3.3.12 \(\int \frac {(a+b \log (c x^n)) \text {Li}_2(e x)}{x^2} \, dx\) [212]

Optimal. Leaf size=142 \[ 2 b e n \log (x)-\frac {1}{2} b e n \log ^2(x)+e \log (x) \left (a+b \log \left (c x^n\right )\right )-2 b e n \log (1-e x)+\frac {2 b n \log (1-e x)}{x}-e \left (a+b \log \left (c x^n\right )\right ) \log (1-e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1-e x)}{x}-b e n \text {Li}_2(e x)-\frac {b n \text {Li}_2(e x)}{x}-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(e x)}{x} \]

[Out]

2*b*e*n*ln(x)-1/2*b*e*n*ln(x)^2+e*ln(x)*(a+b*ln(c*x^n))-2*b*e*n*ln(-e*x+1)+2*b*n*ln(-e*x+1)/x-e*(a+b*ln(c*x^n)
)*ln(-e*x+1)+(a+b*ln(c*x^n))*ln(-e*x+1)/x-b*e*n*polylog(2,e*x)-b*n*polylog(2,e*x)/x-(a+b*ln(c*x^n))*polylog(2,
e*x)/x

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Rubi [A]
time = 0.08, antiderivative size = 142, normalized size of antiderivative = 1.00, 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 = {2432, 2442, 36, 29, 31, 2423, 2338, 2438} \begin {gather*} -\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} \end {gather*}

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 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 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 2338

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 2423

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 2432

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 2438

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]

Rule 2442

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]

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*}

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Mathematica [A]
time = 0.12, size = 115, normalized size = 0.81 \begin {gather*} \frac {\left (a-b n \log (x)+b \log \left (c x^n\right )\right ) (e x \log (x)+(1-e x) \log (1-e x)-\text {Li}_2(e x))}{x}+\frac {b n \left (e x \log ^2(x)-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)) \text {Li}_2(e x)\right )}{2 x} \end {gather*}

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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, e x \right )}{x^{2}}\, dx\]

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 = 0.00, 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((a+b*log(c*x^n))*polylog(2,e*x)/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.39, size = 144, normalized size = 1.01 \begin {gather*} \frac {b n x e \log \left (x\right )^{2} - 2 \, {\left (b n x e + b n + a\right )} {\rm Li}_2\left (x e\right ) - 2 \, {\left ({\left (2 \, b n + a\right )} x e - 2 \, b n - a\right )} \log \left (-x e + 1\right ) - 2 \, {\left (b {\rm Li}_2\left (x e\right ) + {\left (b x e - b\right )} \log \left (-x e + 1\right )\right )} \log \left (c\right ) + 2 \, {\left (b x e \log \left (c\right ) - b n {\rm Li}_2\left (x e\right ) + {\left (2 \, b n + a\right )} x e - {\left (b n x e - b n\right )} \log \left (-x e + 1\right )\right )} \log \left (x\right )}{2 \, x} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \operatorname {Li}_{2}\left (e x\right )}{x^{2}}\, dx \end {gather*}

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]

Integral((a + b*log(c*x**n))*polylog(2, e*x)/x**2, x)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {polylog}\left (2,e\,x\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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