3.389 \(\int \frac{\log (x) \log (d+e x^m)}{x} \, dx\)

Optimal. Leaf size=69 \[ \frac{\text{PolyLog}\left (3,-\frac{e x^m}{d}\right )}{m^2}-\frac{\log (x) \text{PolyLog}\left (2,-\frac{e x^m}{d}\right )}{m}+\frac{1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac{1}{2} \log ^2(x) \log \left (\frac{e x^m}{d}+1\right ) \]

[Out]

(Log[x]^2*Log[d + e*x^m])/2 - (Log[x]^2*Log[1 + (e*x^m)/d])/2 - (Log[x]*PolyLog[2, -((e*x^m)/d)])/m + PolyLog[
3, -((e*x^m)/d)]/m^2

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Rubi [A]  time = 0.124492, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2375, 2337, 2374, 6589} \[ \frac{\text{PolyLog}\left (3,-\frac{e x^m}{d}\right )}{m^2}-\frac{\log (x) \text{PolyLog}\left (2,-\frac{e x^m}{d}\right )}{m}+\frac{1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac{1}{2} \log ^2(x) \log \left (\frac{e x^m}{d}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(Log[x]*Log[d + e*x^m])/x,x]

[Out]

(Log[x]^2*Log[d + e*x^m])/2 - (Log[x]^2*Log[1 + (e*x^m)/d])/2 - (Log[x]*PolyLog[2, -((e*x^m)/d)])/m + PolyLog[
3, -((e*x^m)/d)]/m^2

Rule 2375

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

Rule 2337

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(
a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 6589

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 (x) \log \left (d+e x^m\right )}{x} \, dx &=\frac{1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac{1}{2} (e m) \int \frac{x^{-1+m} \log ^2(x)}{d+e x^m} \, dx\\ &=\frac{1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac{1}{2} \log ^2(x) \log \left (1+\frac{e x^m}{d}\right )+\int \frac{\log (x) \log \left (1+\frac{e x^m}{d}\right )}{x} \, dx\\ &=\frac{1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac{1}{2} \log ^2(x) \log \left (1+\frac{e x^m}{d}\right )-\frac{\log (x) \text{Li}_2\left (-\frac{e x^m}{d}\right )}{m}+\frac{\int \frac{\text{Li}_2\left (-\frac{e x^m}{d}\right )}{x} \, dx}{m}\\ &=\frac{1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac{1}{2} \log ^2(x) \log \left (1+\frac{e x^m}{d}\right )-\frac{\log (x) \text{Li}_2\left (-\frac{e x^m}{d}\right )}{m}+\frac{\text{Li}_3\left (-\frac{e x^m}{d}\right )}{m^2}\\ \end{align*}

Mathematica [A]  time = 0.0579992, size = 75, normalized size = 1.09 \[ \frac{\text{PolyLog}\left (3,-\frac{d x^{-m}}{e}\right )}{m^2}+\frac{\log (x) \text{PolyLog}\left (2,-\frac{d x^{-m}}{e}\right )}{m}-\frac{1}{6} \log ^2(x) \left (3 \log \left (\frac{d x^{-m}}{e}+1\right )-3 \log \left (d+e x^m\right )+m \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[x]*Log[d + e*x^m])/x,x]

[Out]

-(Log[x]^2*(m*Log[x] + 3*Log[1 + d/(e*x^m)] - 3*Log[d + e*x^m]))/6 + (Log[x]*PolyLog[2, -(d/(e*x^m))])/m + Pol
yLog[3, -(d/(e*x^m))]/m^2

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Maple [A]  time = 2.256, size = 66, normalized size = 1. \begin{align*}{\frac{ \left ( \ln \left ( x \right ) \right ) ^{2}\ln \left ( d+e{x}^{m} \right ) }{2}}-{\frac{ \left ( \ln \left ( x \right ) \right ) ^{2}}{2}\ln \left ( 1+{\frac{e{x}^{m}}{d}} \right ) }-{\frac{\ln \left ( x \right ) }{m}{\it polylog} \left ( 2,-{\frac{e{x}^{m}}{d}} \right ) }+{\frac{1}{{m}^{2}}{\it polylog} \left ( 3,-{\frac{e{x}^{m}}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)*ln(d+e*x^m)/x,x)

[Out]

1/2*ln(x)^2*ln(d+e*x^m)-1/2*ln(x)^2*ln(1+e*x^m/d)-ln(x)*polylog(2,-e*x^m/d)/m+polylog(3,-e*x^m/d)/m^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{6} \, m \log \left (x\right )^{3} + d m \int \frac{\log \left (x\right )^{2}}{2 \,{\left (e x x^{m} + d x\right )}}\,{d x} + \frac{1}{2} \, \log \left (e x^{m} + d\right ) \log \left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*m*log(x)^3 + d*m*integrate(1/2*log(x)^2/(e*x*x^m + d*x), x) + 1/2*log(e*x^m + d)*log(x)^2

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Fricas [C]  time = 1.72426, size = 185, normalized size = 2.68 \begin{align*} \frac{m^{2} \log \left (e x^{m} + d\right ) \log \left (x\right )^{2} - m^{2} \log \left (x\right )^{2} \log \left (\frac{e x^{m} + d}{d}\right ) - 2 \, m{\rm Li}_2\left (-\frac{e x^{m} + d}{d} + 1\right ) \log \left (x\right ) + 2 \,{\rm polylog}\left (3, -\frac{e x^{m}}{d}\right )}{2 \, m^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(m^2*log(e*x^m + d)*log(x)^2 - m^2*log(x)^2*log((e*x^m + d)/d) - 2*m*dilog(-(e*x^m + d)/d + 1)*log(x) + 2*
polylog(3, -e*x^m/d))/m^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(x)*ln(d+e*x**m)/x,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(e*x^m + d)*log(x)/x, x)