Integrand size = 15, antiderivative size = 161 \[ \int (d x)^{3/2} \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=\frac {128 d \sqrt {d x}}{125 a}+\frac {128 (d x)^{5/2}}{625 d}-\frac {64 d^{3/2} \arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{125 a^{5/4}}-\frac {64 d^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{125 a^{5/4}}-\frac {32 (d x)^{5/2} \log \left (1-a x^2\right )}{125 d}-\frac {8 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{25 d}+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (3,a x^2\right )}{5 d} \]
128/625*(d*x)^(5/2)/d-64/125*d^(3/2)*arctan(a^(1/4)*(d*x)^(1/2)/d^(1/2))/a ^(5/4)-64/125*d^(3/2)*arctanh(a^(1/4)*(d*x)^(1/2)/d^(1/2))/a^(5/4)-32/125* (d*x)^(5/2)*ln(-a*x^2+1)/d-8/25*(d*x)^(5/2)*polylog(2,a*x^2)/d+2/5*(d*x)^( 5/2)*polylog(3,a*x^2)/d+128/125*d*(d*x)^(1/2)/a
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.55 \[ \int (d x)^{3/2} \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=-\frac {9 d \sqrt {d x} \operatorname {Gamma}\left (\frac {9}{4}\right ) \left (-320-64 a x^2+320 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},a x^2\right )+80 a x^2 \log \left (1-a x^2\right )+100 a x^2 \operatorname {PolyLog}\left (2,a x^2\right )-125 a x^2 \operatorname {PolyLog}\left (3,a x^2\right )\right )}{1250 a \operatorname {Gamma}\left (\frac {13}{4}\right )} \]
(-9*d*Sqrt[d*x]*Gamma[9/4]*(-320 - 64*a*x^2 + 320*Hypergeometric2F1[1/4, 1 , 5/4, a*x^2] + 80*a*x^2*Log[1 - a*x^2] + 100*a*x^2*PolyLog[2, a*x^2] - 12 5*a*x^2*PolyLog[3, a*x^2]))/(1250*a*Gamma[13/4])
Time = 0.42 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {7145, 7145, 25, 2905, 8, 262, 262, 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.
| \(\displaystyle \int (d x)^{3/2} \operatorname {PolyLog}\left (3,a x^2\right ) \, dx\) |
| \(\Big \downarrow \) 7145 |
| \(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (3,a x^2\right )}{5 d}-\frac {4}{5} \int (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )dx\) |
| \(\Big \downarrow \) 7145 |
| \(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (3,a x^2\right )}{5 d}-\frac {4}{5} \left (\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}-\frac {4}{5} \int -(d x)^{3/2} \log \left (1-a x^2\right )dx\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (3,a x^2\right )}{5 d}-\frac {4}{5} \left (\frac {4}{5} \int (d x)^{3/2} \log \left (1-a x^2\right )dx+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\right )\) |
| \(\Big \downarrow \) 2905 |
| \(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (3,a x^2\right )}{5 d}-\frac {4}{5} \left (\frac {4}{5} \left (\frac {4 a \int \frac {x (d x)^{5/2}}{1-a x^2}dx}{5 d}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\right )\) |
| \(\Big \downarrow \) 8 |
| \(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (3,a x^2\right )}{5 d}-\frac {4}{5} \left (\frac {4}{5} \left (\frac {4 a \int \frac {(d x)^{7/2}}{1-a x^2}dx}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (3,a x^2\right )}{5 d}-\frac {4}{5} \left (\frac {4}{5} \left (\frac {4 a \left (\frac {d^2 \int \frac {(d x)^{3/2}}{1-a x^2}dx}{a}-\frac {2 d (d x)^{5/2}}{5 a}\right )}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (3,a x^2\right )}{5 d}-\frac {4}{5} \left (\frac {4}{5} \left (\frac {4 a \left (\frac {d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {d x} \left (1-a x^2\right )}dx}{a}-\frac {2 d \sqrt {d x}}{a}\right )}{a}-\frac {2 d (d x)^{5/2}}{5 a}\right )}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (3,a x^2\right )}{5 d}-\frac {4}{5} \left (\frac {4}{5} \left (\frac {4 a \left (\frac {d^2 \left (\frac {2 d \int \frac {1}{1-a x^2}d\sqrt {d x}}{a}-\frac {2 d \sqrt {d x}}{a}\right )}{a}-\frac {2 d (d x)^{5/2}}{5 a}\right )}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (3,a x^2\right )}{5 d}-\frac {4}{5} \left (\frac {4}{5} \left (\frac {4 a \left (\frac {d^2 \left (\frac {2 d \left (\frac {1}{2} d \int \frac {1}{d-\sqrt {a} d x}d\sqrt {d x}+\frac {1}{2} d \int \frac {1}{\sqrt {a} x d+d}d\sqrt {d x}\right )}{a}-\frac {2 d \sqrt {d x}}{a}\right )}{a}-\frac {2 d (d x)^{5/2}}{5 a}\right )}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (3,a x^2\right )}{5 d}-\frac {4}{5} \left (\frac {4}{5} \left (\frac {4 a \left (\frac {d^2 \left (\frac {2 d \left (\frac {1}{2} d \int \frac {1}{d-\sqrt {a} d x}d\sqrt {d x}+\frac {\sqrt {d} \arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{a}}\right )}{a}-\frac {2 d \sqrt {d x}}{a}\right )}{a}-\frac {2 d (d x)^{5/2}}{5 a}\right )}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (3,a x^2\right )}{5 d}-\frac {4}{5} \left (\frac {4}{5} \left (\frac {4 a \left (\frac {d^2 \left (\frac {2 d \left (\frac {\sqrt {d} \arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{a}}+\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{a}}\right )}{a}-\frac {2 d \sqrt {d x}}{a}\right )}{a}-\frac {2 d (d x)^{5/2}}{5 a}\right )}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\right )\) |
(-4*((4*((4*a*((-2*d*(d*x)^(5/2))/(5*a) + (d^2*((-2*d*Sqrt[d*x])/a + (2*d* ((Sqrt[d]*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(2*a^(1/4)) + (Sqrt[d]*ArcT anh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(2*a^(1/4))))/a))/a))/(5*d^2) + (2*(d*x) ^(5/2)*Log[1 - a*x^2])/(5*d)))/5 + (2*(d*x)^(5/2)*PolyLog[2, a*x^2])/(5*d) ))/5 + (2*(d*x)^(5/2)*PolyLog[3, a*x^2])/(5*d)
3.1.79.3.1 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*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && 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 = 0.15 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.96
| method | result | size |
| meijerg | \(-\frac {\left (d x \right )^{\frac {3}{2}} \left (\frac {4 \sqrt {x}\, \left (-a \right )^{\frac {9}{4}} \left (576 a \,x^{2}+2880\right )}{5625 a^{2}}+\frac {64 \sqrt {x}\, \left (-a \right )^{\frac {9}{4}} \left (\ln \left (1-\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (a \,x^{2}\right )^{\frac {1}{4}}\right )\right )}{125 a^{2} \left (a \,x^{2}\right )^{\frac {1}{4}}}-\frac {64 x^{\frac {5}{2}} \left (-a \right )^{\frac {9}{4}} \ln \left (-a \,x^{2}+1\right )}{125 a}-\frac {16 x^{\frac {5}{2}} \left (-a \right )^{\frac {9}{4}} \operatorname {polylog}\left (2, a \,x^{2}\right )}{25 a}+\frac {4 x^{\frac {5}{2}} \left (-a \right )^{\frac {9}{4}} \operatorname {polylog}\left (3, a \,x^{2}\right )}{5 a}\right )}{2 x^{\frac {3}{2}} \left (-a \right )^{\frac {5}{4}}}\) | \(155\) |
-1/2*(d*x)^(3/2)/x^(3/2)/(-a)^(5/4)*(4/5625*x^(1/2)*(-a)^(9/4)*(576*a*x^2+ 2880)/a^2+64/125*x^(1/2)*(-a)^(9/4)/a^2/(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)))-64/125*x^(5/2)*(-a)^(9/4)*ln (-a*x^2+1)/a-16/25*x^(5/2)*(-a)^(9/4)/a*polylog(2,a*x^2)+4/5*x^(5/2)*(-a)^ (9/4)/a*polylog(3,a*x^2))
