3.1.83 \(\int \frac {a+b \log (-1+e x)}{x} \, dx\) [83]

Optimal. Leaf size=26 \[ \log (e x) (a+b \log (-1+e x))+b \text {Li}_2(1-e x) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2441, 2352} \begin {gather*} b \text {PolyLog}(2,1-e x)+\log (e x) (a+b \log (e x-1)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[-1 + e*x])/x,x]

[Out]

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

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 {a+b \log (-1+e x)}{x} \, dx &=\log (e x) (a+b \log (-1+e x))-(b e) \int \frac {\log (e x)}{-1+e x} \, dx\\ &=\log (e x) (a+b \log (-1+e x))+b \text {Li}_2(1-e x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.04 \begin {gather*} a \log (x)+b \log (e x) \log (-1+e x)+b \text {Li}_2(1-e x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[-1 + e*x])/x,x]

[Out]

a*Log[x] + b*Log[e*x]*Log[-1 + e*x] + b*PolyLog[2, 1 - e*x]

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Maple [A]
time = 0.16, size = 26, normalized size = 1.00

method	result	size
risch	\(\ln \left (x \right ) a +\ln \left (e x \right ) \ln \left (e x -1\right ) b +\dilog \left (e x \right ) b\)	\(24\)
derivativedivides	\(a \ln \left (e x \right )+\ln \left (e x \right ) \ln \left (e x -1\right ) b +\dilog \left (e x \right ) b\)	\(26\)
default	\(a \ln \left (e x \right )+\ln \left (e x \right ) \ln \left (e x -1\right ) b +\dilog \left (e x \right ) b\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(e*x-1))/x,x,method=_RETURNVERBOSE)

[Out]

a*ln(e*x)+ln(e*x)*ln(e*x-1)*b+dilog(e*x)*b

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Maxima [A]
time = 0.33, size = 29, normalized size = 1.12 \begin {gather*} {\left (\log \left (x e - 1\right ) \log \left (x e\right ) + {\rm Li}_2\left (-x e + 1\right )\right )} b + a \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(e*x-1))/x,x, algorithm="maxima")

[Out]

(log(x*e - 1)*log(x*e) + dilog(-x*e + 1))*b + a*log(x)

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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((a+b*log(e*x-1))/x,x, algorithm="fricas")

[Out]

integral((b*log(x*e - 1) + a)/x, x)

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Sympy [A]
time = 2.35, size = 66, normalized size = 2.54 \begin {gather*} a \log {\left (x \right )} + b \left (\begin {cases} - \operatorname {Li}_{2}\left (e x\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\i \pi \log {\left (x \right )} - \operatorname {Li}_{2}\left (e x\right ) & \text {for}\: \left |{x}\right | < 1 \\- i \pi \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (e x\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- i \pi {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} + i \pi {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} - \operatorname {Li}_{2}\left (e x\right ) & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(e*x-1))/x,x)

[Out]

a*log(x) + b*Piecewise((-polylog(2, e*x), (Abs(x) < 1) & (1/Abs(x) < 1)), (I*pi*log(x) - polylog(2, e*x), Abs(
x) < 1), (-I*pi*log(1/x) - polylog(2, e*x), 1/Abs(x) < 1), (-I*pi*meijerg(((), (1, 1)), ((0, 0), ()), x) + I*p
i*meijerg(((1, 1), ()), ((), (0, 0)), x) - polylog(2, e*x), 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((a+b*log(e*x-1))/x,x, algorithm="giac")

[Out]

integrate((b*log(x*e - 1) + a)/x, x)

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Mupad [B]
time = 0.16, size = 23, normalized size = 0.88 \begin {gather*} b\,{\mathrm {Li}}_{\mathrm {2}}\left (e\,x\right )+a\,\ln \left (x\right )+b\,\ln \left (e\,x-1\right )\,\ln \left (e\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(e*x - 1))/x,x)

[Out]

b*dilog(e*x) + a*log(x) + b*log(e*x - 1)*log(e*x)

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