Integrand size = 11, antiderivative size = 70 \[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{x^4} \, dx=-\frac {8 a}{27 x}+\frac {8}{27} a^{3/2} \text {arctanh}\left (\sqrt {a} x\right )+\frac {4 \log \left (1-a x^2\right )}{27 x^3}-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{9 x^3}-\frac {\operatorname {PolyLog}\left (3,a x^2\right )}{3 x^3} \]
-8/27*a/x+8/27*a^(3/2)*arctanh(x*a^(1/2))+4/27*ln(-a*x^2+1)/x^3-2/9*polylo g(2,a*x^2)/x^3-1/3*polylog(3,a*x^2)/x^3
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.87 \[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{x^4} \, dx=-\frac {8 a x^2-8 a^{3/2} x^3 \text {arctanh}\left (\sqrt {a} x\right )-4 \log \left (1-a x^2\right )+6 \operatorname {PolyLog}\left (2,a x^2\right )+9 \operatorname {PolyLog}\left (3,a x^2\right )}{27 x^3} \]
-1/27*(8*a*x^2 - 8*a^(3/2)*x^3*ArcTanh[Sqrt[a]*x] - 4*Log[1 - a*x^2] + 6*P olyLog[2, a*x^2] + 9*PolyLog[3, a*x^2])/x^3
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.14, 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, 264, 219}
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.
\(\displaystyle \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {2}{3} \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{x^4}dx-\frac {\operatorname {PolyLog}\left (3,a x^2\right )}{3 x^3}\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {2}{3} \left (\frac {2}{3} \int -\frac {\log \left (1-a x^2\right )}{x^4}dx-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{3 x^3}\right )-\frac {\operatorname {PolyLog}\left (3,a x^2\right )}{3 x^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2}{3} \left (-\frac {2}{3} \int \frac {\log \left (1-a x^2\right )}{x^4}dx-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{3 x^3}\right )-\frac {\operatorname {PolyLog}\left (3,a x^2\right )}{3 x^3}\) |
\(\Big \downarrow \) 2905 |
\(\displaystyle \frac {2}{3} \left (-\frac {2}{3} \left (-\frac {2}{3} a \int \frac {1}{x^2 \left (1-a x^2\right )}dx-\frac {\log \left (1-a x^2\right )}{3 x^3}\right )-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{3 x^3}\right )-\frac {\operatorname {PolyLog}\left (3,a x^2\right )}{3 x^3}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {2}{3} \left (-\frac {2}{3} \left (-\frac {2}{3} a \left (a \int \frac {1}{1-a x^2}dx-\frac {1}{x}\right )-\frac {\log \left (1-a x^2\right )}{3 x^3}\right )-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{3 x^3}\right )-\frac {\operatorname {PolyLog}\left (3,a x^2\right )}{3 x^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2}{3} \left (-\frac {2}{3} \left (-\frac {2}{3} a \left (\sqrt {a} \text {arctanh}\left (\sqrt {a} x\right )-\frac {1}{x}\right )-\frac {\log \left (1-a x^2\right )}{3 x^3}\right )-\frac {\operatorname {PolyLog}\left (2,a x^2\right )}{3 x^3}\right )-\frac {\operatorname {PolyLog}\left (3,a x^2\right )}{3 x^3}\) |
(2*((-2*((-2*a*(-x^(-1) + Sqrt[a]*ArcTanh[Sqrt[a]*x]))/3 - Log[1 - a*x^2]/ (3*x^3)))/3 - PolyLog[2, a*x^2]/(3*x^3)))/3 - PolyLog[3, a*x^2]/(3*x^3)
3.1.43.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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. \(124\) vs. \(2(56)=112\).
