrm {2}}\left (\frac {e\,x}{2}\right )+\ln \left (e\,x-2\right )\,\ln \left (\frac {e\,x}{2}\right ) \]

3.1.78 \(\int \frac {\log (-2+e x)}{x} \, dx\) [78]

3.1.78.1 Optimal result