Integrand size = 13, antiderivative size = 80 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{\sqrt {d x}} \, dx=-\frac {8 \sqrt {d x}}{d}+\frac {8 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {a} \sqrt {d}}+\frac {4 \sqrt {d x} \log (1-a x)}{d}+\frac {2 \sqrt {d x} \operatorname {PolyLog}(2,a x)}{d} \]
8*arctanh(a^(1/2)*(d*x)^(1/2)/d^(1/2))/a^(1/2)/d^(1/2)-8*(d*x)^(1/2)/d+4*l n(-a*x+1)*(d*x)^(1/2)/d+2*polylog(2,a*x)*(d*x)^(1/2)/d
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.79 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{\sqrt {d x}} \, dx=\frac {8 \sqrt {x} \text {arctanh}\left (\sqrt {a} \sqrt {x}\right )+4 \sqrt {a} x (-2+\log (1-a x))+2 \sqrt {a} x \operatorname {PolyLog}(2,a x)}{\sqrt {a} \sqrt {d x}} \]
(8*Sqrt[x]*ArcTanh[Sqrt[a]*Sqrt[x]] + 4*Sqrt[a]*x*(-2 + Log[1 - a*x]) + 2* Sqrt[a]*x*PolyLog[2, a*x])/(Sqrt[a]*Sqrt[d*x])
Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {7145, 25, 2842, 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.
\(\displaystyle \int \frac {\operatorname {PolyLog}(2,a x)}{\sqrt {d x}} \, dx\) |
\(\Big \downarrow \) 7145 |
\(\displaystyle \frac {2 \sqrt {d x} \operatorname {PolyLog}(2,a x)}{d}-2 \int -\frac {\log (1-a x)}{\sqrt {d x}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 \int \frac {\log (1-a x)}{\sqrt {d x}}dx+\frac {2 \sqrt {d x} \operatorname {PolyLog}(2,a x)}{d}\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle 2 \left (\frac {2 a \int \frac {\sqrt {d x}}{1-a x}dx}{d}+\frac {2 \sqrt {d x} \log (1-a x)}{d}\right )+\frac {2 \sqrt {d x} \operatorname {PolyLog}(2,a x)}{d}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle 2 \left (\frac {2 a \left (\frac {d \int \frac {1}{\sqrt {d x} (1-a x)}dx}{a}-\frac {2 \sqrt {d x}}{a}\right )}{d}+\frac {2 \sqrt {d x} \log (1-a x)}{d}\right )+\frac {2 \sqrt {d x} \operatorname {PolyLog}(2,a x)}{d}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle 2 \left (\frac {2 a \left (\frac {2 \int \frac {1}{1-a x}d\sqrt {d x}}{a}-\frac {2 \sqrt {d x}}{a}\right )}{d}+\frac {2 \sqrt {d x} \log (1-a x)}{d}\right )+\frac {2 \sqrt {d x} \operatorname {PolyLog}(2,a x)}{d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (\frac {2 a \left (\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d x}}{\sqrt {d}}\right )}{a^{3/2}}-\frac {2 \sqrt {d x}}{a}\right )}{d}+\frac {2 \sqrt {d x} \log (1-a x)}{d}\right )+\frac {2 \sqrt {d x} \operatorname {PolyLog}(2,a x)}{d}\) |
2*((2*a*((-2*Sqrt[d*x])/a + (2*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d*x])/Sqrt[d] ])/a^(3/2)))/d + (2*Sqrt[d*x]*Log[1 - a*x])/d) + (2*Sqrt[d*x]*PolyLog[2, a *x])/d
3.1.61.3.1 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 = 0.71 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {2 \sqrt {d x}\, \operatorname {polylog}\left (2, a x \right )+4 \sqrt {d x}\, \ln \left (\frac {-a d x +d}{d}\right )+8 a \left (-\frac {\sqrt {d x}}{a}+\frac {d \,\operatorname {arctanh}\left (\frac {a \sqrt {d x}}{\sqrt {a d}}\right )}{a \sqrt {a d}}\right )}{d}\) | \(74\) |
