3.2.71 \(\int \frac {\log (2 (3+e x^n))}{x} \, dx\) [171]

Optimal. Leaf size=21 \[ \log (6) \log (x)-\frac {\text {Li}_2\left (-\frac {e x^n}{3}\right )}{n} \]

[Out]

ln(6)*ln(x)-polylog(2,-1/3*e*x^n)/n

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Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2504, 2439, 2438} \begin {gather*} \log (6) \log (x)-\frac {\text {PolyLog}\left (2,-\frac {e x^n}{3}\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[2*(3 + e*x^n)]/x,x]

[Out]

Log[6]*Log[x] - PolyLog[2, -1/3*(e*x^n)]/n

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]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps

\begin {align*} \int \frac {\log \left (2 \left (3+e x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \frac {\log (2 (3+e x))}{x} \, dx,x,x^n\right )}{n}\\ &=\log (6) \log (x)+\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{3}\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\log (6) \log (x)-\frac {\text {Li}_2\left (-\frac {e x^n}{3}\right )}{n}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} \log (6) \log (x)-\frac {\text {Li}_2\left (-\frac {e x^n}{3}\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[2*(3 + e*x^n)]/x,x]

[Out]

Log[6]*Log[x] - PolyLog[2, -1/3*(e*x^n)]/n

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(45\) vs. \(2(19)=38\).
time = 1.18, size = 46, normalized size = 2.19

method	result	size
risch	\(\ln \left (x \right ) \ln \left (6+2 e \,x^{n}\right )-\frac {\dilog \left (\frac {e \,x^{n}}{3}+1\right )}{n}-\ln \left (x \right ) \ln \left (\frac {e \,x^{n}}{3}+1\right )\)	\(41\)
derivativedivides	\(\frac {\left (\ln \left (6+2 e \,x^{n}\right )-\ln \left (\frac {e \,x^{n}}{3}+1\right )\right ) \ln \left (-\frac {e \,x^{n}}{3}\right )-\dilog \left (\frac {e \,x^{n}}{3}+1\right )}{n}\)	\(46\)
default	\(\frac {\left (\ln \left (6+2 e \,x^{n}\right )-\ln \left (\frac {e \,x^{n}}{3}+1\right )\right ) \ln \left (-\frac {e \,x^{n}}{3}\right )-\dilog \left (\frac {e \,x^{n}}{3}+1\right )}{n}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(6+2*e*x^n)/x,x,method=_RETURNVERBOSE)

[Out]

1/n*((ln(6+2*e*x^n)-ln(1/3*e*x^n+1))*ln(-1/3*e*x^n)-dilog(1/3*e*x^n+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(6+2*e*x^n)/x,x, algorithm="maxima")

[Out]

-1/2*n*log(x)^2 + 3*n*integrate(log(x)/(x*e^(n*log(x) + 1) + 3*x), x) + log(2)*log(x) + log(x)*log(e^(n*log(x)
+ 1) + 3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
time = 0.37, size = 44, normalized size = 2.10 \begin {gather*} \frac {n \log \left (2 \, x^{n} e + 6\right ) \log \left (x\right ) - n \log \left (\frac {1}{3} \, x^{n} e + 1\right ) \log \left (x\right ) - {\rm Li}_2\left (-\frac {1}{3} \, x^{n} e\right )}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(6+2*e*x^n)/x,x, algorithm="fricas")

[Out]

(n*log(2*x^n*e + 6)*log(x) - n*log(1/3*x^n*e + 1)*log(x) - dilog(-1/3*x^n*e))/n

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Sympy [C] Result contains complex when optimal does not.
time = 1.63, size = 100, normalized size = 4.76 \begin {gather*} \begin {cases} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{n} e^{i \pi }}{3}\right )}{n} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (6 \right )} \log {\left (x \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{n} e^{i \pi }}{3}\right )}{n} & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (6 \right )} \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{n} e^{i \pi }}{3}\right )}{n} & \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 (6 \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (6 \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{n} e^{i \pi }}{3}\right )}{n} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(6+2*e*x**n)/x,x)

[Out]

Piecewise((-polylog(2, e*x**n*exp_polar(I*pi)/3)/n, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(6)*log(x) - polylog(2
, e*x**n*exp_polar(I*pi)/3)/n, Abs(x) < 1), (-log(6)*log(1/x) - polylog(2, e*x**n*exp_polar(I*pi)/3)/n, 1/Abs(
x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(6) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(6) - pol
ylog(2, e*x**n*exp_polar(I*pi)/3)/n, 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(6+2*e*x^n)/x,x, algorithm="giac")

[Out]

integrate(log(2*x^n*e + 6)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\ln \left (2\,e\,x^n+6\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(2*e*x^n + 6)/x,x)

[Out]

int(log(2*e*x^n + 6)/x, x)

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