3.140 \(\int \text{PolyLog}(2,c (a+b x)) \, dx\)

Optimal. Leaf size=60 \[ x \text{PolyLog}(2,c (a+b x))+\frac{a \text{PolyLog}(2,c (a+b x))}{b}-\frac{(-a c-b c x+1) \log (-a c-b c x+1)}{b c}-x \]

[Out]

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

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Rubi [A]  time = 0.0519429, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778, Rules used = {6595, 2444, 2389, 2295, 2421, 2393, 2391} \[ x \text{PolyLog}(2,c (a+b x))+\frac{a \text{PolyLog}(2,c (a+b x))}{b}-\frac{(-a c-b c x+1) \log (-a c-b c x+1)}{b c}-x \]

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, c*(a + b*x)],x]

[Out]

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

Rule 6595

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)], x_Symbol] :> Simp[x*PolyLog[n, c*(a + b*x)^p], x] + (-Dist[
p, Int[PolyLog[n - 1, c*(a + b*x)^p], x], x] + Dist[a*p, Int[PolyLog[n - 1, c*(a + b*x)^p]/(a + b*x), x], x])
/; FreeQ[{a, b, c, p}, x] && GtQ[n, 0]

Rule 2444

Int[((a_.) + Log[(c_.)*(v_)^(n_.)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Int[u*(a + b*Log[c*ExpandToSum[v, x]^n])^p
, x] /; FreeQ[{a, b, c, n, p}, x] && LinearQ[v, x] &&  !LinearMatchQ[v, x] &&  !(EqQ[n, 1] && MatchQ[c*v, (e_.
)*((f_) + (g_.)*x) /; FreeQ[{e, f, g}, x]])

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2421

Int[((a_.) + Log[(c_.)*(v_)^(n_.)]*(b_.))^(p_.)*(u_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^q*(a + b*Log[c*
ExpandToSum[v, x]^n])^p, x] /; FreeQ[{a, b, c, n, p, q}, x] && BinomialQ[u, x] && LinearQ[v, x] &&  !(Binomial
MatchQ[u, x] && LinearMatchQ[v, x])

Rule 2393

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

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

Mathematica [A]  time = 0.0160818, size = 53, normalized size = 0.88 \[ \frac{c (a+b x) \text{PolyLog}(2,c (a+b x))-c (a+b x)+(c (a+b x)-1) \log (1-c (a+b x))}{b c} \]

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, c*(a + b*x)],x]

[Out]

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

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Maple [A]  time = 0.006, size = 96, normalized size = 1.6 \begin{align*} \ln \left ( -xbc-ac+1 \right ) x+{\it polylog} \left ( 2,xbc+ac \right ) x+{\frac{\ln \left ( -xbc-ac+1 \right ) a}{b}}+{\frac{{\it polylog} \left ( 2,xbc+ac \right ) a}{b}}-x-{\frac{a}{b}}-{\frac{\ln \left ( -xbc-ac+1 \right ) }{bc}}+{\frac{1}{bc}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00266, size = 122, normalized size = 2.03 \begin{align*} -\frac{{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) +{\rm Li}_2\left (-b c x - a c + 1\right )\right )} a}{b} + \frac{b c x{\rm Li}_2\left (b c x + a c\right ) - b c x +{\left (b c x + a c - 1\right )} \log \left (-b c x - a c + 1\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(b*c*x + a*c)*log(-b*c*x - a*c + 1) + dilog(-b*c*x - a*c + 1))*a/b + (b*c*x*dilog(b*c*x + a*c) - b*c*x +
(b*c*x + a*c - 1)*log(-b*c*x - a*c + 1))/(b*c)

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Fricas [A]  time = 2.43648, size = 126, normalized size = 2.1 \begin{align*} -\frac{b c x -{\left (b c x + a c\right )}{\rm Li}_2\left (b c x + a c\right ) -{\left (b c x + a c - 1\right )} \log \left (-b c x - a c + 1\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*c*x - (b*c*x + a*c)*dilog(b*c*x + a*c) - (b*c*x + a*c - 1)*log(-b*c*x - a*c + 1))/(b*c)

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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)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(dilog((b*x + a)*c), x)