Integrand size = 13, antiderivative size = 121 \[ \int \sqrt {d x} \operatorname {PolyLog}(3,a x) \, dx=\frac {16 \sqrt {d x}}{27 a}+\frac {16 (d x)^{3/2}}{81 d}-\frac {16 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{27 a^{3/2}}-\frac {8 (d x)^{3/2} \log (1-a x)}{27 d}-\frac {4 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{9 d}+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(3,a x)}{3 d} \]
16/81*(d*x)^(3/2)/d-8/27*(d*x)^(3/2)*ln(-a*x+1)/d-4/9*(d*x)^(3/2)*polylog( 2,a*x)/d+2/3*(d*x)^(3/2)*polylog(3,a*x)/d-16/27*arctanh(a^(1/2)*(d*x)^(1/2 )/d^(1/2))*d^(1/2)/a^(3/2)+16/27*(d*x)^(1/2)/a
Time = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.60 \[ \int \sqrt {d x} \operatorname {PolyLog}(3,a x) \, dx=\frac {2}{81} \sqrt {d x} \left (4 \left (\frac {6}{a}+2 x-\frac {6 \text {arctanh}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2} \sqrt {x}}-3 x \log (1-a x)\right )-18 x \operatorname {PolyLog}(2,a x)+27 x \operatorname {PolyLog}(3,a x)\right ) \]
(2*Sqrt[d*x]*(4*(6/a + 2*x - (6*ArcTanh[Sqrt[a]*Sqrt[x]])/(a^(3/2)*Sqrt[x] ) - 3*x*Log[1 - a*x]) - 18*x*PolyLog[2, a*x] + 27*x*PolyLog[3, a*x]))/81
Time = 0.35 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {7145, 7145, 25, 2842, 60, 60, 73, 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.
| \(\displaystyle \int \sqrt {d x} \operatorname {PolyLog}(3,a x) \, dx\) |
| \(\Big \downarrow \) 7145 |
| \(\displaystyle \frac {2 (d x)^{3/2} \operatorname {PolyLog}(3,a x)}{3 d}-\frac {2}{3} \int \sqrt {d x} \operatorname {PolyLog}(2,a x)dx\) |
| \(\Big \downarrow \) 7145 |
| \(\displaystyle \frac {2 (d x)^{3/2} \operatorname {PolyLog}(3,a x)}{3 d}-\frac {2}{3} \left (\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}-\frac {2}{3} \int -\sqrt {d x} \log (1-a x)dx\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (d x)^{3/2} \operatorname {PolyLog}(3,a x)}{3 d}-\frac {2}{3} \left (\frac {2}{3} \int \sqrt {d x} \log (1-a x)dx+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}\right )\) |
| \(\Big \downarrow \) 2842 |
| \(\displaystyle \frac {2 (d x)^{3/2} \operatorname {PolyLog}(3,a x)}{3 d}-\frac {2}{3} \left (\frac {2}{3} \left (\frac {2 a \int \frac {(d x)^{3/2}}{1-a x}dx}{3 d}+\frac {2 (d x)^{3/2} \log (1-a x)}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (d x)^{3/2} \operatorname {PolyLog}(3,a x)}{3 d}-\frac {2}{3} \left (\frac {2}{3} \left (\frac {2 a \left (\frac {d \int \frac {\sqrt {d x}}{1-a x}dx}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{3 d}+\frac {2 (d x)^{3/2} \log (1-a x)}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (d x)^{3/2} \operatorname {PolyLog}(3,a x)}{3 d}-\frac {2}{3} \left (\frac {2}{3} \left (\frac {2 a \left (\frac {d \left (\frac {d \int \frac {1}{\sqrt {d x} (1-a x)}dx}{a}-\frac {2 \sqrt {d x}}{a}\right )}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{3 d}+\frac {2 (d x)^{3/2} \log (1-a x)}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 (d x)^{3/2} \operatorname {PolyLog}(3,a x)}{3 d}-\frac {2}{3} \left (\frac {2}{3} \left (\frac {2 a \left (\frac {d \left (\frac {2 \int \frac {1}{1-a x}d\sqrt {d x}}{a}-\frac {2 \sqrt {d x}}{a}\right )}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{3 d}+\frac {2 (d x)^{3/2} \log (1-a x)}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 (d x)^{3/2} \operatorname {PolyLog}(3,a x)}{3 d}-\frac {2}{3} \left (\frac {2}{3} \left (\frac {2 a \left (\frac {d \left (\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{a^{3/2}}-\frac {2 \sqrt {d x}}{a}\right )}{a}-\frac {2 (d x)^{3/2}}{3 a}\right )}{3 d}+\frac {2 (d x)^{3/2} \log (1-a x)}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}(2,a x)}{3 d}\right )\) |
