Integrand size = 11, antiderivative size = 77 \[ \int x^2 \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=\frac {8 x}{27 a}+\frac {8 x^3}{81}-\frac {8 \text {arctanh}\left (\sqrt {a} x\right )}{27 a^{3/2}}-\frac {4}{27} x^3 \log \left (1-a x^2\right )-\frac {2}{9} x^3 \operatorname {PolyLog}\left (2,a x^2\right )+\frac {1}{3} x^3 \operatorname {PolyLog}\left (3,a x^2\right ) \]
8/27*x/a+8/81*x^3-8/27*arctanh(x*a^(1/2))/a^(3/2)-4/27*x^3*ln(-a*x^2+1)-2/ 9*x^3*polylog(2,a*x^2)+1/3*x^3*polylog(3,a*x^2)
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90 \[ \int x^2 \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=\frac {1}{81} \left (\frac {24 x}{a}+8 x^3-\frac {24 \text {arctanh}\left (\sqrt {a} x\right )}{a^{3/2}}-12 x^3 \log \left (1-a x^2\right )-18 x^3 \operatorname {PolyLog}\left (2,a x^2\right )+27 x^3 \operatorname {PolyLog}\left (3,a x^2\right )\right ) \]
((24*x)/a + 8*x^3 - (24*ArcTanh[Sqrt[a]*x])/a^(3/2) - 12*x^3*Log[1 - a*x^2 ] - 18*x^3*PolyLog[2, a*x^2] + 27*x^3*PolyLog[3, a*x^2])/81
Time = 0.32 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {7145, 7145, 25, 2905, 254, 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.
\(\displaystyle \int x^2 \operatorname {PolyLog}\left (3,a x^2\right ) \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {1}{3} x^3 \operatorname {PolyLog}\left (3,a x^2\right )-\frac {2}{3} \int x^2 \operatorname {PolyLog}\left (2,a x^2\right )dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {1}{3} x^3 \operatorname {PolyLog}\left (3,a x^2\right )-\frac {2}{3} \left (\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^2\right )-\frac {2}{3} \int -x^2 \log \left (1-a x^2\right )dx\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} x^3 \operatorname {PolyLog}\left (3,a x^2\right )-\frac {2}{3} \left (\frac {2}{3} \int x^2 \log \left (1-a x^2\right )dx+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^2\right )\right )\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle \frac {1}{3} x^3 \operatorname {PolyLog}\left (3,a x^2\right )-\frac {2}{3} \left (\frac {2}{3} \left (\frac {2}{3} a \int \frac {x^4}{1-a x^2}dx+\frac {1}{3} x^3 \log \left (1-a x^2\right )\right )+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^2\right )\right )\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {1}{3} x^3 \operatorname {PolyLog}\left (3,a x^2\right )-\frac {2}{3} \left (\frac {2}{3} \left (\frac {2}{3} a \int \left (-\frac {x^2}{a}+\frac {1}{a^2 \left (1-a x^2\right )}-\frac {1}{a^2}\right )dx+\frac {1}{3} x^3 \log \left (1-a x^2\right )\right )+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^2\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} x^3 \operatorname {PolyLog}\left (3,a x^2\right )-\frac {2}{3} \left (\frac {2}{3} \left (\frac {2}{3} a \left (\frac {\text {arctanh}\left (\sqrt {a} x\right )}{a^{5/2}}-\frac {x}{a^2}-\frac {x^3}{3 a}\right )+\frac {1}{3} x^3 \log \left (1-a x^2\right )\right )+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^2\right )\right )\) |
(-2*((2*((2*a*(-(x/a^2) - x^3/(3*a) + ArcTanh[Sqrt[a]*x]/a^(5/2)))/3 + (x^ 3*Log[1 - a*x^2])/3))/3 + (x^3*PolyLog[2, a*x^2])/3))/3 + (x^3*PolyLog[3, a*x^2])/3
3.1.40.3.1 Defintions of rubi rules used
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^ (m_.), x_Symbol] :> Simp[(f*x)^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Simp[b*e*n*(p/(f*(m + 1))) Int[x^(n - 1)*((f*x)^(m + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -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]
Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs. \(2(61)=122\).
