3.212 \(\int \frac{(a+b \log (c x^n)) \text{PolyLog}(2,e x)}{x^2} \, dx\)

Optimal. Leaf size=142 \[ -\frac{\text{PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}-b e n \text{PolyLog}(2,e x)-\frac{b n \text{PolyLog}(2,e x)}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac{\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{1}{2} b e n \log ^2(x)+2 b e n \log (x)-2 b e n \log (1-e x)+\frac{2 b n \log (1-e x)}{x} \]

[Out]

2*b*e*n*Log[x] - (b*e*n*Log[x]^2)/2 + e*Log[x]*(a + b*Log[c*x^n]) - 2*b*e*n*Log[1 - e*x] + (2*b*n*Log[1 - e*x]
)/x - e*(a + b*Log[c*x^n])*Log[1 - e*x] + ((a + b*Log[c*x^n])*Log[1 - e*x])/x - b*e*n*PolyLog[2, e*x] - (b*n*P
olyLog[2, e*x])/x - ((a + b*Log[c*x^n])*PolyLog[2, e*x])/x

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Rubi [A]  time = 0.113685, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {2385, 2395, 36, 29, 31, 2376, 2301, 2391} \[ -\frac{\text{PolyLog}(2,e x) \left (a+b \log \left (c x^n\right )\right )}{x}-b e n \text{PolyLog}(2,e x)-\frac{b n \text{PolyLog}(2,e x)}{x}+e \log (x) \left (a+b \log \left (c x^n\right )\right )-e \log (1-e x) \left (a+b \log \left (c x^n\right )\right )+\frac{\log (1-e x) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{1}{2} b e n \log ^2(x)+2 b e n \log (x)-2 b e n \log (1-e x)+\frac{2 b n \log (1-e x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*b*e*n*Log[x] - (b*e*n*Log[x]^2)/2 + e*Log[x]*(a + b*Log[c*x^n]) - 2*b*e*n*Log[1 - e*x] + (2*b*n*Log[1 - e*x]
)/x - e*(a + b*Log[c*x^n])*Log[1 - e*x] + ((a + b*Log[c*x^n])*Log[1 - e*x])/x - b*e*n*PolyLog[2, e*x] - (b*n*P
olyLog[2, e*x])/x - ((a + b*Log[c*x^n])*PolyLog[2, e*x])/x

Rule 2385

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

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2391

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*} \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)}{x^2} \, dx &=-\frac{b n \text{Li}_2(e x)}{x}-\frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)}{x}-(b n) \int \frac{\log (1-e x)}{x^2} \, dx-\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x^2} \, dx\\ &=e \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-\frac{b n \text{Li}_2(e x)}{x}-\frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)}{x}+(b n) \int \left (-\frac{e \log (x)}{x}-\frac{\log (1-e x)}{x^2}+\frac{e \log (1-e x)}{x}\right ) \, dx+(b e n) \int \frac{1}{x (1-e x)} \, dx\\ &=e \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac{b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-\frac{b n \text{Li}_2(e x)}{x}-\frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)}{x}-(b n) \int \frac{\log (1-e x)}{x^2} \, dx+(b e n) \int \frac{1}{x} \, dx-(b e n) \int \frac{\log (x)}{x} \, dx+(b e n) \int \frac{\log (1-e x)}{x} \, dx+\left (b e^2 n\right ) \int \frac{1}{1-e x} \, dx\\ &=b e n \log (x)-\frac{1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )-b e n \log (1-e x)+\frac{2 b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-b e n \text{Li}_2(e x)-\frac{b n \text{Li}_2(e x)}{x}-\frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)}{x}+(b e n) \int \frac{1}{x (1-e x)} \, dx\\ &=b e n \log (x)-\frac{1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )-b e n \log (1-e x)+\frac{2 b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-b e n \text{Li}_2(e x)-\frac{b n \text{Li}_2(e x)}{x}-\frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)}{x}+(b e n) \int \frac{1}{x} \, dx+\left (b e^2 n\right ) \int \frac{1}{1-e x} \, dx\\ &=2 b e n \log (x)-\frac{1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )-2 b e n \log (1-e x)+\frac{2 b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-b e n \text{Li}_2(e x)-\frac{b n \text{Li}_2(e x)}{x}-\frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(e x)}{x}\\ \end{align*}

