3.169 \(\int \frac{\log (1+e x^n)}{x} \, dx\)

Optimal. Leaf size=13 \[ -\frac{\text{PolyLog}\left (2,-e x^n\right )}{n} \]

[Out]

-(PolyLog[2, -(e*x^n)]/n)

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Rubi [A]  time = 0.0086505, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2391} \[ -\frac{\text{PolyLog}\left (2,-e x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[Log[1 + e*x^n]/x,x]

[Out]

-(PolyLog[2, -(e*x^n)]/n)

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

Mathematica [A]  time = 0.0026189, size = 13, normalized size = 1. \[ -\frac{\text{PolyLog}\left (2,-e x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[1 + e*x^n]/x,x]

[Out]

-(PolyLog[2, -(e*x^n)]/n)

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Maple [A]  time = 0.07, size = 14, normalized size = 1.1 \begin{align*} -{\frac{{\it dilog} \left ( 1+e{x}^{n} \right ) }{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.01841, size = 24, normalized size = 1.85 \begin{align*} -\frac{{\rm Li}_2\left (-e x^{n}\right )}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-dilog(-e*x^n)/n

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Sympy [C]  time = 4.06251, size = 14, normalized size = 1.08 \begin{align*} - \frac{\operatorname{Li}_{2}\left (e x^{n} e^{i \pi }\right )}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-polylog(2, e*x**n*exp_polar(I*pi))/n

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(e*x^n + 1)/x, x)