3.1.74 \(\int \frac {\log (2+e x)}{x} \, dx\) [74]

Optimal. Leaf size=16 \[ \log (2) \log (x)-\text {Li}_2\left (-\frac {e x}{2}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 16, 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 = {2439, 2438} \begin {gather*} \log (2) \log (x)-\text {PolyLog}\left (2,-\frac {e x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[2 + e*x]/x,x]

[Out]

Log[2]*Log[x] - PolyLog[2, -1/2*(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]

Rule 2439

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*d])*Log[x], x] + Dist[
b, Int[Log[1 + e*(x/d)]/x, x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[c*d, 0]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \log (2) \log (x)-\text {Li}_2\left (-\frac {e x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[2 + e*x]/x,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(32\) vs. \(2(14)=28\).
time = 0.11, size = 33, normalized size = 2.06

method	result	size
derivativedivides	\(\left (\ln \left (e x +2\right )-\ln \left (\frac {e x}{2}+1\right )\right ) \ln \left (-\frac {e x}{2}\right )-\dilog \left (\frac {e x}{2}+1\right )\)	\(33\)
default	\(\left (\ln \left (e x +2\right )-\ln \left (\frac {e x}{2}+1\right )\right ) \ln \left (-\frac {e x}{2}\right )-\dilog \left (\frac {e x}{2}+1\right )\)	\(33\)
risch	\(\left (\ln \left (e x +2\right )-\ln \left (\frac {e x}{2}+1\right )\right ) \ln \left (-\frac {e x}{2}\right )-\dilog \left (\frac {e x}{2}+1\right )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 23, normalized size = 1.44 \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 = 1.39, size = 87, normalized size = 5.44 \begin {gather*} \begin {cases} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{2}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (2 \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{2}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (2 \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{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 )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (2 \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{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*exp_polar(I*pi)/2), (Abs(x) < 1) & (1/Abs(x) < 1)), (log(2)*log(x) - polylog(2, e*x
*exp_polar(I*pi)/2), Abs(x) < 1), (-log(2)*log(1/x) - polylog(2, e*x*exp_polar(I*pi)/2), 1/Abs(x) < 1), (-meij
erg(((), (1, 1)), ((0, 0), ()), x)*log(2) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(2) - polylog(2, e*x*exp
_polar(I*pi)/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 = 1.12 \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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