3.6 \(\int \frac{(a+b \log (c x^n)) \log (1+e x)}{x} \, dx\)

Optimal. Leaf size=28 \[ b n \text{PolyLog}(3,-e x)-\text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right ) \]

[Out]

-((a + b*Log[c*x^n])*PolyLog[2, -(e*x)]) + b*n*PolyLog[3, -(e*x)]

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Rubi [A]  time = 0.0275801, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2374, 6589} \[ b n \text{PolyLog}(3,-e x)-\text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Log[c*x^n])*Log[1 + e*x])/x,x]

[Out]

-((a + b*Log[c*x^n])*PolyLog[2, -(e*x)]) + b*n*PolyLog[3, -(e*x)]

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{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx &=-\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)+(b n) \int \frac{\text{Li}_2(-e x)}{x} \, dx\\ &=-\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)+b n \text{Li}_3(-e x)\\ \end{align*}

Mathematica [A]  time = 0.0089549, size = 34, normalized size = 1.21 \[ -a \text{PolyLog}(2,-e x)-b \text{PolyLog}(2,-e x) \log \left (c x^n\right )+b n \text{PolyLog}(3,-e x) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Log[c*x^n])*Log[1 + e*x])/x,x]

[Out]

-(a*PolyLog[2, -(e*x)]) - b*Log[c*x^n]*PolyLog[2, -(e*x)] + b*n*PolyLog[3, -(e*x)]

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Maple [F]  time = 0.187, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( ex+1 \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))*ln(e*x+1)/x,x)

[Out]

int((a+b*ln(c*x^n))*ln(e*x+1)/x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left (e x + 1\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(e*x+1)/x,x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)*log(e*x + 1)/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a \log \left (e x + 1\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n)*log(e*x + 1) + a*log(e*x + 1))/x, x)

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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((a+b*ln(c*x**n))*ln(e*x+1)/x,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left (e x + 1\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)*log(e*x + 1)/x, x)