3.141 \(\int \frac{\text{PolyLog}(2,c (a+b x))}{d+e x} \, dx\)

Optimal. Leaf size=591 \[ -\frac{\text{PolyLog}\left (3,-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{e}+\frac{\text{PolyLog}\left (3,\frac{(1-c (a+b x)) (b d-a e)}{b (d+e x)}\right )}{e}-\frac{\log \left (\frac{b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right ) \text{PolyLog}\left (2,-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{e}+\frac{\log \left (\frac{b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right ) \text{PolyLog}\left (2,\frac{(1-c (a+b x)) (b d-a e)}{b (d+e x)}\right )}{e}+\frac{\log (d+e x) \text{PolyLog}(2,c (a+b x))}{e}+\frac{\text{PolyLog}\left (2,\frac{b (d+e x)}{b d-a e}\right ) \left (\log \left (\frac{b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )+\log (1-c (a+b x))\right )}{e}+\frac{\text{PolyLog}(2,1-c (a+b x)) \left (\log (d+e x)-\log \left (\frac{b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )\right )}{e}-\frac{\text{PolyLog}(3,1-c (a+b x))}{e}-\frac{\text{PolyLog}\left (3,\frac{b (d+e x)}{b d-a e}\right )}{e}+\frac{\left (\log \left (\frac{-a c e+b c d+e}{b c (d+e x)}\right )-\log \left (\frac{(a+b x) (-a c e+b c d+e)}{b (d+e x)}\right )+\log (c (a+b x))\right ) \log ^2\left (\frac{b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )}{2 e}-\frac{\left (\log (c (a+b x))-\log \left (-\frac{e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac{b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )+\log (1-c (a+b x))\right )^2}{2 e}+\frac{\log (d+e x) \log (c (a+b x)) \log (1-c (a+b x))}{e} \]

[Out]

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

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Rubi [A]  time = 0.517702, antiderivative size = 591, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {6597, 2440, 2435} \[ -\frac{\text{PolyLog}\left (3,-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{e}+\frac{\text{PolyLog}\left (3,\frac{(1-c (a+b x)) (b d-a e)}{b (d+e x)}\right )}{e}-\frac{\log \left (\frac{b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right ) \text{PolyLog}\left (2,-\frac{e (1-c (a+b x))}{b c (d+e x)}\right )}{e}+\frac{\log \left (\frac{b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right ) \text{PolyLog}\left (2,\frac{(1-c (a+b x)) (b d-a e)}{b (d+e x)}\right )}{e}+\frac{\log (d+e x) \text{PolyLog}(2,c (a+b x))}{e}+\frac{\text{PolyLog}\left (2,\frac{b (d+e x)}{b d-a e}\right ) \left (\log \left (\frac{b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )+\log (1-c (a+b x))\right )}{e}+\frac{\text{PolyLog}(2,1-c (a+b x)) \left (\log (d+e x)-\log \left (\frac{b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )\right )}{e}-\frac{\text{PolyLog}(3,1-c (a+b x))}{e}-\frac{\text{PolyLog}\left (3,\frac{b (d+e x)}{b d-a e}\right )}{e}+\frac{\left (\log \left (\frac{-a c e+b c d+e}{b c (d+e x)}\right )-\log \left (\frac{(a+b x) (-a c e+b c d+e)}{b (d+e x)}\right )+\log (c (a+b x))\right ) \log ^2\left (\frac{b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )}{2 e}-\frac{\left (\log (c (a+b x))-\log \left (-\frac{e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac{b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )+\log (1-c (a+b x))\right )^2}{2 e}+\frac{\log (d+e x) \log (c (a+b x)) \log (1-c (a+b x))}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6597

Int[PolyLog[2, (c_.)*((a_.) + (b_.)*(x_))]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*PolyLog[2, c*
(a + b*x)])/e, x] + Dist[b/e, Int[(Log[d + e*x]*Log[1 - a*c - b*c*x])/(a + b*x), x], x] /; FreeQ[{a, b, c, d,
e}, x] && NeQ[c*(b*d - a*e) + e, 0]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))
*((k_) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/l, Subst[Int[x^r*(a + b*Log[c*(-((e*k - d*l)/l) + (e*x)/l)^n])
*(f + g*Log[h*(-((j*k - i*l)/l) + (j*x)/l)^m]), x], x, k + l*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k,
l, m, n}, x] && IntegerQ[r]

