3.1.75 \(\int \frac {\log (1+e x)}{x} \, dx\) [75]

Optimal. Leaf size=8 \[ -\text {Li}_2(-e x) \]

[Out]

-polylog(2,-e*x)

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Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2438} \begin {gather*} -\text {PolyLog}(2,-e x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[1 + e*x]/x,x]

[Out]

-PolyLog[2, -(e*x)]

Rule 2438

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*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} -\text {Li}_2(-e x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[1 + e*x]/x,x]

[Out]

-PolyLog[2, -(e*x)]

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Maple [A]
time = 0.09, size = 9, normalized size = 1.12

method	result	size
derivativedivides	\(-\dilog \left (e x +1\right )\)	\(9\)
default	\(-\dilog \left (e x +1\right )\)	\(9\)
meijerg	\(-\polylog \left (2, -e x \right )\)	\(9\)
risch	\(-\dilog \left (e x +1\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-dilog(e*x+1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (8) = 16\).
time = 0.31, size = 22, normalized size = 2.75 \begin {gather*} \log \left (x e + 1\right ) \log \left (-x e\right ) + {\rm Li}_2\left (x e + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 8, normalized size = 1.00 \begin {gather*} -{\rm Li}_2\left (-x e\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-dilog(-x*e)

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Sympy [C] Result contains complex when optimal does not.
time = 1.32, size = 10, normalized size = 1.25 \begin {gather*} - \operatorname {Li}_{2}\left (e x e^{i \pi }\right ) \end {gather*}

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 = 0.00, 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+1)/x,x, algorithm="giac")

[Out]

integrate(log(x*e + 1)/x, x)

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Mupad [B]
time = 0.03, size = 18, normalized size = 2.25 \begin {gather*} {\mathrm {Li}}_{\mathrm {2}}\left (-e\,x\right )+\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(log(e*x + 1)/x,x)

[Out]

dilog(-e*x) + log(e*x + 1)*log(-e*x)

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