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.29 \[ \int (d x)^{3/2} \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=\frac {2 \, {\left (125 \, \sqrt {d x} a d x^{2} {\rm polylog}\left (3, a x^{2}\right ) - 80 \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}} \log \left (32 \, \sqrt {d x} d + 32 \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}}\right ) - 80 i \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}} \log \left (32 \, \sqrt {d x} d + 32 i \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}}\right ) + 80 i \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}} \log \left (32 \, \sqrt {d x} d - 32 i \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}}\right ) + 80 \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}} \log \left (32 \, \sqrt {d x} d - 32 \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}}\right ) - 4 \, {\left (25 \, a d x^{2} {\rm Li}_2\left (a x^{2}\right ) + 20 \, a d x^{2} \log \left (-a x^{2} + 1\right ) - 16 \, a d x^{2} - 80 \, d\right )} \sqrt {d x}\right )}}{625 \, a} \]
2/625*(125*sqrt(d*x)*a*d*x^2*polylog(3, a*x^2) - 80*a*(d^6/a^5)^(1/4)*log( 32*sqrt(d*x)*d + 32*a*(d^6/a^5)^(1/4)) - 80*I*a*(d^6/a^5)^(1/4)*log(32*sqr t(d*x)*d + 32*I*a*(d^6/a^5)^(1/4)) + 80*I*a*(d^6/a^5)^(1/4)*log(32*sqrt(d* x)*d - 32*I*a*(d^6/a^5)^(1/4)) + 80*a*(d^6/a^5)^(1/4)*log(32*sqrt(d*x)*d - 32*a*(d^6/a^5)^(1/4)) - 4*(25*a*d*x^2*dilog(a*x^2) + 20*a*d*x^2*log(-a*x^ 2 + 1) - 16*a*d*x^2 - 80*d)*sqrt(d*x))/a
\[ \int (d x)^{3/2} \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=\int \left (d x\right )^{\frac {3}{2}} \operatorname {Li}_{3}\left (a x^{2}\right )\, dx \]
Time = 0.26 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09 \[ \int (d x)^{3/2} \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=-\frac {2 \, {\left (\frac {100 \, \left (d x\right )^{\frac {5}{2}} a {\rm Li}_2\left (a x^{2}\right ) + 80 \, \left (d x\right )^{\frac {5}{2}} a \log \left (-a d^{2} x^{2} + d^{2}\right ) - 125 \, \left (d x\right )^{\frac {5}{2}} a {\rm Li}_{3}(a x^{2}) - 32 \, \left (d x\right )^{\frac {5}{2}} {\left (5 \, a \log \left (d\right ) + 2 \, a\right )} - 320 \, \sqrt {d x} d^{2}}{a} + \frac {80 \, {\left (\frac {2 \, d^{3} \arctan \left (\frac {\sqrt {d x} \sqrt {a}}{\sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d}} - \frac {d^{3} \log \left (\frac {\sqrt {d x} \sqrt {a} - \sqrt {\sqrt {a} d}}{\sqrt {d x} \sqrt {a} + \sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d}}\right )}}{a}\right )}}{625 \, d} \]
-2/625*((100*(d*x)^(5/2)*a*dilog(a*x^2) + 80*(d*x)^(5/2)*a*log(-a*d^2*x^2 + d^2) - 125*(d*x)^(5/2)*a*polylog(3, a*x^2) - 32*(d*x)^(5/2)*(5*a*log(d) + 2*a) - 320*sqrt(d*x)*d^2)/a + 80*(2*d^3*arctan(sqrt(d*x)*sqrt(a)/sqrt(sq rt(a)*d))/sqrt(sqrt(a)*d) - d^3*log((sqrt(d*x)*sqrt(a) - sqrt(sqrt(a)*d))/ (sqrt(d*x)*sqrt(a) + sqrt(sqrt(a)*d)))/sqrt(sqrt(a)*d))/a)/d
\[ \int (d x)^{3/2} \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=\int { \left (d x\right )^{\frac {3}{2}} {\rm Li}_{3}(a x^{2}) \,d x } \]
Timed out. \[ \int (d x)^{3/2} \operatorname {PolyLog}\left (3,a x^2\right ) \, dx=\int \mathrm {polylog}\left (3,a\,x^2\right )\,{\left (d\,x\right )}^{3/2} \,d x \]