∫ PolyLog(3,ax) ______________________

Optimal. Leaf size=80 \[ -\frac {a}{54 x^2}-\frac {a^2}{27 x}+\frac {1}{27} a^3 \log (x)-\frac {1}{27} a^3 \log (1-a x)+\frac {\log (1-a x)}{27 x^3}-\frac {\text {PolyLog}(2,a x)}{9 x^3}-\frac {\text {PolyLog}(3,a x)}{3 x^3} \]

-1/54*a/x^2-1/27*a^2/x+1/27*a^3*ln(x)-1/27*a^3*ln(-a*x+1)+1/27*ln(-a*x+1)/x^3-1/9*polylog(2,a*x)/x^3-1/3*polyl

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 80, 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}{27} a^3 \log (x)-\frac {1}{27} a^3 \log (1-a x)-\frac {a^2}{27 x}-\frac {\text {Li}_2(a x)}{9 x^3}-\frac {\text {Li}_3(a x)}{3 x^3}+\frac {\log (1-a x)}{27 x^3}-\frac {a}{54 x^2} \end {gather*}

-1/54*a/x^2 - a^2/(27*x) + (a^3*Log[x])/27 - (a^3*Log[1 - a*x])/27 + Log[1 - a*x]/(27*x^3) - PolyLog[2, a*x]/(

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^4} \, dx &=-\frac {\text {Li}_3(a x)}{3 x^3}+\frac {1}{3} \int \frac {\text {Li}_2(a x)}{x^4} \, dx\\ &=-\frac {\text {Li}_2(a x)}{9 x^3}-\frac {\text {Li}_3(a x)}{3 x^3}-\frac {1}{9} \int \frac {\log (1-a x)}{x^4} \, dx\\ &=\frac {\log (1-a x)}{27 x^3}-\frac {\text {Li}_2(a x)}{9 x^3}-\frac {\text {Li}_3(a x)}{3 x^3}+\frac {1}{27} a \int \frac {1}{x^3 (1-a x)} \, dx\\ &=\frac {\log (1-a x)}{27 x^3}-\frac {\text {Li}_2(a x)}{9 x^3}-\frac {\text {Li}_3(a x)}{3 x^3}+\frac {1}{27} a \int \left (\frac {1}{x^3}+\frac {a}{x^2}+\frac {a^2}{x}-\frac {a^3}{-1+a x}\right ) \, dx\\ &=-\frac {a}{54 x^2}-\frac {a^2}{27 x}+\frac {1}{27} a^3 \log (x)-\frac {1}{27} a^3 \log (1-a x)+\frac {\log (1-a x)}{27 x^3}-\frac {\text {Li}_2(a x)}{9 x^3}-\frac {\text {Li}_3(a x)}{3 x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 0.01, size = 25, normalized size = 0.31 \begin {gather*} \frac {G_{5,5}^{2,4}\left (-a x\left |\begin {array}{c} 1,1,1,1,4 \\ 1,3,0,0,0 \\\end {array}\right .\right )}{x^3} \end {gather*}

________________________________________________________________________________________


method	result	size

meijerg	\(a^{3} \left (\frac {64 a^{2} x^{2}+152 a x +832}{1728 a^{2} x^{2}}+\frac {\left (-64 a^{3} x^{3}+64\right ) \ln \left (-a x +1\right )}{1728 a^{3} x^{3}}-\frac {\polylog \left (2, a x \right )}{9 a^{3} x^{3}}-\frac {\polylog \left (3, a x \right )}{3 a^{3} x^{3}}-\frac {1}{27}+\frac {\ln \left (x \right )}{27}+\frac {\ln \left (-a \right )}{27}-\frac {1}{2 a^{2} x^{2}}-\frac {1}{8 a x}\right )\)	\(106\)

a^3*(1/1728/a^2/x^2*(64*a^2*x^2+152*a*x+832)+1/1728/a^3/x^3*(-64*a^3*x^3+64)*ln(-a*x+1)-1/9/a^3/x^3*polylog(2,

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 56, normalized size = 0.70 \begin {gather*} \frac {1}{27} \, a^{3} \log \left (x\right ) - \frac {2 \, a^{2} x^{2} + a x + 2 \, {\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right ) + 6 \, {\rm Li}_2\left (a x\right ) + 18 \, {\rm Li}_{3}(a x)}{54 \, x^{3}} \end {gather*}

1/27*a^3*log(x) - 1/54*(2*a^2*x^2 + a*x + 2*(a^3*x^3 - 1)*log(-a*x + 1) + 6*dilog(a*x) + 18*polylog(3, a*x))/x

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 63, normalized size = 0.79 \begin {gather*} -\frac {2 \, a^{3} x^{3} \log \left (a x - 1\right ) - 2 \, a^{3} x^{3} \log \left (x\right ) + 2 \, a^{2} x^{2} + a x + 6 \, {\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right ) + 18 \, {\rm polylog}\left (3, a x\right )}{54 \, x^{3}} \end {gather*}

-1/54*(2*a^3*x^3*log(a*x - 1) - 2*a^3*x^3*log(x) + 2*a^2*x^2 + a*x + 6*dilog(a*x) - 2*log(-a*x + 1) + 18*polyl

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {Li}_{3}\left (a x\right )}{x^{4}}\, dx \end {gather*}

________________________________________________________________________________________

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

________________________________________________________________________________________

Mupad [B]
time = 1.50, size = 62, normalized size = 0.78 \begin {gather*} \frac {\ln \left (1-a\,x\right )}{27\,x^3}-\frac {\mathrm {polylog}\left (2,a\,x\right )}{9\,x^3}-\frac {\mathrm {polylog}\left (3,a\,x\right )}{3\,x^3}-\frac {x\,a^2+\frac {a}{2}}{27\,x^2}-\frac {a^3\,\mathrm {atan}\left (a\,x\,2{}\mathrm {i}-\mathrm {i}\right )\,2{}\mathrm {i}}{27} \end {gather*}

log(1 - a*x)/(27*x^3) - (a^3*atan(a*x*2i - 1i)*2i)/27 - polylog(2, a*x)/(9*x^3) - polylog(3, a*x)/(3*x^3) - (a

________________________________________________________________________________________

3.1.18 \(\int \frac {\text {PolyLog}(3,a x)}{x^4} \, dx\) [18]