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

Optimal. Leaf size=25 \[ \log \left (\frac {e x}{2}\right ) \log (-2+e x)+\text {Li}_2\left (1-\frac {e x}{2}\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2441, 2352} \begin {gather*} \text {PolyLog}\left (2,1-\frac {e x}{2}\right )+\log \left (\frac {e x}{2}\right ) \log (e x-2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[-2 + e*x]/x,x]

[Out]

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

Rule 2352

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

Rule 2441

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] - Dist[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]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.08 \begin {gather*} \log \left (\frac {e x}{2}\right ) \log (-2+e x)+\text {Li}_2\left (\frac {1}{2} (2-e x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[-2 + e*x]/x,x]

[Out]

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

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Maple [A]
time = 0.10, size = 19, normalized size = 0.76

method	result	size
derivativedivides	\(\dilog \left (\frac {e x}{2}\right )+\ln \left (\frac {e x}{2}\right ) \ln \left (e x -2\right )\)	\(19\)
default	\(\dilog \left (\frac {e x}{2}\right )+\ln \left (\frac {e x}{2}\right ) \ln \left (e x -2\right )\)	\(19\)
risch	\(\dilog \left (\frac {e x}{2}\right )+\ln \left (\frac {e x}{2}\right ) \ln \left (e x -2\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 23, normalized size = 0.92 \begin {gather*} \log \left (x e - 2\right ) \log \left (\frac {1}{2} \, x e\right ) + {\rm Li}_2\left (-\frac {1}{2} \, x e + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log(x*e - 2)/x, x)

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Sympy [C] Result contains complex when optimal does not.
time = 1.99, size = 102, normalized size = 4.08 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(x*e - 2)/x, x)

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Mupad [B]
time = 0.03, size = 18, normalized size = 0.72 \begin {gather*} {\mathrm {Li}}_{\mathrm {2}}\left (\frac {e\,x}{2}\right )+\ln \left (e\,x-2\right )\,\ln \left (\frac {e\,x}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(e*x - 2)/x,x)

[Out]

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

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