Integrand size = 15, antiderivative size = 147 \[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{(d x)^{9/2}} \, dx=-\frac {128 a}{1029 d^3 (d x)^{3/2}}+\frac {64 a^{7/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{343 d^{9/2}}+\frac {64 a^{7/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{343 d^{9/2}}+\frac {32 \log \left (1-a x^2\right )}{343 d (d x)^{7/2}}-\frac {8 \operatorname {PolyLog}\left (2,a x^2\right )}{49 d (d x)^{7/2}}-\frac {2 \operatorname {PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}} \]
-128/1029*a/d^3/(d*x)^(3/2)+64/343*a^(7/4)*arctan(a^(1/4)*(d*x)^(1/2)/d^(1 /2))/d^(9/2)+64/343*a^(7/4)*arctanh(a^(1/4)*(d*x)^(1/2)/d^(1/2))/d^(9/2)+3 2/343*ln(-a*x^2+1)/d/(d*x)^(7/2)-8/49*polylog(2,a*x^2)/d/(d*x)^(7/2)-2/7*p olylog(3,a*x^2)/d/(d*x)^(7/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.57 \[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{(d x)^{9/2}} \, dx=-\frac {\sqrt {d x} \operatorname {Gamma}\left (-\frac {3}{4}\right ) \left (-64 a x^2+192 a^2 x^4 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},a x^2\right )+48 \log \left (1-a x^2\right )-84 \operatorname {PolyLog}\left (2,a x^2\right )-147 \operatorname {PolyLog}\left (3,a x^2\right )\right )}{686 d^5 x^4 \operatorname {Gamma}\left (\frac {1}{4}\right )} \]
-1/686*(Sqrt[d*x]*Gamma[-3/4]*(-64*a*x^2 + 192*a^2*x^4*Hypergeometric2F1[1 /4, 1, 5/4, a*x^2] + 48*Log[1 - a*x^2] - 84*PolyLog[2, a*x^2] - 147*PolyLo g[3, a*x^2]))/(d^5*x^4*Gamma[1/4])
Time = 0.40 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {7145, 7145, 25, 2905, 8, 264, 266, 756, 218, 221}
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.
| \(\displaystyle \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{(d x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 7145 |
| \(\displaystyle \frac {4}{7} \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{(d x)^{9/2}}dx-\frac {2 \operatorname {PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 7145 |
| \(\displaystyle \frac {4}{7} \left (\frac {4}{7} \int -\frac {\log \left (1-a x^2\right )}{(d x)^{9/2}}dx-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{7} \left (-\frac {4}{7} \int \frac {\log \left (1-a x^2\right )}{(d x)^{9/2}}dx-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 2905 |
| \(\displaystyle \frac {4}{7} \left (-\frac {4}{7} \left (-\frac {4 a \int \frac {x}{(d x)^{7/2} \left (1-a x^2\right )}dx}{7 d}-\frac {2 \log \left (1-a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 8 |
| \(\displaystyle \frac {4}{7} \left (-\frac {4}{7} \left (-\frac {4 a \int \frac {1}{(d x)^{5/2} \left (1-a x^2\right )}dx}{7 d^2}-\frac {2 \log \left (1-a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {4}{7} \left (-\frac {4}{7} \left (-\frac {4 a \left (\frac {a \int \frac {1}{\sqrt {d x} \left (1-a x^2\right )}dx}{d^2}-\frac {2}{3 d (d x)^{3/2}}\right )}{7 d^2}-\frac {2 \log \left (1-a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {4}{7} \left (-\frac {4}{7} \left (-\frac {4 a \left (\frac {2 a \int \frac {1}{1-a x^2}d\sqrt {d