3.83 \(\int \frac{a+b \log (-1+e x)}{x} \, dx\)

Optimal. Leaf size=26 \[ b \text{PolyLog}(2,1-e x)+\log (e x) (a+b \log (e x-1)) \]

[Out]

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

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Rubi [A]  time = 0.0222327, antiderivative size = 26, normalized size of antiderivative = 1., 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 = {2394, 2315} \[ b \text{PolyLog}(2,1-e x)+\log (e x) (a+b \log (e x-1)) \]

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 2394

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]

Rule 2315

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

Mathematica [A]  time = 0.0022989, size = 27, normalized size = 1.04 \[ b \text{PolyLog}(2,1-e x)+a \log (x)+b \log (e x) \log (e x-1) \]

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 = 0.065, size = 26, normalized size = 1. \begin{align*} a\ln \left ( ex \right ) +\ln \left ( ex \right ) \ln \left ( ex-1 \right ) b+{\it dilog} \left ( ex \right ) b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.73951, size = 35, normalized size = 1.35 \begin{align*}{\left (\log \left (e x - 1\right ) \log \left (e x\right ) +{\rm Li}_2\left (-e x + 1\right )\right )} b + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (e x - 1\right ) + a}{x}, x\right ) \end{align*}

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(e*x - 1) + a)/x, x)

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Sympy [A]  time = 4.64911, size = 54, normalized size = 2.08 \begin{align*} a \log{\left (x \right )} + b \left (\begin{cases} 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{align*}

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((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*pi*meijerg(((1, 1), ()), ((), (0, 0)), x) - polylog(
2, e*x), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (e x - 1\right ) + a}{x}\,{d x} \end{align*}

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(e*x - 1) + a)/x, x)