Integrand size = 13, antiderivative size = 85 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{3/2}} \, dx=\frac {16 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {8 \log (1-a x)}{d \sqrt {d x}}-\frac {4 \operatorname {PolyLog}(2,a x)}{d \sqrt {d x}}-\frac {2 \operatorname {PolyLog}(3,a x)}{d \sqrt {d x}} \]
16*arctanh(a^(1/2)*(d*x)^(1/2)/d^(1/2))*a^(1/2)/d^(3/2)+8*ln(-a*x+1)/d/(d* x)^(1/2)-4*polylog(2,a*x)/d/(d*x)^(1/2)-2*polylog(3,a*x)/d/(d*x)^(1/2)
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.68 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{3/2}} \, dx=\frac {2 x \left (8 \sqrt {a} \sqrt {x} \text {arctanh}\left (\sqrt {a} \sqrt {x}\right )+4 \log (1-a x)-2 \operatorname {PolyLog}(2,a x)-\operatorname {PolyLog}(3,a x)\right )}{(d x)^{3/2}} \]
(2*x*(8*Sqrt[a]*Sqrt[x]*ArcTanh[Sqrt[a]*Sqrt[x]] + 4*Log[1 - a*x] - 2*Poly Log[2, a*x] - PolyLog[3, a*x]))/(d*x)^(3/2)
Time = 0.33 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {7145, 7145, 25, 2842, 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.
\(\displaystyle \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle 2 \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{3/2}}dx-\frac {2 \operatorname {PolyLog}(3,a x)}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle 2 \left (2 \int -\frac {\log (1-a x)}{(d x)^{3/2}}dx-\frac {2 \operatorname {PolyLog}(2,a x)}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}(3,a x)}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 \left (-2 \int \frac {\log (1-a x)}{(d x)^{3/2}}dx-\frac {2 \operatorname {PolyLog}(2,a x)}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}(3,a x)}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle 2 \left (-2 \left (-\frac {2 a \int \frac {1}{\sqrt {d x} (1-a x)}dx}{d}-\frac {2 \log (1-a x)}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}(3,a x)}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle 2 \left (-2 \left (-\frac {4 a \int \frac {1}{1-a x}d\sqrt {d x}}{d^2}-\frac {2 \log (1-a x)}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}(3,a x)}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (-2 \left (-\frac {4 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \log (1-a x)}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}(3,a x)}{d \sqrt {d x}}\) |
2*(-2*((-4*Sqrt[a]*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d]])/d^(3/2) - (2*Log[ 1 - a*x])/(d*Sqrt[d*x])) - (2*PolyLog[2, a*x])/(d*Sqrt[d*x])) - (2*PolyLog [3, a*x])/(d*Sqrt[d*x])
3.1.69.3.1 Defintions of rubi rules used
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 = 111, normalized size of antiderivative = 1.31
method | result | size |
meijerg | \(\frac {x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2}} \left (-\frac {8 \sqrt {x}\, \sqrt {-a}\, \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{\sqrt {a x}}+\frac {8 \sqrt {-a}\, \ln \left (-a x +1\right )}{\sqrt {x}\, a}-\frac {4 \sqrt {-a}\, \operatorname {polylog}\left (2, a x \right )}{\sqrt {x}\, a}-\frac {2 \sqrt {-a}\, \operatorname {polylog}\left (3, a x \right )}{\sqrt {x}\, a}\right )}{\left (d x \right )^{\frac {3}{2}} a}\) | \(111\) |
1/(d*x)^(3/2)*x^(3/2)*(-a)^(3/2)/a*(-8*x^(1/2)*(-a)^(1/2)/(a*x)^(1/2)*(ln( 1-(a*x)^(1/2))-ln(1+(a*x)^(1/2)))+8/x^(1/2)*(-a)^(1/2)/a*ln(-a*x+1)-4/x^(1 /2)*(-a)^(1/2)/a*polylog(2,a*x)-2/x^(1/2)*(-a)^(1/2)/a*polylog(3,a*x))
Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.84 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{3/2}} \, dx=\left [\frac {2 \, {\left (4 \, d x \sqrt {\frac {a}{d}} \log \left (\frac {a x + 2 \, \sqrt {d x} \sqrt {\frac {a}{d}} + 1}{a x - 1}\right ) - 2 \, \sqrt {d x} {\left ({\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )} - \sqrt {d x} {\rm polylog}\left (3, a x\right )\right )}}{d^{2} x}, -\frac {2 \, {\left (8 \, d x \sqrt {-\frac {a}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {-\frac {a}{d}}}{a x}\right ) + 2 \, \sqrt {d x} {\left ({\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )} + \sqrt {d x} {\rm polylog}\left (3, a x\right )\right )}}{d^{2} x}\right ] \]
[2*(4*d*x*sqrt(a/d)*log((a*x + 2*sqrt(d*x)*sqrt(a/d) + 1)/(a*x - 1)) - 2*s qrt(d*x)*(dilog(a*x) - 2*log(-a*x + 1)) - sqrt(d*x)*polylog(3, a*x))/(d^2* x), -2*(8*d*x*sqrt(-a/d)*arctan(sqrt(d*x)*sqrt(-a/d)/(a*x)) + 2*sqrt(d*x)* (dilog(a*x) - 2*log(-a*x + 1)) + sqrt(d*x)*polylog(3, a*x))/(d^2*x)]
\[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{3/2}} \, dx=\int \frac {\operatorname {Li}_{3}\left (a x\right )}{\left (d x\right )^{\frac {3}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {4 \, a \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d}} + \frac {2 \, {\rm Li}_2\left (a x\right ) - 4 \, \log \left (-a d x + d\right ) + 4 \, \log \left (d\right ) + {\rm Li}_{3}(a x)}{\sqrt {d x}}\right )}}{d} \]
-2*(4*a*log((sqrt(d*x)*a - sqrt(a*d))/(sqrt(d*x)*a + sqrt(a*d)))/sqrt(a*d) + (2*dilog(a*x) - 4*log(-a*d*x + d) + 4*log(d) + polylog(3, a*x))/sqrt(d* x))/d
\[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{3/2}} \, dx=\int { \frac {{\rm Li}_{3}(a x)}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {PolyLog}(3,a x)}{(d x)^{3/2}} \, dx=\int \frac {\mathrm {polylog}\left (3,a\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]