Rule 2435

Int[(Log[(a_) + (b_.)*(x_)]*Log[(c_) + (d_.)*(x_)])/(x_), x_Symbol] :> Simp[Log[-((b*x)/a)]*Log[a + b*x]*Log[c
+ d*x], x] + (Simp[(1*(Log[-((b*x)/a)] - Log[-(((b*c - a*d)*x)/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d*x)
)])*Log[(a*(c + d*x))/(c*(a + b*x))]^2)/2, x] - Simp[(1*(Log[-((b*x)/a)] - Log[-((d*x)/c)])*(Log[a + b*x] + Lo
g[(a*(c + d*x))/(c*(a + b*x))])^2)/2, x] + Simp[(Log[c + d*x] - Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1
+ (b*x)/a], x] + Simp[(Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (d*x)/c], x] + Simp[Lo
g[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[Log[(a*(c + d*x))/(c*(a + b*
x))]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))], x] - Simp[PolyLog[3, 1 + (b*x)/a], x] - Simp[PolyLog[3, 1 + (d*x
)/c], x] + Simp[PolyLog[3, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[PolyLog[3, (d*(a + b*x))/(b*(c + d*x))], x]
) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.307802, size = 622, normalized size = 1.05 \[ \frac{-\text{PolyLog}\left (3,\frac{b c (d+e x)}{e (a c+b c x-1)}\right )+\text{PolyLog}\left (3,-\frac{b (d+e x)}{(a c+b c x-1) (b d-a e)}\right )+\log \left (-\frac{b (d+e x)}{(a c+b c x-1) (b d-a e)}\right ) \left (\text{PolyLog}\left (2,\frac{b c (d+e x)}{e (a c+b c x-1)}\right )-\text{PolyLog}\left (2,-\frac{b (d+e x)}{(a c+b c x-1) (b d-a e)}\right )\right )+\log (d+e x) \text{PolyLog}(2,c (a+b x))+\text{PolyLog}(2,-a c-b c x+1) \left (\log (d+e x)-\log \left (-\frac{b (d+e x)}{(a c+b c x-1) (b d-a e)}\right )\right )+\text{PolyLog}\left (2,\frac{b (d+e x)}{b d-a e}\right ) \left (\log \left (-\frac{b (d+e x)}{(a c+b c x-1) (b d-a e)}\right )+\log (-a c-b c x+1)\right )-\text{PolyLog}(3,-a c-b c x+1)-\text{PolyLog}\left (3,\frac{b (d+e x)}{b d-a e}\right )+\frac{1}{2} \left (-\log \left (\frac{(a+b x) (-a c e+b c d+e)}{(a c+b c x-1) (b d-a e)}\right )+\log \left (\frac{-a c e+b c d+e}{-a c e-b c e x+e}\right )+\log (c (a+b x))\right ) \log ^2\left (-\frac{b (d+e x)}{(a c+b c x-1) (b d-a e)}\right )+\log \left (\frac{b (d+e x)}{b d-a e}\right ) \left (\log \left (\frac{e (a+b x)}{a e-b d}\right )-\log (c (a+b x))\right ) \log \left (-\frac{b (d+e x)}{(a c+b c x-1) (b d-a e)}\right )+\log (d+e x) \log (c (a+b x)) \log (-a c-b c x+1)+\frac{1}{2} \log \left (\frac{b (d+e x)}{b d-a e}\right ) \left (\log (c (a+b x))-\log \left (\frac{e (a+b x)}{a e-b d}\right )\right ) \left (\log \left (\frac{b (d+e x)}{b d-a e}\right )-2 \log (-a c-b c x+1)\right )}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.401, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it polylog} \left ( 2,c \left ( bx+a \right ) \right ) }{ex+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,c*(b*x+a))/(e*x+d),x)

[Out]

int(polylog(2,c*(b*x+a))/(e*x+d),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(dilog((b*x + a)*c)/(e*x + d), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\rm Li}_2\left (b c x + a c\right )}{e x + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(dilog(b*c*x + a*c)/(e*x + d), 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(polylog(2,c*(b*x+a))/(e*x+d),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(dilog((b*x + a)*c)/(e*x + d), x)