3.170 \(\int \frac{\log (2+e x^n)}{x} \, dx\)

Optimal. Leaf size=21 \[ \log (2) \log (x)-\frac{\text{PolyLog}\left (2,-\frac{e x^n}{2}\right )}{n} \]

[Out]

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

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Rubi [A]  time = 0.027229, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2454, 2392, 2391} \[ \log (2) \log (x)-\frac{\text{PolyLog}\left (2,-\frac{e x^n}{2}\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[Log[2 + e*x^n]/x,x]

[Out]

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

Rule 2454

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])

Rule 2392

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

Mathematica [A]  time = 0.0026883, size = 21, normalized size = 1. \[ \log (2) \log (x)-\frac{\text{PolyLog}\left (2,-\frac{e x^n}{2}\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[2 + e*x^n]/x,x]

[Out]

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

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Maple [B]  time = 0.072, size = 56, normalized size = 2.7 \begin{align*} -{\frac{1}{n}\ln \left ( -{\frac{e{x}^{n}}{2}} \right ) \ln \left ({\frac{e{x}^{n}}{2}}+1 \right ) }+{\frac{\ln \left ( 2+e{x}^{n} \right ) }{n}\ln \left ( -{\frac{e{x}^{n}}{2}} \right ) }-{\frac{1}{n}{\it dilog} \left ({\frac{e{x}^{n}}{2}}+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(2+e*x^n)/x,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, n \log \left (x\right )^{2} + 2 \, n \int \frac{\log \left (x\right )}{e x x^{n} + 2 \, x}\,{d x} + \log \left (e x^{n} + 2\right ) \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.01465, size = 107, normalized size = 5.1 \begin{align*} \frac{n \log \left (e x^{n} + 2\right ) \log \left (x\right ) - n \log \left (\frac{1}{2} \, e x^{n} + 1\right ) \log \left (x\right ) -{\rm Li}_2\left (-\frac{1}{2} \, e x^{n}\right )}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 4.60811, size = 78, normalized size = 3.71 \begin{align*} \begin{cases} \log{\left (2 \right )} \log{\left (x \right )} - \frac{\operatorname{Li}_{2}\left (\frac{e x^{n} e^{i \pi }}{2}\right )}{n} & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (2 \right )} \log{\left (\frac{1}{x} \right )} - \frac{\operatorname{Li}_{2}\left (\frac{e x^{n} e^{i \pi }}{2}\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 (2 \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (2 \right )} - \frac{\operatorname{Li}_{2}\left (\frac{e x^{n} e^{i \pi }}{2}\right )}{n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(2)*log(x) - polylog(2, e*x**n*exp_polar(I*pi)/2)/n, Abs(x) < 1), (-log(2)*log(1/x) - polylog(2,
e*x**n*exp_polar(I*pi)/2)/n, 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(2) + meijerg(((1, 1)
, ()), ((), (0, 0)), x)*log(2) - polylog(2, e*x**n*exp_polar(I*pi)/2)/n, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (e x^{n} + 2\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(e*x^n + 2)/x, x)