Time = 0.20 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.79
method | result | size |
meijerg | \(-\frac {a^{2} \left (-\frac {16}{27 x \sqrt {-a}}-\frac {8 x a \left (\ln \left (1-\sqrt {a \,x^{2}}\right )-\ln \left (1+\sqrt {a \,x^{2}}\right )\right )}{27 \sqrt {-a}\, \sqrt {a \,x^{2}}}+\frac {8 \ln \left (-a \,x^{2}+1\right )}{27 x^{3} \sqrt {-a}\, a}-\frac {4 \operatorname {polylog}\left (2, a \,x^{2}\right )}{9 x^{3} \sqrt {-a}\, a}-\frac {2 \operatorname {polylog}\left (3, a \,x^{2}\right )}{3 x^{3} \sqrt {-a}\, a}\right )}{2 \sqrt {-a}}\) | \(125\) |
-1/2*a^2/(-a)^(1/2)*(-16/27/x/(-a)^(1/2)-8/27*x/(-a)^(1/2)*a/(a*x^2)^(1/2) *(ln(1-(a*x^2)^(1/2))-ln(1+(a*x^2)^(1/2)))+8/27/x^3/(-a)^(1/2)*ln(-a*x^2+1 )/a-4/9/x^3/(-a)^(1/2)/a*polylog(2,a*x^2)-2/3/x^3/(-a)^(1/2)/a*polylog(3,a *x^2))
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.89 \[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{x^4} \, dx=\left [\frac {4 \, a^{\frac {3}{2}} x^{3} \log \left (\frac {a x^{2} + 2 \, \sqrt {a} x + 1}{a x^{2} - 1}\right ) - 8 \, a x^{2} - 6 \, {\rm Li}_2\left (a x^{2}\right ) + 4 \, \log \left (-a x^{2} + 1\right ) - 9 \, {\rm polylog}\left (3, a x^{2}\right )}{27 \, x^{3}}, -\frac {8 \, \sqrt {-a} a x^{3} \arctan \left (\sqrt {-a} x\right ) + 8 \, a x^{2} + 6 \, {\rm Li}_2\left (a x^{2}\right ) - 4 \, \log \left (-a x^{2} + 1\right ) + 9 \, {\rm polylog}\left (3, a x^{2}\right )}{27 \, x^{3}}\right ] \]
[1/27*(4*a^(3/2)*x^3*log((a*x^2 + 2*sqrt(a)*x + 1)/(a*x^2 - 1)) - 8*a*x^2 - 6*dilog(a*x^2) + 4*log(-a*x^2 + 1) - 9*polylog(3, a*x^2))/x^3, -1/27*(8* sqrt(-a)*a*x^3*arctan(sqrt(-a)*x) + 8*a*x^2 + 6*dilog(a*x^2) - 4*log(-a*x^ 2 + 1) + 9*polylog(3, a*x^2))/x^3]
\[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{x^4} \, dx=\int \frac {\operatorname {Li}_{3}\left (a x^{2}\right )}{x^{4}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{x^4} \, dx=-\frac {4}{27} \, a^{\frac {3}{2}} \log \left (\frac {a x - \sqrt {a}}{a x + \sqrt {a}}\right ) - \frac {8 \, a x^{2} + 6 \, {\rm Li}_2\left (a x^{2}\right ) - 4 \, \log \left (-a x^{2} + 1\right ) + 9 \, {\rm Li}_{3}(a x^{2})}{27 \, x^{3}} \]
-4/27*a^(3/2)*log((a*x - sqrt(a))/(a*x + sqrt(a))) - 1/27*(8*a*x^2 + 6*dil og(a*x^2) - 4*log(-a*x^2 + 1) + 9*polylog(3, a*x^2))/x^3
\[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{x^4} \, dx=\int { \frac {{\rm Li}_{3}(a x^{2})}{x^{4}} \,d x } \]
Time = 5.42 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84 \[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{x^4} \, dx=\frac {4\,\ln \left (1-a\,x^2\right )}{27\,x^3}-\frac {\mathrm {polylog}\left (3,a\,x^2\right )}{3\,x^3}-\frac {8\,a}{27\,x}-\frac {2\,\mathrm {polylog}\left (2,a\,x^2\right )}{9\,x^3}-\frac {a^{3/2}\,\mathrm {atan}\left (\sqrt {a}\,x\,1{}\mathrm {i}\right )\,8{}\mathrm {i}}{27} \]