x}}{d^3}-\frac {2}{3 d (d x)^{3/2}}\right )}{7 d^2}-\frac {2 \log \left (1-a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {4}{7} \left (-\frac {4}{7} \left (-\frac {4 a \left (\frac {2 a \left (\frac {1}{2} d \int \frac {1}{d-\sqrt {a} d x}d\sqrt {d x}+\frac {1}{2} d \int \frac {1}{\sqrt {a} x d+d}d\sqrt {d x}\right )}{d^3}-\frac {2}{3 d (d x)^{3/2}}\right )}{7 d^2}-\frac {2 \log \left (1-a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {4}{7} \left (-\frac {4}{7} \left (-\frac {4 a \left (\frac {2 a \left (\frac {1}{2} d \int \frac {1}{d-\sqrt {a} d x}d\sqrt {d x}+\frac {\sqrt {d} \arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{a}}\right )}{d^3}-\frac {2}{3 d (d x)^{3/2}}\right )}{7 d^2}-\frac {2 \log \left (1-a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4}{7} \left (-\frac {4}{7} \left (-\frac {4 a \left (\frac {2 a \left (\frac {\sqrt {d} \arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{a}}+\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{a}}\right )}{d^3}-\frac {2}{3 d (d x)^{3/2}}\right )}{7 d^2}-\frac {2 \log \left (1-a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{7 d (d x)^{7/2}}\right )-\frac {2 \operatorname {PolyLog}\left (3,a x^2\right )}{7 d (d x)^{7/2}}\) |
(4*((-4*((-4*a*(-2/(3*d*(d*x)^(3/2)) + (2*a*((Sqrt[d]*ArcTan[(a^(1/4)*Sqrt [d*x])/Sqrt[d]])/(2*a^(1/4)) + (Sqrt[d]*ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d ]])/(2*a^(1/4))))/d^3))/(7*d^2) - (2*Log[1 - a*x^2])/(7*d*(d*x)^(7/2))))/7 - (2*PolyLog[2, a*x^2])/(7*d*(d*x)^(7/2))))/7 - (2*PolyLog[3, a*x^2])/(7* d*(d*x)^(7/2))
3.1.85.3.1 Defintions of rubi rules used
Int[(u_.)*(x_)^(m_.)*((a_.)*(x_))^(p_), x_Symbol] :> Simp[1/a^m Int[u*(a* x)^(m + p), x], x] /; FreeQ[{a, m, p}, x] && IntegerQ[m]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
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 = 0.16 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.97
| method | result | size |
| meijerg | \(-\frac {x^{\frac {9}{2}} \left (-a \right )^{\frac {7}{4}} \left (-\frac {256}{1029 x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{4}}}-\frac {64 \sqrt {x}\, a \left (\ln \left (1-\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (a \,x^{2}\right )^{\frac {1}{4}}\right )\right )}{343 \left (-a \right )^{\frac {3}{4}} \left (a \,x^{2}\right )^{\frac {1}{4}}}+\frac {64 \ln \left (-a \,x^{2}+1\right )}{343 x^{\frac {7}{2}} \left (-a \right )^{\frac {3}{4}} a}-\frac {16 \operatorname {polylog}\left (2, a \,x^{2}\right )}{49 x^{\frac {7}{2}} \left (-a \right )^{\frac {3}{4}} a}-\frac {4 \operatorname {polylog}\left (3, a \,x^{2}\right )}{7 x^{\frac {7}{2}} \left (-a \right )^{\frac {3}{4}} a}\right )}{2 \left (d x \right )^{\frac {9}{2}}}\) | \(142\) |
