\(\int \frac {\log (f x^p) \log (1+e x^m)}{x} \, dx\) [619]

Optimal result

Integrand size = 18, antiderivative size = 33 \[ \int \frac {\log \left (f x^p\right ) \log \left (1+e x^m\right )}{x} \, dx=-\frac {\log \left (f x^p\right ) \operatorname {PolyLog}\left (2,-e x^m\right )}{m}+\frac {p \operatorname {PolyLog}\left (3,-e x^m\right )}{m^2} \]

[Out]

-ln(f*x^p)*polylog(2,-e*x^m)/m+p*polylog(3,-e*x^m)/m^2

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2421, 6724} \[ \int \frac {\log \left (f x^p\right ) \log \left (1+e x^m\right )}{x} \, dx=\frac {p \operatorname {PolyLog}\left (3,-e x^m\right )}{m^2}-\frac {\operatorname {PolyLog}\left (2,-e x^m\right ) \log \left (f x^p\right )}{m} \]

[In]

Int[(Log[f*x^p]*Log[1 + e*x^m])/x,x]

[Out]

-((Log[f*x^p]*PolyLog[2, -(e*x^m)])/m) + (p*PolyLog[3, -(e*x^m)])/m^2

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (f x^p\right ) \text {Li}_2\left (-e x^m\right )}{m}+\frac {p \int \frac {\text {Li}_2\left (-e x^m\right )}{x} \, dx}{m} \\ & = -\frac {\log \left (f x^p\right ) \text {Li}_2\left (-e x^m\right )}{m}+\frac {p \text {Li}_3\left (-e x^m\right )}{m^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (f x^p\right ) \log \left (1+e x^m\right )}{x} \, dx=-\frac {\log \left (f x^p\right ) \operatorname {PolyLog}\left (2,-e x^m\right )}{m}+\frac {p \operatorname {PolyLog}\left (3,-e x^m\right )}{m^2} \]

[In]

Integrate[(Log[f*x^p]*Log[1 + e*x^m])/x,x]

[Out]

-((Log[f*x^p]*PolyLog[2, -(e*x^m)])/m) + (p*PolyLog[3, -(e*x^m)])/m^2

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.54 (sec) , antiderivative size = 148, normalized size of antiderivative = 4.48

method	result	size
risch	\(-\frac {p \ln \left (x \right ) \operatorname {Li}_{2}\left (-e \,x^{m}\right )}{m}+\frac {p \,\operatorname {Li}_{3}\left (-e \,x^{m}\right )}{m^{2}}-\frac {\left (\ln \left (x^{p}\right )-p \ln \left (x \right )\right ) \operatorname {dilog}\left (1+e \,x^{m}\right )}{m}-\frac {\left (-\frac {i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{p}\right ) \operatorname {csgn}\left (i f \,x^{p}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{p}\right )^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i x^{p}\right ) \operatorname {csgn}\left (i f \,x^{p}\right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (i f \,x^{p}\right )^{3}}{2}+\ln \left (f \right )\right ) \operatorname {dilog}\left (1+e \,x^{m}\right )}{m}\)	\(148\)

[In]

int(ln(f*x^p)*ln(1+e*x^m)/x,x,method=_RETURNVERBOSE)

[Out]

-p/m*ln(x)*polylog(2,-e*x^m)+p*polylog(3,-e*x^m)/m^2-1/m*(ln(x^p)-p*ln(x))*dilog(1+e*x^m)-(-1/2*I*Pi*csgn(I*f)
*csgn(I*x^p)*csgn(I*f*x^p)+1/2*I*Pi*csgn(I*f)*csgn(I*f*x^p)^2+1/2*I*Pi*csgn(I*x^p)*csgn(I*f*x^p)^2-1/2*I*Pi*cs
gn(I*f*x^p)^3+ln(f))/m*dilog(1+e*x^m)

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {\log \left (f x^p\right ) \log \left (1+e x^m\right )}{x} \, dx=-\frac {{\left (m p \log \left (x\right ) + m \log \left (f\right )\right )} {\rm Li}_2\left (-e x^{m}\right ) - p {\rm polylog}\left (3, -e x^{m}\right )}{m^{2}} \]

[In]

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

[Out]

-((m*p*log(x) + m*log(f))*dilog(-e*x^m) - p*polylog(3, -e*x^m))/m^2

Sympy [F(-2)]

Exception generated. \[ \int \frac {\log \left (f x^p\right ) \log \left (1+e x^m\right )}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(ln(f*x**p)*ln(1+e*x**m)/x,x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

Maxima [F]

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

[In]

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

[Out]

-1/2*(p*log(x)^2 - 2*log(f)*log(x) - 2*log(x)*log(x^p))*log(e*x^m + 1) - integrate(1/2*(2*e*m*x^m*log(x)*log(x
^p) - (e*m*p*log(x)^2 - 2*e*m*log(f)*log(x))*x^m)/(e*x*x^m + x), x)

Giac [F]

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

[In]

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

[Out]

integrate(log(e*x^m + 1)*log(f*x^p)/x, x)

Mupad [F(-1)]

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

[In]

int((log(f*x^p)*log(e*x^m + 1))/x,x)

[Out]

int((log(f*x^p)*log(e*x^m + 1))/x, x)