(-2*((2*((2*a*((-2*(d*x)^(3/2))/(3*a) + (d*((-2*Sqrt[d*x])/a + (2*Sqrt[d]* ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/a^(3/2)))/a))/(3*d) + (2*(d*x)^(3/2) *Log[1 - a*x])/(3*d)))/3 + (2*(d*x)^(3/2)*PolyLog[2, a*x])/(3*d)))/3 + (2* (d*x)^(3/2)*PolyLog[3, a*x])/(3*d)
3.1.67.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ ))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1))) Int[(f + g*x)^(q + 1)/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -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 = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10
| method | result | size |
| meijerg | \(\frac {\sqrt {d x}\, \left (\frac {2 \sqrt {x}\, \left (-a \right )^{\frac {5}{2}} \left (40 a x +120\right )}{405 a^{2}}+\frac {8 \sqrt {x}\, \left (-a \right )^{\frac {5}{2}} \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{27 a^{2} \sqrt {a x}}-\frac {8 x^{\frac {3}{2}} \left (-a \right )^{\frac {5}{2}} \ln \left (-a x +1\right )}{27 a}-\frac {4 x^{\frac {3}{2}} \left (-a \right )^{\frac {5}{2}} \operatorname {polylog}\left (2, a x \right )}{9 a}+\frac {2 x^{\frac {3}{2}} \left (-a \right )^{\frac {5}{2}} \operatorname {polylog}\left (3, a x \right )}{3 a}\right )}{\sqrt {x}\, \sqrt {-a}\, a}\) | \(133\) |
(d*x)^(1/2)/x^(1/2)/(-a)^(1/2)/a*(2/405*x^(1/2)*(-a)^(5/2)*(40*a*x+120)/a^ 2+8/27*x^(1/2)*(-a)^(5/2)/a^2/(a*x)^(1/2)*(ln(1-(a*x)^(1/2))-ln(1+(a*x)^(1 /2)))-8/27*x^(3/2)*(-a)^(5/2)/a*ln(-a*x+1)-4/9*x^(3/2)*(-a)^(5/2)/a*polylo g(2,a*x)+2/3*x^(3/2)*(-a)^(5/2)/a*polylog(3,a*x))
Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.43 \[ \int \sqrt {d x} \operatorname {PolyLog}(3,a x) \, dx=\left [\frac {2 \, {\left (27 \, \sqrt {d x} a x {\rm polylog}\left (3, a x\right ) - 2 \, {\left (9 \, a x {\rm Li}_2\left (a x\right ) + 6 \, a x \log \left (-a x + 1\right ) - 4 \, a x - 12\right )} \sqrt {d x} + 12 \, \sqrt {\frac {d}{a}} \log \left (\frac {a d x - 2 \, \sqrt {d x} a \sqrt {\frac {d}{a}} + d}{a x - 1}\right )\right )}}{81 \, a}, \frac {2 \, {\left (27 \, \sqrt {d x} a x {\rm polylog}\left (3, a x\right ) - 2 \, {\left (9 \, a x {\rm Li}_2\left (a x\right ) + 6 \, a x \log \left (-a x + 1\right ) - 4 \, a x - 12\right )} \sqrt {d x} + 24 \, \sqrt {-\frac {d}{a}} \arctan \left (\frac {\sqrt {d x} a \sqrt {-\frac {d}{a}}}{d}\right )\right )}}{81 \, a}\right ] \]
[2/81*(27*sqrt(d*x)*a*x*polylog(3, a*x) - 2*(9*a*x*dilog(a*x) + 6*a*x*log( -a*x + 1) - 4*a*x - 12)*sqrt(d*x) + 12*sqrt(d/a)*log((a*d*x - 2*sqrt(d*x)* a*sqrt(d/a) + d)/(a*x - 1)))/a, 2/81*(27*sqrt(d*x)*a*x*polylog(3, a*x) - 2 *(9*a*x*dilog(a*x) + 6*a*x*log(-a*x + 1) - 4*a*x - 12)*sqrt(d*x) + 24*sqrt (-d/a)*arctan(sqrt(d*x)*a*sqrt(-d/a)/d))/a]
\[ \int \sqrt {d x} \operatorname {PolyLog}(3,a x) \, dx=\int \sqrt {d x} \operatorname {Li}_{3}\left (a x\right )\, dx \]
Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.01 \[ \int \sqrt {d x} \operatorname {PolyLog}(3,a x) \, dx=\frac {2 \, {\left (\frac {12 \, d^{2} \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d} a} - \frac {18 \, \left (d x\right )^{\frac {3}{2}} a {\rm Li}_2\left (a x\right ) + 12 \, \left (d x\right )^{\frac {3}{2}} a \log \left (-a d x + d\right ) - 27 \, \left (d x\right )^{\frac {3}{2}} a {\rm Li}_{3}(a x) - 4 \, \left (d x\right )^{\frac {3}{2}} {\left (3 \, a \log \left (d\right ) + 2 \, a\right )} - 24 \, \sqrt {d x} d}{a}\right )}}{81 \, d} \]
2/81*(12*d^2*log((sqrt(d*x)*a - sqrt(a*d))/(sqrt(d*x)*a + sqrt(a*d)))/(sqr t(a*d)*a) - (18*(d*x)^(3/2)*a*dilog(a*x) + 12*(d*x)^(3/2)*a*log(-a*d*x + d ) - 27*(d*x)^(3/2)*a*polylog(3, a*x) - 4*(d*x)^(3/2)*(3*a*log(d) + 2*a) - 24*sqrt(d*x)*d)/a)/d
\[ \int \sqrt {d x} \operatorname {PolyLog}(3,a x) \, dx=\int { \sqrt {d x} {\rm Li}_{3}(a x) \,d x } \]
Timed out. \[ \int \sqrt {d x} \operatorname {PolyLog}(3,a x) \, dx=\int \sqrt {d\,x}\,\mathrm {polylog}\left (3,a\,x\right ) \,d x \]