3.75 \(\int \frac{\log (1+e x)}{x} \, dx\)

Optimal. Leaf size=8 \[ -\text{PolyLog}(2,-e x) \]

[Out]

-PolyLog[2, -(e*x)]

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Rubi [A]  time = 0.007346, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2391} \[ -\text{PolyLog}(2,-e x) \]

Antiderivative was successfully verified.

[In]

Int[Log[1 + e*x]/x,x]

[Out]

-PolyLog[2, -(e*x)]

Rule 2391

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]

Rubi steps

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

Mathematica [A]  time = 0.0011049, size = 8, normalized size = 1. \[ -\text{PolyLog}(2,-e x) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[1 + e*x]/x,x]

[Out]

-PolyLog[2, -(e*x)]

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Maple [A]  time = 0.059, size = 9, normalized size = 1.1 \begin{align*} -{\it dilog} \left ( ex+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(e*x+1)/x,x)

[Out]

-dilog(e*x+1)

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Maxima [B]  time = 1.26031, size = 26, normalized size = 3.25 \begin{align*} \log \left (e x + 1\right ) \log \left (-e x\right ) +{\rm Li}_2\left (e x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.9614, size = 19, normalized size = 2.38 \begin{align*} -{\rm Li}_2\left (-e x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*x+1)/x,x, algorithm="fricas")

[Out]

-dilog(-e*x)

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Sympy [C]  time = 2.52224, size = 10, normalized size = 1.25 \begin{align*} - \operatorname{Li}_{2}\left (e x e^{i \pi }\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(e*x+1)/x,x)

[Out]

-polylog(2, e*x*exp_polar(I*pi))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*x+1)/x,x, algorithm="giac")

[Out]

integrate(log(e*x + 1)/x, x)