-1/2/(d*x)^(9/2)*x^(9/2)*(-a)^(7/4)*(-256/1029/x^(3/2)/(-a)^(3/4)-64/343*x ^(1/2)/(-a)^(3/4)*a/(a*x^2)^(1/4)*(ln(1-(a*x^2)^(1/4))-ln(1+(a*x^2)^(1/4)) -2*arctan((a*x^2)^(1/4)))+64/343/x^(7/2)/(-a)^(3/4)*ln(-a*x^2+1)/a-16/49/x ^(7/2)/(-a)^(3/4)/a*polylog(2,a*x^2)-4/7/x^(7/2)/(-a)^(3/4)/a*polylog(3,a* x^2))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.55 \[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{(d x)^{9/2}} \, dx=\frac {2 \, {\left (48 \, d^{5} x^{4} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} \log \left (32 \, d^{5} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} + 32 \, \sqrt {d x} a^{2}\right ) + 48 i \, d^{5} x^{4} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} \log \left (32 i \, d^{5} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} + 32 \, \sqrt {d x} a^{2}\right ) - 48 i \, d^{5} x^{4} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} \log \left (-32 i \, d^{5} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} + 32 \, \sqrt {d x} a^{2}\right ) - 48 \, d^{5} x^{4} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} \log \left (-32 \, d^{5} \left (\frac {a^{7}}{d^{18}}\right )^{\frac {1}{4}} + 32 \, \sqrt {d x} a^{2}\right ) - 4 \, {\left (16 \, a x^{2} + 21 \, {\rm Li}_2\left (a x^{2}\right ) - 12 \, \log \left (-a x^{2} + 1\right )\right )} \sqrt {d x} - 147 \, \sqrt {d x} {\rm polylog}\left (3, a x^{2}\right )\right )}}{1029 \, d^{5} x^{4}} \]
2/1029*(48*d^5*x^4*(a^7/d^18)^(1/4)*log(32*d^5*(a^7/d^18)^(1/4) + 32*sqrt( d*x)*a^2) + 48*I*d^5*x^4*(a^7/d^18)^(1/4)*log(32*I*d^5*(a^7/d^18)^(1/4) + 32*sqrt(d*x)*a^2) - 48*I*d^5*x^4*(a^7/d^18)^(1/4)*log(-32*I*d^5*(a^7/d^18) ^(1/4) + 32*sqrt(d*x)*a^2) - 48*d^5*x^4*(a^7/d^18)^(1/4)*log(-32*d^5*(a^7/ d^18)^(1/4) + 32*sqrt(d*x)*a^2) - 4*(16*a*x^2 + 21*dilog(a*x^2) - 12*log(- a*x^2 + 1))*sqrt(d*x) - 147*sqrt(d*x)*polylog(3, a*x^2))/(d^5*x^4)
Timed out. \[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{(d x)^{9/2}} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.14 \[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{(d x)^{9/2}} \, dx=\frac {2 \, {\left (\frac {48 \, {\left (\frac {2 \, a^{2} \arctan \left (\frac {\sqrt {d x} \sqrt {a}}{\sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} d} - \frac {a^{2} \log \left (\frac {\sqrt {d x} \sqrt {a} - \sqrt {\sqrt {a} d}}{\sqrt {d x} \sqrt {a} + \sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} d}\right )}}{d^{2}} - \frac {64 \, a d^{2} x^{2} + 84 \, d^{2} {\rm Li}_2\left (a x^{2}\right ) - 48 \, d^{2} \log \left (-a d^{2} x^{2} + d^{2}\right ) + 96 \, d^{2} \log \left (d\right ) + 147 \, d^{2} {\rm Li}_{3}(a x^{2})}{\left (d x\right )^{\frac {7}{2}} d^{2}}\right )}}{1029 \, d} \]
2/1029*(48*(2*a^2*arctan(sqrt(d*x)*sqrt(a)/sqrt(sqrt(a)*d))/(sqrt(sqrt(a)* d)*d) - a^2*log((sqrt(d*x)*sqrt(a) - sqrt(sqrt(a)*d))/(sqrt(d*x)*sqrt(a) + sqrt(sqrt(a)*d)))/(sqrt(sqrt(a)*d)*d))/d^2 - (64*a*d^2*x^2 + 84*d^2*dilog (a*x^2) - 48*d^2*log(-a*d^2*x^2 + d^2) + 96*d^2*log(d) + 147*d^2*polylog(3 , a*x^2))/((d*x)^(7/2)*d^2))/d
\[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{(d x)^{9/2}} \, dx=\int { \frac {{\rm Li}_{3}(a x^{2})}{\left (d x\right )^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {PolyLog}\left (3,a x^2\right )}{(d x)^{9/2}} \, dx=\int \frac {\mathrm {polylog}\left (3,a\,x^2\right )}{{\left (d\,x\right )}^{9/2}} \,d x \]