3.2.41 \(\int \frac {\text {PolyLog}(2,c (a+b x))}{d+e x} \, dx\) [141]

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

[Out]

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

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Rubi [A]
time = 0.35, antiderivative size = 591, normalized size of antiderivative = 1.00, 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 = {6732, 2490, 2485} \begin {gather*} -\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}-\frac {\text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right ) \log \left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )}{e}+\frac {\text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right ) \log \left (\frac {b (d+e x)}{(1-c (a+b x)) (b d-a e)}\right )}{e}+\frac {\log (d+e x) \text {Li}_2(c (a+b x))}{e}+\frac {\text {Li}_2\left (\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 {Li}_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 {\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}-\frac {\text {Li}_3(1-c (a+b x))}{e}-\frac {\text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{e} \end {gather*}

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 2485

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

Rule 2490

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 6732

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]

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 {\text {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*}

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Mathematica [A]
time = 0.17, size = 622, normalized size = 1.05 \begin {gather*} \frac {\log (c (a+b x)) \log (1-a c-b c x) \log (d+e x)+\frac {1}{2} \left (\log (c (a+b x))-\log \left (\frac {e (a+b x)}{-b d+a e}\right )\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right ) \left (-2 \log (1-a c-b c x)+\log \left (\frac {b (d+e x)}{b d-a e}\right )\right )+\left (-\log (c (a+b x))+\log \left (\frac {e (a+b x)}{-b d+a e}\right )\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right ) \log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )+\frac {1}{2} \log ^2\left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right ) \left (\log (c (a+b x))-\log \left (\frac {(b c d+e-a c e) (a+b x)}{(b d-a e) (-1+a c+b c x)}\right )+\log \left (\frac {b c d+e-a c e}{e-a c e-b c e x}\right )\right )+\log (d+e x) \text {PolyLog}(2,c (a+b x))+\left (\log (d+e x)-\log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right ) \text {PolyLog}(2,1-a c-b c x)+\left (\log (1-a c-b c x)+\log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right ) \text {PolyLog}\left (2,\frac {b (d+e x)}{b d-a e}\right )+\log \left (-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right ) \left (\text {PolyLog}\left (2,\frac {b c (d+e x)}{e (-1+a c+b c x)}\right )-\text {PolyLog}\left (2,-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )\right )-\text {PolyLog}(3,1-a c-b c x)-\text {PolyLog}\left (3,\frac {b (d+e x)}{b d-a e}\right )-\text {PolyLog}\left (3,\frac {b c (d+e x)}{e (-1+a c+b c x)}\right )+\text {PolyLog}\left (3,-\frac {b (d+e x)}{(b d-a e) (-1+a c+b c x)}\right )}{e} \end {gather*}

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 = 0.82, size = 0, normalized size = 0.00 \[\int \frac {\polylog \left (2, c \left (b x +a \right )\right )}{e x +d}\, dx\]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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