Mathematica [A]  time = 0.153703, size = 115, normalized size = 0.81 \[ \frac{(-\text{PolyLog}(2,e x)+e x \log (x)+(1-e x) \log (1-e x)) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{x}+\frac{b n \left (-2 (e x+\log (x)+1) \text{PolyLog}(2,e x)+e x \log ^2(x)+\log (x) (4 e x+(2-2 e x) \log (1-e x))-4 (e x-1) \log (1-e x)\right )}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b*n*Log[x] + b*Log[c*x^n])*(e*x*Log[x] + (1 - e*x)*Log[1 - e*x] - PolyLog[2, e*x]))/x + (b*n*(e*x*Log[x]
^2 - 4*(-1 + e*x)*Log[1 - e*x] + Log[x]*(4*e*x + (2 - 2*e*x)*Log[1 - e*x]) - 2*(1 + e*x + Log[x])*PolyLog[2, e
*x]))/(2*x)

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Maple [F]  time = 0.145, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ){\it polylog} \left ( 2,ex \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))*polylog(2,e*x)/x^2,x)

[Out]

int((a+b*ln(c*x^n))*polylog(2,e*x)/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (e \log \left (x\right ) - \frac{{\left (e x - 1\right )} \log \left (-e x + 1\right ) +{\rm Li}_2\left (e x\right )}{x}\right )} a - b{\left (\frac{{\left (n + \log \left (c\right ) + \log \left (x^{n}\right )\right )}{\rm Li}_2\left (e x\right ) -{\left (e n x \log \left (x\right ) + 2 \, n + \log \left (c\right )\right )} \log \left (-e x + 1\right ) -{\left (e x \log \left (x\right ) -{\left (e x - 1\right )} \log \left (-e x + 1\right )\right )} \log \left (x^{n}\right )}{x} + \int \frac{2 \, e n + e \log \left (c\right ) +{\left (2 \, e^{2} n x - e n\right )} \log \left (x\right )}{e x^{2} - x}\,{d x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*polylog(2,e*x)/x^2,x, algorithm="maxima")

[Out]

(e*log(x) - ((e*x - 1)*log(-e*x + 1) + dilog(e*x))/x)*a - b*(((n + log(c) + log(x^n))*dilog(e*x) - (e*n*x*log(
x) + 2*n + log(c))*log(-e*x + 1) - (e*x*log(x) - (e*x - 1)*log(-e*x + 1))*log(x^n))/x + integrate((2*e*n + e*l
og(c) + (2*e^2*n*x - e*n)*log(x))/(e*x^2 - x), x))

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Fricas [A]  time = 0.974024, size = 351, normalized size = 2.47 \begin{align*} \frac{b e n x \log \left (x\right )^{2} - 2 \,{\left (b e n x + b n + a\right )}{\rm Li}_2\left (e x\right ) + 2 \,{\left (2 \, b n -{\left (2 \, b e n + a e\right )} x + a\right )} \log \left (-e x + 1\right ) - 2 \,{\left (b{\rm Li}_2\left (e x\right ) +{\left (b e x - b\right )} \log \left (-e x + 1\right )\right )} \log \left (c\right ) + 2 \,{\left (b e x \log \left (c\right ) - b n{\rm Li}_2\left (e x\right ) +{\left (2 \, b e n + a e\right )} x -{\left (b e n x - b n\right )} \log \left (-e x + 1\right )\right )} \log \left (x\right )}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*polylog(2,e*x)/x^2,x, algorithm="fricas")

[Out]

1/2*(b*e*n*x*log(x)^2 - 2*(b*e*n*x + b*n + a)*dilog(e*x) + 2*(2*b*n - (2*b*e*n + a*e)*x + a)*log(-e*x + 1) - 2
*(b*dilog(e*x) + (b*e*x - b)*log(-e*x + 1))*log(c) + 2*(b*e*x*log(c) - b*n*dilog(e*x) + (2*b*e*n + a*e)*x - (b
*e*n*x - b*n)*log(-e*x + 1))*log(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))*polylog(2,e*x)/x**2,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 )}{\rm Li}_2\left (e x\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*polylog(2,e*x)/x^2,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*dilog(e*x)/x^2, x)