3.1.5 \(\int \text {PolyLog}(2,a x) \, dx\) [5]

Optimal. Leaf size=29 \[ -x-\frac {(1-a x) \log (1-a x)}{a}+x \text {PolyLog}(2,a x) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, 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)-\frac {(1-a x) \log (1-a x)}{a}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[PolyLog[2, a*x],x]

[Out]

-x - ((1 - a*x)*Log[1 - a*x])/a + x*PolyLog[2, 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}_2(a x) \, dx &=x \text {Li}_2(a x)+\int \log (1-a x) \, dx\\ &=x \text {Li}_2(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)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.90 \begin {gather*} -x+\left (-\frac {1}{a}+x\right ) \log (1-a x)+x \text {PolyLog}(2,a x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[PolyLog[2, a*x],x]

[Out]

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

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Maple [A]
time = 0.29, size = 34, normalized size = 1.17

method	result	size
meijerg	\(\frac {-a x -\frac {\left (-2 a x +2\right ) \ln \left (-a x +1\right )}{2}+a x \polylog \left (2, a x \right )}{a}\)	\(33\)
derivativedivides	\(\frac {a x \polylog \left (2, a x \right )-\ln \left (-a x +1\right ) \left (-a x +1\right )+1-a x}{a}\)	\(34\)
default	\(\frac {a x \polylog \left (2, a x \right )-\ln \left (-a x +1\right ) \left (-a x +1\right )+1-a x}{a}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2,a*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 29, normalized size = 1.00 \begin {gather*} \frac {a x {\rm Li}_2\left (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(2,a*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 29, normalized size = 1.00 \begin {gather*} \frac {a x {\rm Li}_2\left (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(2,a*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.25, size = 22, normalized size = 0.76 \begin {gather*} \begin {cases} - x \operatorname {Li}_{1}\left (a x\right ) + x \operatorname {Li}_{2}\left (a x\right ) - x + \frac {\operatorname {Li}_{1}\left (a x\right )}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(polylog(2,a*x),x)

[Out]

Piecewise((-x*polylog(1, a*x) + x*polylog(2, a*x) - x + polylog(1, a*x)/a, Ne(a, 0)), (0, True))

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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,a*x),x, algorithm="giac")

[Out]

integrate(dilog(a*x), x)

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Mupad [B]
time = 0.26, size = 32, normalized size = 1.10 \begin {gather*} x\,\mathrm {polylog}\left (2,a\,x\right )-\frac {\ln \left (1-a\,x\right )}{a}-x+x\,\ln \left (1-a\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(2, a*x),x)

[Out]

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

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