Integrand size = 9, antiderivative size = 80 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{x^4} \, dx=-\frac {a}{54 x^2}-\frac {a^2}{27 x}+\frac {1}{27} a^3 \log (x)-\frac {1}{27} a^3 \log (1-a x)+\frac {\log (1-a x)}{27 x^3}-\frac {\operatorname {PolyLog}(2,a x)}{9 x^3}-\frac {\operatorname {PolyLog}(3,a x)}{3 x^3} \] Output:
-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*polylog(3,a*x)/x^3
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.31 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{x^4} \, dx=\frac {G_{5,5}^{2,4}\left (-a x\left |\begin {array}{c} 1,1,1,1,4 \\ 1,3,0,0,0 \\\end {array}\right .\right )}{x^3} \] Input:
Integrate[PolyLog[3, a*x]/x^4,x]
Output:
MeijerG[{{1, 1, 1, 1}, {4}}, {{1, 3}, {0, 0, 0}}, -(a*x)]/x^3
Time = 0.32 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {7145, 7145, 25, 2842, 54, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\operatorname {PolyLog}(3,a x)}{x^4} \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {1}{3} \int \frac {\operatorname {PolyLog}(2,a x)}{x^4}dx-\frac {\operatorname {PolyLog}(3,a x)}{3 x^3}\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \int -\frac {\log (1-a x)}{x^4}dx-\frac {\operatorname {PolyLog}(2,a x)}{3 x^3}\right )-\frac {\operatorname {PolyLog}(3,a x)}{3 x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} \left (-\frac {1}{3} \int \frac {\log (1-a x)}{x^4}dx-\frac {\operatorname {PolyLog}(2,a x)}{3 x^3}\right )-\frac {\operatorname {PolyLog}(3,a x)}{3 x^3}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {1}{3} a \int \frac {1}{x^3 (1-a x)}dx+\frac {\log (1-a x)}{3 x^3}\right )-\frac {\operatorname {PolyLog}(2,a x)}{3 x^3}\right )-\frac {\operatorname {PolyLog}(3,a x)}{3 x^3}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {1}{3} a \int \left (-\frac {a^3}{a x-1}+\frac {a^2}{x}+\frac {a}{x^2}+\frac {1}{x^3}\right )dx+\frac {\log (1-a x)}{3 x^3}\right )-\frac {\operatorname {PolyLog}(2,a x)}{3 x^3}\right )-\frac {\operatorname {PolyLog}(3,a x)}{3 x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{3} \left (\frac {1}{3} a \left (a^2 \log (x)-a^2 \log (1-a x)-\frac {a}{x}-\frac {1}{2 x^2}\right )+\frac {\log (1-a x)}{3 x^3}\right )-\frac {\operatorname {PolyLog}(2,a x)}{3 x^3}\right )-\frac {\operatorname {PolyLog}(3,a x)}{3 x^3}\) |
Input:
Int[PolyLog[3, a*x]/x^4,x]
Output:
((Log[1 - a*x]/(3*x^3) + (a*(-1/2*1/x^2 - a/x + a^2*Log[x] - a^2*Log[1 - a *x]))/3)/3 - PolyLog[2, a*x]/(3*x^3))/3 - PolyLog[3, a*x]/(3*x^3)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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] - Simp[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_Symbo l] :> Simp[(d*x)^(m + 1)*(PolyLog[n, a*(b*x^p)^q]/(d*(m + 1))), x] - Simp[p *(q/(m + 1)) Int[(d*x)^m*PolyLog[n - 1, a*(b*x^p)^q], x], x] /; FreeQ[{a, b, d, m, p, q}, x] && NeQ[m, -1] && GtQ[n, 0]
Time = 0.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.32
method | result | size |
meijerg | \(a^{3} \left (-\frac {1}{2 x^{2} a^{2}}-\frac {1}{8 a x}-\frac {1}{27}+\frac {\ln \left (x \right )}{27}+\frac {\ln \left (-a \right )}{27}+\frac {64 a^{2} x^{2}+152 x a +832}{1728 x^{2} a^{2}}+\frac {\left (-64 x^{3} a^{3}+64\right ) \ln \left (-x a +1\right )}{1728 x^{3} a^{3}}-\frac {\operatorname {polylog}\left (2, x a \right )}{9 x^{3} a^{3}}-\frac {\operatorname {polylog}\left (3, x a \right )}{3 x^{3} a^{3}}\right )\) | \(106\) |
Input:
int(polylog(3,x*a)/x^4,x,method=_RETURNVERBOSE)
Output:
a^3*(-1/2/x^2/a^2-1/8/a/x-1/27+1/27*ln(x)+1/27*ln(-a)+1/1728/x^2/a^2*(64*a ^2*x^2+152*a*x+832)+1/1728/x^3/a^3*(-64*a^3*x^3+64)*ln(-a*x+1)-1/9/x^3/a^3 *polylog(2,x*a)-1/3/x^3/a^3*polylog(3,x*a))
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{x^4} \, dx=-\frac {2 \, a^{3} x^{3} \log \left (a x - 1\right ) - 2 \, a^{3} x^{3} \log \left (x\right ) + 2 \, a^{2} x^{2} + a x + 6 \, {\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right ) + 18 \, {\rm polylog}\left (3, a x\right )}{54 \, x^{3}} \] Input:
integrate(polylog(3,a*x)/x^4,x, algorithm="fricas")
Output:
-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*dil og(a*x) - 2*log(-a*x + 1) + 18*polylog(3, a*x))/x^3
\[ \int \frac {\operatorname {PolyLog}(3,a x)}{x^4} \, dx=\int \frac {\operatorname {Li}_{3}\left (a x\right )}{x^{4}}\, dx \] Input:
integrate(polylog(3,a*x)/x**4,x)
Output:
Integral(polylog(3, a*x)/x**4, x)
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.70 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{x^4} \, dx=\frac {1}{27} \, a^{3} \log \left (x\right ) - \frac {2 \, a^{2} x^{2} + a x + 2 \, {\left (a^{3} x^{3} - 1\right )} \log \left (-a x + 1\right ) + 6 \, {\rm Li}_2\left (a x\right ) + 18 \, {\rm Li}_{3}(a x)}{54 \, x^{3}} \] Input:
integrate(polylog(3,a*x)/x^4,x, algorithm="maxima")
Output:
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^3
\[ \int \frac {\operatorname {PolyLog}(3,a x)}{x^4} \, dx=\int { \frac {{\rm Li}_{3}(a x)}{x^{4}} \,d x } \] Input:
integrate(polylog(3,a*x)/x^4,x, algorithm="giac")
Output:
integrate(polylog(3, a*x)/x^4, x)
Time = 0.89 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.78 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{x^4} \, dx=\frac {\ln \left (1-a\,x\right )}{27\,x^3}-\frac {\mathrm {polylog}\left (2,a\,x\right )}{9\,x^3}-\frac {\mathrm {polylog}\left (3,a\,x\right )}{3\,x^3}-\frac {x\,a^2+\frac {a}{2}}{27\,x^2}-\frac {a^3\,\mathrm {atan}\left (a\,x\,2{}\mathrm {i}-\mathrm {i}\right )\,2{}\mathrm {i}}{27} \] Input:
int(polylog(3, a*x)/x^4,x)
Output:
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/2 + a^2*x)/(27*x^2)
\[ \int \frac {\operatorname {PolyLog}(3,a x)}{x^4} \, dx=\int \frac {\mathit {polylog}\left (3, a x \right )}{x^{4}}d x \] Input:
int(polylog(3,a*x)/x^4,x)
Output:
int(polylog(3,a*x)/x**4,x)