Integrand size = 13, antiderivative size = 68 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{3/2}} \, dx=\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {4 \log (1-a x)}{d \sqrt {d x}}-\frac {2 \operatorname {PolyLog}(2,a x)}{d \sqrt {d x}} \]
8*arctanh(a^(1/2)*(d*x)^(1/2)/d^(1/2))*a^(1/2)/d^(3/2)+4*ln(-a*x+1)/d/(d*x )^(1/2)-2*polylog(2,a*x)/d/(d*x)^(1/2)
Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.75 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{3/2}} \, dx=\frac {2 x \left (4 \sqrt {a} \sqrt {x} \text {arctanh}\left (\sqrt {a} \sqrt {x}\right )+2 \log (1-a x)-\operatorname {PolyLog}(2,a x)\right )}{(d x)^{3/2}} \]
(2*x*(4*Sqrt[a]*Sqrt[x]*ArcTanh[Sqrt[a]*Sqrt[x]] + 2*Log[1 - a*x] - PolyLo g[2, a*x]))/(d*x)^(3/2)
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {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.
\(\displaystyle \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle 2 \int -\frac {\log (1-a x)}{(d x)^{3/2}}dx-\frac {2 \operatorname {PolyLog}(2,a x)}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {\log (1-a x)}{(d x)^{3/2}}dx-\frac {2 \operatorname {PolyLog}(2,a x)}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle -2 \left (-\frac {2 a \int \frac {1}{\sqrt {d x} (1-a x)}dx}{d}-\frac {2 \log (1-a x)}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -2 \left (-\frac {4 a \int \frac {1}{1-a x}d\sqrt {d x}}{d^2}-\frac {2 \log (1-a x)}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{d \sqrt {d x}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 \left (-\frac {4 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}-\frac {2 \log (1-a x)}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}(2,a x)}{d \sqrt {d x}}\) |
-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])
3.1.62.3.1 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 = 0.70 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {-\frac {2 \operatorname {polylog}\left (2, a x \right )}{\sqrt {d x}}+\frac {4 \ln \left (\frac {-a d x +d}{d}\right )}{\sqrt {d x}}+\frac {8 a \,\operatorname {arctanh}\left (\frac {a \sqrt {d x}}{\sqrt {a d}}\right )}{\sqrt {a d}}}{d}\) | \(59\) |
default | \(\frac {-\frac {2 \operatorname {polylog}\left (2, a x \right )}{\sqrt {d x}}+\frac {4 \ln \left (\frac {-a d x +d}{d}\right )}{\sqrt {d x}}+\frac {8 a \,\operatorname {arctanh}\left (\frac {a \sqrt {d x}}{\sqrt {a d}}\right )}{\sqrt {a d}}}{d}\) | \(59\) |
meijerg | \(\frac {x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{2}} \left (-\frac {4 \sqrt {x}\, \sqrt {-a}\, \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{\sqrt {a x}}+\frac {4 \sqrt {-a}\, \ln \left (-a x +1\right )}{\sqrt {x}\, a}-\frac {2 \sqrt {-a}\, \operatorname {polylog}\left (2, a x \right )}{\sqrt {x}\, a}\right )}{\left (d x \right )^{\frac {3}{2}} a}\) | \(93\) |
2/d*(-polylog(2,a*x)/(d*x)^(1/2)+2/(d*x)^(1/2)*ln((-a*d*x+d)/d)+4*a/(a*d)^ (1/2)*arctanh(a*(d*x)^(1/2)/(a*d)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.94 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{3/2}} \, dx=\left [\frac {2 \, {\left (2 \, d x \sqrt {\frac {a}{d}} \log \left (\frac {a x + 2 \, \sqrt {d x} \sqrt {\frac {a}{d}} + 1}{a x - 1}\right ) - \sqrt {d x} {\left ({\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )}\right )}}{d^{2} x}, -\frac {2 \, {\left (4 \, d x \sqrt {-\frac {a}{d}} \arctan \left (\frac {\sqrt {d x} \sqrt {-\frac {a}{d}}}{a x}\right ) + \sqrt {d x} {\left ({\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a x + 1\right )\right )}\right )}}{d^{2} x}\right ] \]
[2*(2*d*x*sqrt(a/d)*log((a*x + 2*sqrt(d*x)*sqrt(a/d) + 1)/(a*x - 1)) - sqr t(d*x)*(dilog(a*x) - 2*log(-a*x + 1)))/(d^2*x), -2*(4*d*x*sqrt(-a/d)*arcta n(sqrt(d*x)*sqrt(-a/d)/(a*x)) + sqrt(d*x)*(dilog(a*x) - 2*log(-a*x + 1)))/ (d^2*x)]
\[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{3/2}} \, dx=\int \frac {\operatorname {Li}_{2}\left (a x\right )}{\left (d x\right )^{\frac {3}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.04 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{3/2}} \, dx=-\frac {2 \, {\left (\frac {2 \, a \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d}} + \frac {{\rm Li}_2\left (a x\right ) - 2 \, \log \left (-a d x + d\right ) + 2 \, \log \left (d\right )}{\sqrt {d x}}\right )}}{d} \]
-2*(2*a*log((sqrt(d*x)*a - sqrt(a*d))/(sqrt(d*x)*a + sqrt(a*d)))/sqrt(a*d) + (dilog(a*x) - 2*log(-a*d*x + d) + 2*log(d))/sqrt(d*x))/d
\[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{3/2}} \, dx=\int { \frac {{\rm Li}_2\left (a x\right )}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {PolyLog}(2,a x)}{(d x)^{3/2}} \, dx=\int \frac {\mathrm {polylog}\left (2,a\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]