Time = 0.19 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.77
method | result | size |
meijerg | \(\frac {\frac {2 x \left (-a \right )^{\frac {5}{2}} \left (40 a \,x^{2}+120\right )}{405 a^{2}}+\frac {8 x \left (-a \right )^{\frac {5}{2}} \left (\ln \left (1-\sqrt {a \,x^{2}}\right )-\ln \left (1+\sqrt {a \,x^{2}}\right )\right )}{27 a^{2} \sqrt {a \,x^{2}}}-\frac {8 x^{3} \left (-a \right )^{\frac {5}{2}} \ln \left (-a \,x^{2}+1\right )}{27 a}-\frac {4 x^{3} \left (-a \right )^{\frac {5}{2}} \operatorname {polylog}\left (2, a \,x^{2}\right )}{9 a}+\frac {2 x^{3} \left (-a \right )^{\frac {5}{2}} \operatorname {polylog}\left (3, a \,x^{2}\right )}{3 a}}{2 a \sqrt {-a}}\) | \(136\) |
1/2/a/(-a)^(1/2)*(2/405*x*(-a)^(5/2)*(40*a*x^2+120)/a^2+8/27*x*(-a)^(5/2)/ a^2/(a*x^2)^(1/2)*(ln(1-(a*x^2)^(1/2))-ln(1+(a*x^2)^(1/2)))-8/27*x^3*(-a)^ (5/2)*ln(-a*x^2+1)/a-4/9*x^3*(-a)^(5/2)/a*polylog(2,a*x^2)+2/3*x^3*(-a)^(5 /2)/a*polylog(3,a*x^2))
Time = 0.26 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.25 \[ \int x^2 \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=\left [-\frac {18 \, a^{2} x^{3} {\rm Li}_2\left (a x^{2}\right ) + 12 \, a^{2} x^{3} \log \left (-a x^{2} + 1\right ) - 27 \, a^{2} x^{3} {\rm polylog}\left (3, a x^{2}\right ) - 8 \, a^{2} x^{3} - 24 \, a x - 12 \, \sqrt {a} \log \left (\frac {a x^{2} - 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right )}{81 \, a^{2}}, -\frac {18 \, a^{2} x^{3} {\rm Li}_2\left (a x^{2}\right ) + 12 \, a^{2} x^{3} \log \left (-a x^{2} + 1\right ) - 27 \, a^{2} x^{3} {\rm polylog}\left (3, a x^{2}\right ) - 8 \, a^{2} x^{3} - 24 \, a x - 24 \, \sqrt {-a} \arctan \left (\sqrt {-a} x\right )}{81 \, a^{2}}\right ] \]
[-1/81*(18*a^2*x^3*dilog(a*x^2) + 12*a^2*x^3*log(-a*x^2 + 1) - 27*a^2*x^3* polylog(3, a*x^2) - 8*a^2*x^3 - 24*a*x - 12*sqrt(a)*log((a*x^2 - 2*sqrt(a) *x + 1)/(a*x^2 - 1)))/a^2, -1/81*(18*a^2*x^3*dilog(a*x^2) + 12*a^2*x^3*log (-a*x^2 + 1) - 27*a^2*x^3*polylog(3, a*x^2) - 8*a^2*x^3 - 24*a*x - 24*sqrt (-a)*arctan(sqrt(-a)*x))/a^2]
\[ \int x^2 \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=\int x^{2} \operatorname {Li}_{3}\left (a x^{2}\right )\, dx \]
Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.05 \[ \int x^2 \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=-\frac {18 \, a x^{3} {\rm Li}_2\left (a x^{2}\right ) + 12 \, a x^{3} \log \left (-a x^{2} + 1\right ) - 27 \, a x^{3} {\rm Li}_{3}(a x^{2}) - 8 \, a x^{3} - 24 \, x}{81 \, a} + \frac {4 \, \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right )}{27 \, a^{\frac {3}{2}}} \]
-1/81*(18*a*x^3*dilog(a*x^2) + 12*a*x^3*log(-a*x^2 + 1) - 27*a*x^3*polylog (3, a*x^2) - 8*a*x^3 - 24*x)/a + 4/27*log((a*x - sqrt(a))/(a*x + sqrt(a))) /a^(3/2)
\[ \int x^2 \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=\int { x^{2} {\rm Li}_{3}(a x^{2}) \,d x } \]
Time = 5.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.83 \[ \int x^2 \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=\frac {x^3\,\mathrm {polylog}\left (3,a\,x^2\right )}{3}-\frac {2\,x^3\,\mathrm {polylog}\left (2,a\,x^2\right )}{9}+\frac {8\,x}{27\,a}-\frac {4\,x^3\,\ln \left (1-a\,x^2\right )}{27}+\frac {8\,x^3}{81}+\frac {\mathrm {atan}\left (\sqrt {a}\,x\,1{}\mathrm {i}\right )\,8{}\mathrm {i}}{27\,a^{3/2}} \]