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

Optimal. Leaf size=33 \[ -\frac {\log \left (f x^p\right ) \text {Li}_2\left (-e x^m\right )}{m}+\frac {p \text {Li}_3\left (-e x^m\right )}{m^2} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2421, 6724} \begin {gather*} \frac {p \text {PolyLog}\left (3,-e x^m\right )}{m^2}-\frac {\log \left (f x^p\right ) \text {PolyLog}\left (2,-e x^m\right )}{m} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.45, size = 191, normalized size = 5.79

method	result	size
risch	\(-\frac {p \ln \left (x \right ) \polylog \left (2, -e \,x^{m}\right )}{m}+\frac {p \polylog \left (3, -e \,x^{m}\right )}{m^{2}}-\frac {\left (\ln \left (x^{p}\right )-\ln \left (x \right ) p \right ) \dilog \left (1+e \,x^{m}\right )}{m}+\frac {i \dilog \left (1+e \,x^{m}\right ) \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x^{p}\right ) \mathrm {csgn}\left (i f \,x^{p}\right )}{2 m}-\frac {i \dilog \left (1+e \,x^{m}\right ) \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f \,x^{p}\right )^{2}}{2 m}-\frac {i \dilog \left (1+e \,x^{m}\right ) \pi \,\mathrm {csgn}\left (i x^{p}\right ) \mathrm {csgn}\left (i f \,x^{p}\right )^{2}}{2 m}+\frac {i \dilog \left (1+e \,x^{m}\right ) \pi \mathrm {csgn}\left (i f \,x^{p}\right )^{3}}{2 m}-\frac {\dilog \left (1+e \,x^{m}\right ) \ln \left (f \right )}{m}\)	\(191\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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Fricas [A]
time = 0.37, size = 37, normalized size = 1.12 \begin {gather*} -\frac {{\left (m p \log \left (x\right ) + m \log \left (f\right )\right )} {\rm Li}_2\left (-x^{m} e\right ) - p {\rm polylog}\left (3, -x^{m} e\right )}{m^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (f\,x^p\right )\,\ln \left (e\,x^m+1\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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