3.1.14 \(\int \text {PolyLog}(3,a x) \, dx\) [14]

Optimal. Leaf size=34 \[ x+\frac {(1-a x) \log (1-a x)}{a}-x \text {PolyLog}(2,a x)+x \text {PolyLog}(3,a x) \]

[Out]

x+(-a*x+1)*ln(-a*x+1)/a-x*polylog(2,a*x)+x*polylog(3,a*x)

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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6721, 2436, 2332} \begin {gather*} x (-\text {Li}_2(a x))+x \text {Li}_3(a x)+\frac {(1-a x) \log (1-a x)}{a}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[PolyLog[3, a*x],x]

[Out]

x + ((1 - a*x)*Log[1 - a*x])/a - x*PolyLog[2, a*x] + x*PolyLog[3, a*x]

Rule 2332

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

Rule 2436

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 6721

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

Rubi steps

\begin {align*} \int \text {Li}_3(a x) \, dx &=x \text {Li}_3(a x)-\int \text {Li}_2(a x) \, dx\\ &=-x \text {Li}_2(a x)+x \text {Li}_3(a x)-\int \log (1-a x) \, dx\\ &=-x \text {Li}_2(a x)+x \text {Li}_3(a x)+\frac {\text {Subst}(\int \log (x) \, dx,x,1-a x)}{a}\\ &=x+\frac {(1-a x) \log (1-a x)}{a}-x \text {Li}_2(a x)+x \text {Li}_3(a x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 39, normalized size = 1.15 \begin {gather*} x \left (1-\log (1-a x)+\frac {\log (1-a x)}{a x}-\text {PolyLog}(2,a x)+\text {PolyLog}(3,a x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[3, a*x],x]

[Out]

x*(1 - Log[1 - a*x] + Log[1 - a*x]/(a*x) - PolyLog[2, a*x] + PolyLog[3, a*x])

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Maple [A]
time = 0.09, size = 41, normalized size = 1.21

method	result	size
meijerg	\(\frac {a x +\frac {\left (-2 a x +2\right ) \ln \left (-a x +1\right )}{2}-a x \polylog \left (2, a x \right )+a x \polylog \left (3, a x \right )}{a}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3,a*x),x,method=_RETURNVERBOSE)

[Out]

1/a*(a*x+1/2*(-2*a*x+2)*ln(-a*x+1)-a*x*polylog(2,a*x)+a*x*polylog(3,a*x))

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Maxima [A]
time = 0.27, size = 39, normalized size = 1.15 \begin {gather*} -\frac {a x {\rm Li}_2\left (a x\right ) - a x {\rm Li}_{3}(a x) - a x + {\left (a x - 1\right )} \log \left (-a x + 1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x),x, algorithm="maxima")

[Out]

-(a*x*dilog(a*x) - a*x*polylog(3, a*x) - a*x + (a*x - 1)*log(-a*x + 1))/a

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Fricas [A]
time = 0.33, size = 39, normalized size = 1.15 \begin {gather*} -\frac {a x {\rm Li}_2\left (a x\right ) - a x {\rm polylog}\left (3, a x\right ) - a x + {\left (a x - 1\right )} \log \left (-a x + 1\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x),x, algorithm="fricas")

[Out]

-(a*x*dilog(a*x) - a*x*polylog(3, a*x) - a*x + (a*x - 1)*log(-a*x + 1))/a

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {Li}_{3}\left (a x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(3,a*x),x)

[Out]

Integral(polylog(3, a*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(3,a*x),x, algorithm="giac")

[Out]

integrate(polylog(3, a*x), x)

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Mupad [B]
time = 0.84, size = 37, normalized size = 1.09 \begin {gather*} x+\frac {\ln \left (a\,x-1\right )}{a}-x\,\mathrm {polylog}\left (2,a\,x\right )+x\,\mathrm {polylog}\left (3,a\,x\right )-x\,\ln \left (1-a\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3, a*x),x)

[Out]

x + log(a*x - 1)/a - x*polylog(2, a*x) + x*polylog(3, a*x) - x*log(1 - a*x)

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