∫ PolyLog(3,ax) ______________________

Optimal. Leaf size=70 \[ -\frac {a}{8 x}+\frac {1}{8} a^2 \log (x)-\frac {1}{8} a^2 \log (1-a x)+\frac {\log (1-a x)}{8 x^2}-\frac {\text {PolyLog}(2,a x)}{4 x^2}-\frac {\text {PolyLog}(3,a x)}{2 x^2} \]

-1/8*a/x+1/8*a^2*ln(x)-1/8*a^2*ln(-a*x+1)+1/8*ln(-a*x+1)/x^2-1/4*polylog(2,a*x)/x^2-1/2*polylog(3,a*x)/x^2

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Rubi [A]
time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6726, 2442, 46} \begin {gather*} \frac {1}{8} a^2 \log (x)-\frac {1}{8} a^2 \log (1-a x)-\frac {\text {Li}_2(a x)}{4 x^2}-\frac {\text {Li}_3(a x)}{2 x^2}+\frac {\log (1-a x)}{8 x^2}-\frac {a}{8 x} \end {gather*}

-1/8*a/x + (a^2*Log[x])/8 - (a^2*Log[1 - a*x])/8 + Log[1 - a*x]/(8*x^2) - PolyLog[2, a*x]/(4*x^2) - PolyLog[3,

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x

)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && Lt

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*

x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +

e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Int[((d_.)*(x_))^(m_.)*PolyLog[n_, (a_.)*((b_.)*(x_)^(p_.))^(q_.)], x_Symbol] :> Simp[(d*x)^(m + 1)*(PolyLog[n

, a*(b*x^p)^q]/(d*(m + 1))), x] - Dist[p*(q/(m + 1)), Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ

\begin {align*} \int \frac {\text {Li}_3(a x)}{x^3} \, dx &=-\frac {\text {Li}_3(a x)}{2 x^2}+\frac {1}{2} \int \frac {\text {Li}_2(a x)}{x^3} \, dx\\ &=-\frac {\text {Li}_2(a x)}{4 x^2}-\frac {\text {Li}_3(a x)}{2 x^2}-\frac {1}{4} \int \frac {\log (1-a x)}{x^3} \, dx\\ &=\frac {\log (1-a x)}{8 x^2}-\frac {\text {Li}_2(a x)}{4 x^2}-\frac {\text {Li}_3(a x)}{2 x^2}+\frac {1}{8} a \int \frac {1}{x^2 (1-a x)} \, dx\\ &=\frac {\log (1-a x)}{8 x^2}-\frac {\text {Li}_2(a x)}{4 x^2}-\frac {\text {Li}_3(a x)}{2 x^2}+\frac {1}{8} a \int \left (\frac {1}{x^2}+\frac {a}{x}-\frac {a^2}{-1+a x}\right ) \, dx\\ &=-\frac {a}{8 x}+\frac {1}{8} a^2 \log (x)-\frac {1}{8} a^2 \log (1-a x)+\frac {\log (1-a x)}{8 x^2}-\frac {\text {Li}_2(a x)}{4 x^2}-\frac {\text {Li}_3(a x)}{2 x^2}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 0.01, size = 25, normalized size = 0.36 \begin {gather*} \frac {G_{5,5}^{2,4}\left (-a x\left |\begin {array}{c} 1,1,1,1,3 \\ 1,2,0,0,0 \\\end {array}\right .\right )}{x^2} \end {gather*}

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method	result	size

meijerg	\(-a^{2} \left (-\frac {81 a x +378}{432 a x}-\frac {\left (-27 a^{2} x^{2}+27\right ) \ln \left (-a x +1\right )}{216 a^{2} x^{2}}+\frac {\polylog \left (2, a x \right )}{4 a^{2} x^{2}}+\frac {\polylog \left (3, a x \right )}{2 a^{2} x^{2}}+\frac {3}{16}-\frac {\ln \left (x \right )}{8}-\frac {\ln \left (-a \right )}{8}+\frac {1}{a x}\right )\)	\(90\)

-a^2*(-1/432/a/x*(81*a*x+378)-1/216/a^2/x^2*(-27*a^2*x^2+27)*ln(-a*x+1)+1/4/a^2/x^2*polylog(2,a*x)+1/2/a^2/x^2

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Maxima [A]
time = 0.25, size = 47, normalized size = 0.67 \begin {gather*} \frac {1}{8} \, a^{2} \log \left (x\right ) - \frac {a x + {\left (a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right ) + 2 \, {\rm Li}_2\left (a x\right ) + 4 \, {\rm Li}_{3}(a x)}{8 \, x^{2}} \end {gather*}

1/8*a^2*log(x) - 1/8*(a*x + (a^2*x^2 - 1)*log(-a*x + 1) + 2*dilog(a*x) + 4*polylog(3, a*x))/x^2

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Fricas [A]
time = 0.36, size = 54, normalized size = 0.77 \begin {gather*} -\frac {a^{2} x^{2} \log \left (a x - 1\right ) - a^{2} x^{2} \log \left (x\right ) + a x + 2 \, {\rm Li}_2\left (a x\right ) - \log \left (-a x + 1\right ) + 4 \, {\rm polylog}\left (3, a x\right )}{8 \, x^{2}} \end {gather*}

-1/8*(a^2*x^2*log(a*x - 1) - a^2*x^2*log(x) + a*x + 2*dilog(a*x) - log(-a*x + 1) + 4*polylog(3, a*x))/x^2

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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Mupad [B]
time = 1.27, size = 46, normalized size = 0.66 \begin {gather*} \frac {a^2\,\mathrm {atanh}\left (2\,a\,x-1\right )}{4}-\frac {\frac {a\,x}{8}-\frac {\ln \left (1-a\,x\right )}{8}+\frac {\mathrm {polylog}\left (2,a\,x\right )}{4}+\frac {\mathrm {polylog}\left (3,a\,x\right )}{2}}{x^2} \end {gather*}

(a^2*atanh(2*a*x - 1))/4 - ((a*x)/8 - log(1 - a*x)/8 + polylog(2, a*x)/4 + polylog(3, a*x)/2)/x^2

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3.1.17 \(\int \frac {\text {PolyLog}(3,a x)}{x^3} \, dx\) [17]