3.2.69 \(\int \frac {\log (1+e x^n)}{x} \, dx\) [169]

Optimal. Leaf size=13 \[ -\frac {\text {Li}_2\left (-e x^n\right )}{n} \]

[Out]

-polylog(2,-e*x^n)/n

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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 = {2438} \begin {gather*} -\frac {\text {PolyLog}\left (2,-e x^n\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[1 + e*x^n]/x,x]

[Out]

-(PolyLog[2, -(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]

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*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\text {Li}_2\left (-e x^n\right )}{n} \end {gather*}

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 = 1.16, size = 14, normalized size = 1.08

method	result	size
derivativedivides	\(-\frac {\dilog \left (1+e \,x^{n}\right )}{n}\)	\(14\)
default	\(-\frac {\dilog \left (1+e \,x^{n}\right )}{n}\)	\(14\)
meijerg	\(-\frac {\polylog \left (2, -e \,x^{n}\right )}{n}\)	\(14\)
risch	\(-\frac {\dilog \left (1+e \,x^{n}\right )}{n}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(1+e*x^n)/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 13, normalized size = 1.00 \begin {gather*} -\frac {{\rm Li}_2\left (-x^{n} e\right )}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 1.48, size = 14, normalized size = 1.08 \begin {gather*} - \frac {\operatorname {Li}_{2}\left (e x^{n} e^{i \pi }\right )}{n} \end {gather*}

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 = 0.00, 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(1+e*x^n)/x,x, algorithm="giac")

[Out]

integrate(log(x^n*e + 1)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int \frac {\ln \left (e\,x^n+1\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(e*x^n + 1)/x,x)

[Out]

int(log(e*x^n + 1)/x, x)

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