default | \(\frac {2 \sqrt {d x}\, \operatorname {polylog}\left (2, a x \right )+4 \sqrt {d x}\, \ln \left (\frac {-a d x +d}{d}\right )+8 a \left (-\frac {\sqrt {d x}}{a}+\frac {d \,\operatorname {arctanh}\left (\frac {a \sqrt {d x}}{\sqrt {a d}}\right )}{a \sqrt {a d}}\right )}{d}\) | \(74\) |
meijerg | \(\frac {\sqrt {x}\, \sqrt {-a}\, \left (-\frac {8 \sqrt {x}\, \left (-a \right )^{\frac {3}{2}}}{a}-\frac {4 \sqrt {x}\, \left (-a \right )^{\frac {3}{2}} \left (\ln \left (1-\sqrt {a x}\right )-\ln \left (1+\sqrt {a x}\right )\right )}{a \sqrt {a x}}+\frac {4 \sqrt {x}\, \left (-a \right )^{\frac {3}{2}} \ln \left (-a x +1\right )}{a}+\frac {2 \sqrt {x}\, \left (-a \right )^{\frac {3}{2}} \operatorname {polylog}\left (2, a x \right )}{a}\right )}{\sqrt {d x}\, a}\) | \(109\) |
2/d*((d*x)^(1/2)*polylog(2,a*x)+2*(d*x)^(1/2)*ln((-a*d*x+d)/d)+4*a*(-(d*x) ^(1/2)/a+d/a/(a*d)^(1/2)*arctanh(a*(d*x)^(1/2)/(a*d)^(1/2))))
Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.69 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{\sqrt {d x}} \, dx=\left [\frac {2 \, {\left (\sqrt {d x} {\left (a {\rm Li}_2\left (a x\right ) + 2 \, a \log \left (-a x + 1\right ) - 4 \, a\right )} + 2 \, \sqrt {a d} \log \left (\frac {a d x + 2 \, \sqrt {a d} \sqrt {d x} + d}{a x - 1}\right )\right )}}{a d}, \frac {2 \, {\left (\sqrt {d x} {\left (a {\rm Li}_2\left (a x\right ) + 2 \, a \log \left (-a x + 1\right ) - 4 \, a\right )} - 4 \, \sqrt {-a d} \arctan \left (\frac {\sqrt {-a d} \sqrt {d x}}{a d x}\right )\right )}}{a d}\right ] \]
[2*(sqrt(d*x)*(a*dilog(a*x) + 2*a*log(-a*x + 1) - 4*a) + 2*sqrt(a*d)*log(( a*d*x + 2*sqrt(a*d)*sqrt(d*x) + d)/(a*x - 1)))/(a*d), 2*(sqrt(d*x)*(a*dilo g(a*x) + 2*a*log(-a*x + 1) - 4*a) - 4*sqrt(-a*d)*arctan(sqrt(-a*d)*sqrt(d* x)/(a*d*x)))/(a*d)]
\[ \int \frac {\operatorname {PolyLog}(2,a x)}{\sqrt {d x}} \, dx=\int \frac {\operatorname {Li}_{2}\left (a x\right )}{\sqrt {d x}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.04 \[ \int \frac {\operatorname {PolyLog}(2,a x)}{\sqrt {d x}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {d x} {\left (\log \left (d\right ) + 2\right )} - \sqrt {d x} {\rm Li}_2\left (a x\right ) - 2 \, \sqrt {d x} \log \left (-a d x + d\right ) + \frac {2 \, d \log \left (\frac {\sqrt {d x} a - \sqrt {a d}}{\sqrt {d x} a + \sqrt {a d}}\right )}{\sqrt {a d}}\right )}}{d} \]
-2*(2*sqrt(d*x)*(log(d) + 2) - sqrt(d*x)*dilog(a*x) - 2*sqrt(d*x)*log(-a*d *x + d) + 2*d*log((sqrt(d*x)*a - sqrt(a*d))/(sqrt(d*x)*a + sqrt(a*d)))/sqr t(a*d))/d
\[ \int \frac {\operatorname {PolyLog}(2,a x)}{\sqrt {d x}} \, dx=\int { \frac {{\rm Li}_2\left (a x\right )}{\sqrt {d x}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {PolyLog}(2,a x)}{\sqrt {d x}} \, dx=\int \frac {\mathrm {polylog}\left (2,a\,x\right )}{\sqrt {d\,x}} \,d x \]