Integrand size = 11, antiderivative size = 63 \[ \int x^2 \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=-\frac {4 x}{9 a}-\frac {4 x^3}{27}+\frac {4 \text {arctanh}\left (\sqrt {a} x\right )}{9 a^{3/2}}+\frac {2}{9} x^3 \log \left (1-a x^2\right )+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^2\right ) \]
-4/9*x/a-4/27*x^3+4/9*arctanh(x*a^(1/2))/a^(3/2)+2/9*x^3*ln(-a*x^2+1)+1/3* x^3*polylog(2,a*x^2)
Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.90 \[ \int x^2 \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\frac {1}{27} \left (-\frac {12 x}{a}-4 x^3+\frac {12 \text {arctanh}\left (\sqrt {a} x\right )}{a^{3/2}}+6 x^3 \log \left (1-a x^2\right )+9 x^3 \operatorname {PolyLog}\left (2,a x^2\right )\right ) \]
((-12*x)/a - 4*x^3 + (12*ArcTanh[Sqrt[a]*x])/a^(3/2) + 6*x^3*Log[1 - a*x^2 ] + 9*x^3*PolyLog[2, a*x^2])/27
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {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.
\(\displaystyle \int x^2 \operatorname {PolyLog}\left (2,a x^2\right ) \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^2\right )-\frac {2}{3} \int -x^2 \log \left (1-a x^2\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2}{3} \int x^2 \log \left (1-a x^2\right )dx+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^2\right )\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle \frac {2}{3} \left (\frac {2}{3} a \int \frac {x^4}{1-a x^2}dx+\frac {1}{3} x^3 \log \left (1-a x^2\right )\right )+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^2\right )\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {2}{3} \left (\frac {2}{3} a \int \left (-\frac {x^2}{a}+\frac {1}{a^2 \left (1-a x^2\right )}-\frac {1}{a^2}\right )dx+\frac {1}{3} x^3 \log \left (1-a x^2\right )\right )+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^2\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{3} \left (\frac {2}{3} a \left (\frac {\text {arctanh}\left (\sqrt {a} x\right )}{a^{5/2}}-\frac {x}{a^2}-\frac {x^3}{3 a}\right )+\frac {1}{3} x^3 \log \left (1-a x^2\right )\right )+\frac {1}{3} x^3 \operatorname {PolyLog}\left (2,a x^2\right )\) |
(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
3.1.27.3.1 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]
Time = 0.41 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {x^{3} \operatorname {polylog}\left (2, a \,x^{2}\right )}{3}+\frac {2 x^{3} \ln \left (-a \,x^{2}+1\right )}{9}+\frac {4 a \left (-\frac {\frac {1}{3} a \,x^{3}+x}{a^{2}}+\frac {\operatorname {arctanh}\left (x \sqrt {a}\right )}{a^{\frac {5}{2}}}\right )}{9}\) | \(55\) |
parts | \(\frac {x^{3} \operatorname {polylog}\left (2, a \,x^{2}\right )}{3}+\frac {2 x^{3} \ln \left (-a \,x^{2}+1\right )}{9}+\frac {4 a \left (-\frac {\frac {1}{3} a \,x^{3}+x}{a^{2}}+\frac {\operatorname {arctanh}\left (x \sqrt {a}\right )}{a^{\frac {5}{2}}}\right )}{9}\) | \(55\) |
