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

Optimal result

Integrand size = 12, antiderivative size = 13 \[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=-\frac {\operatorname {PolyLog}\left (2,-e x^n\right )}{n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2438} \[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=-\frac {\operatorname {PolyLog}\left (2,-e x^n\right )}{n} \]

[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*} \text {integral}& = -\frac {\text {Li}_2\left (-e x^n\right )}{n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=-\frac {\operatorname {PolyLog}\left (2,-e x^n\right )}{n} \]

[In]

Integrate[Log[1 + e*x^n]/x,x]

[Out]

-(PolyLog[2, -(e*x^n)]/n)

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=-\frac {{\rm Li}_2\left (-e x^{n}\right )}{n} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.46 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=- \frac {\operatorname {Li}_{2}\left (e x^{n} e^{i \pi }\right )}{n} \]

[In]

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

[Out]

-polylog(2, e*x**n*exp_polar(I*pi))/n

Maxima [F]

\[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=\int { \frac {\log \left (e x^{n} + 1\right )}{x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=\int { \frac {\log \left (e x^{n} + 1\right )}{x} \,d x } \]

[In]

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

[Out]

integrate(log(e*x^n + 1)/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (1+e x^n\right )}{x} \, dx=\int \frac {\ln \left (e\,x^n+1\right )}{x} \,d x \]

[In]

int(log(e*x^n + 1)/x,x)

[Out]

int(log(e*x^n + 1)/x, x)