3.102 \(\int \frac{\text{PolyLog}(2,1+\frac{b c-a d}{d (a+b x)})}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=35 \[ -\frac{\text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right )}{b c-a d} \]

[Out]

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

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Rubi [A]  time = 0.0638609, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {6610} \[ -\frac{\text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right )}{b c-a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

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

Mathematica [A]  time = 0.0115641, size = 30, normalized size = 0.86 \[ \frac{\text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{a d-b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

PolyLog[3, (b*(c + d*x))/(d*(a + b*x))]/(-(b*c) + a*d)

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Maple [A]  time = 0.062, size = 36, normalized size = 1. \begin{align*}{\frac{1}{ad-bc}{\it polylog} \left ( 3,1-{\frac{ad-bc}{d \left ( bx+a \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a*d-b*c)*polylog(3,1-(a*d-b*c)/d/(b*x+a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,1+(-a*d+b*c)/d/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

integrate(dilog((b*c - a*d)/((b*x + a)*d) + 1)/((b*x + a)*(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,1+(-a*d+b*c)/d/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

integral(dilog((b*c - a*d)/(b*d*x + a*d) + 1)/(b*d*x^2 + a*c + (b*c + a*d)*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(polylog(2,1+(-a*d+b*c)/d/(b*x+a))/(b*x+a)/(d*x+c),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 (\frac{b c - a d}{{\left (b x + a\right )} d} + 1\right )}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,1+(-a*d+b*c)/d/(b*x+a))/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

integrate(dilog((b*c - a*d)/((b*x + a)*d) + 1)/((b*x + a)*(d*x + c)), x)