meijerg | \(\frac {-\frac {2 x \left (-a \right )^{\frac {5}{2}} \left (20 a \,x^{2}+60\right )}{135 a^{2}}-\frac {4 x \left (-a \right )^{\frac {5}{2}} \left (\ln \left (1-\sqrt {a \,x^{2}}\right )-\ln \left (1+\sqrt {a \,x^{2}}\right )\right )}{9 a^{2} \sqrt {a \,x^{2}}}+\frac {4 x^{3} \left (-a \right )^{\frac {5}{2}} \ln \left (-a \,x^{2}+1\right )}{9 a}+\frac {2 x^{3} \left (-a \right )^{\frac {5}{2}} \operatorname {polylog}\left (2, a \,x^{2}\right )}{3 a}}{2 a \sqrt {-a}}\) | \(116\) |
1/3*x^3*polylog(2,a*x^2)+2/9*x^3*ln(-a*x^2+1)+4/9*a*(-1/a^2*(1/3*a*x^3+x)+ arctanh(x*a^(1/2))/a^(5/2))
Time = 0.26 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.27 \[ \int x^2 \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\left [\frac {9 \, a^{2} x^{3} {\rm Li}_2\left (a x^{2}\right ) + 6 \, a^{2} x^{3} \log \left (-a x^{2} + 1\right ) - 4 \, a^{2} x^{3} - 12 \, a x + 6 \, \sqrt {a} \log \left (\frac {a x^{2} + 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right )}{27 \, a^{2}}, \frac {9 \, a^{2} x^{3} {\rm Li}_2\left (a x^{2}\right ) + 6 \, a^{2} x^{3} \log \left (-a x^{2} + 1\right ) - 4 \, a^{2} x^{3} - 12 \, a x - 12 \, \sqrt {-a} \arctan \left (\sqrt {-a} x\right )}{27 \, a^{2}}\right ] \]
[1/27*(9*a^2*x^3*dilog(a*x^2) + 6*a^2*x^3*log(-a*x^2 + 1) - 4*a^2*x^3 - 12 *a*x + 6*sqrt(a)*log((a*x^2 + 2*sqrt(a)*x + 1)/(a*x^2 - 1)))/a^2, 1/27*(9* a^2*x^3*dilog(a*x^2) + 6*a^2*x^3*log(-a*x^2 + 1) - 4*a^2*x^3 - 12*a*x - 12 *sqrt(-a)*arctan(sqrt(-a)*x))/a^2]
Time = 18.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.32 \[ \int x^2 \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\begin {cases} - \frac {2 x^{3} \operatorname {Li}_{1}\left (a x^{2}\right )}{9} + \frac {x^{3} \operatorname {Li}_{2}\left (a x^{2}\right )}{3} - \frac {4 x^{3}}{27} - \frac {4 x}{9 a} - \frac {4 \log {\left (x - \sqrt {\frac {1}{a}} \right )}}{9 a^{2} \sqrt {\frac {1}{a}}} - \frac {2 \operatorname {Li}_{1}\left (a x^{2}\right )}{9 a^{2} \sqrt {\frac {1}{a}}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((-2*x**3*polylog(1, a*x**2)/9 + x**3*polylog(2, a*x**2)/3 - 4*x* *3/27 - 4*x/(9*a) - 4*log(x - sqrt(1/a))/(9*a**2*sqrt(1/a)) - 2*polylog(1, a*x**2)/(9*a**2*sqrt(1/a)), Ne(a, 0)), (0, True))
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08 \[ \int x^2 \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\frac {9 \, a x^{3} {\rm Li}_2\left (a x^{2}\right ) + 6 \, a x^{3} \log \left (-a x^{2} + 1\right ) - 4 \, a x^{3} - 12 \, x}{27 \, a} - \frac {2 \, \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right )}{9 \, a^{\frac {3}{2}}} \]
1/27*(9*a*x^3*dilog(a*x^2) + 6*a*x^3*log(-a*x^2 + 1) - 4*a*x^3 - 12*x)/a - 2/9*log((a*x - sqrt(a))/(a*x + sqrt(a)))/a^(3/2)
\[ \int x^2 \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\int { x^{2} {\rm Li}_2\left (a x^{2}\right ) \,d x } \]
Time = 5.00 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int x^2 \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\frac {x^3\,\mathrm {polylog}\left (2,a\,x^2\right )}{3}-\frac {4\,x}{9\,a}+\frac {2\,x^3\,\ln \left (1-a\,x^2\right )}{9}-\frac {4\,x^3}{27}-\frac {\mathrm {atan}\left (\sqrt {a}\,x\,1{}\mathrm {i}\right )\,4{}\mathrm {i}}{9\,a^{3/2}} \]