Integrand size = 15, antiderivative size = 125 \[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=-\frac {32 (d x)^{3/2}}{27 d}-\frac {16 \sqrt {d} \arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{9 a^{3/4}}+\frac {16 \sqrt {d} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{9 a^{3/4}}+\frac {8 (d x)^{3/2} \log \left (1-a x^2\right )}{9 d}+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )}{3 d} \]
-32/27*(d*x)^(3/2)/d+8/9*(d*x)^(3/2)*ln(-a*x^2+1)/d+2/3*(d*x)^(3/2)*polylo g(2,a*x^2)/d-16/9*arctan(a^(1/4)*(d*x)^(1/2)/d^(1/2))*d^(1/2)/a^(3/4)+16/9 *arctanh(a^(1/4)*(d*x)^(1/2)/d^(1/2))*d^(1/2)/a^(3/4)
Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.73 \[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\frac {2 \sqrt {d x} \left (\frac {4 \left (-6 \arctan \left (\sqrt [4]{a} \sqrt {x}\right )+6 \text {arctanh}\left (\sqrt [4]{a} \sqrt {x}\right )+a^{3/4} x^{3/2} \left (-4+3 \log \left (1-a x^2\right )\right )\right )}{a^{3/4}}+9 x^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )\right )}{27 \sqrt {x}} \]
(2*Sqrt[d*x]*((4*(-6*ArcTan[a^(1/4)*Sqrt[x]] + 6*ArcTanh[a^(1/4)*Sqrt[x]] + a^(3/4)*x^(3/2)*(-4 + 3*Log[1 - a*x^2])))/a^(3/4) + 9*x^(3/2)*PolyLog[2, a*x^2]))/(27*Sqrt[x])
Time = 0.33 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {7145, 25, 2905, 8, 262, 266, 27, 827, 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.
| \(\displaystyle \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx\) |
| \(\Big \downarrow \) 7145 |
| \(\displaystyle \frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )}{3 d}-\frac {4}{3} \int -\sqrt {d x} \log \left (1-a x^2\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{3} \int \sqrt {d x} \log \left (1-a x^2\right )dx+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )}{3 d}\) |
| \(\Big \downarrow \) 2905 |
| \(\displaystyle \frac {4}{3} \left (\frac {4 a \int \frac {x (d x)^{3/2}}{1-a x^2}dx}{3 d}+\frac {2 (d x)^{3/2} \log \left (1-a x^2\right )}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )}{3 d}\) |
| \(\Big \downarrow \) 8 |
| \(\displaystyle \frac {4}{3} \left (\frac {4 a \int \frac {(d x)^{5/2}}{1-a x^2}dx}{3 d^2}+\frac {2 (d x)^{3/2} \log \left (1-a x^2\right )}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )}{3 d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {4}{3} \left (\frac {4 a \left (\frac {d^2 \int \frac {\sqrt {d x}}{1-a x^2}dx}{a}-\frac {2 d (d x)^{3/2}}{3 a}\right )}{3 d^2}+\frac {2 (d x)^{3/2} \log \left (1-a x^2\right )}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )}{3 d}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {4}{3} \left (\frac {4 a \left (\frac {2 d \int \frac {d^3 x}{d^2-a d^2 x^2}d\sqrt {d x}}{a}-\frac {2 d (d x)^{3/2}}{3 a}\right )}{3 d^2}+\frac {2 (d x)^{3/2} \log \left (1-a x^2\right )}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{3} \left (\frac {4 a \left (\frac {2 d^3 \int \frac {d x}{d^2-a d^2 x^2}d\sqrt {d x}}{a}-\frac {2 d (d x)^{3/2}}{3 a}\right )}{3 d^2}+\frac {2 (d x)^{3/2} \log \left (1-a x^2\right )}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )}{3 d}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {4}{3} \left (\frac {4 a \left (\frac {2 d^3 \left (\frac {\int \frac {1}{d-\sqrt {a} d x}d\sqrt {d x}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a} x d+d}d\sqrt {d x}}{2 \sqrt {a}}\right )}{a}-\frac {2 d (d x)^{3/2}}{3 a}\right )}{3 d^2}+\frac {2 (d x)^{3/2} \log \left (1-a x^2\right )}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )}{3 d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {4}{3} \left (\frac {4 a \left (\frac {2 d^3 \left (\frac {\int \frac {1}{d-\sqrt {a} d x}d\sqrt {d x}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 a^{3/4} \sqrt {d}}\right )}{a}-\frac {2 d (d x)^{3/2}}{3 a}\right )}{3 d^2}+\frac {2 (d x)^{3/2} \log \left (1-a x^2\right )}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )}{3 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4}{3} \left (\frac {4 a \left (\frac {2 d^3 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 a^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 a^{3/4} \sqrt {d}}\right )}{a}-\frac {2 d (d x)^{3/2}}{3 a}\right )}{3 d^2}+\frac {2 (d x)^{3/2} \log \left (1-a x^2\right )}{3 d}\right )+\frac {2 (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right )}{3 d}\) |
(4*((4*a*((-2*d*(d*x)^(3/2))/(3*a) + (2*d^3*(-1/2*ArcTan[(a^(1/4)*Sqrt[d*x ])/Sqrt[d]]/(a^(3/4)*Sqrt[d]) + ArcTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]]/(2*a^ (3/4)*Sqrt[d])))/a))/(3*d^2) + (2*(d*x)^(3/2)*Log[1 - a*x^2])/(3*d)))/3 + (2*(d*x)^(3/2)*PolyLog[2, a*x^2])/(3*d)
3.1.73.3.1 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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) 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 = 0.46 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02
| method | result | size |
| meijerg | \(-\frac {\sqrt {d x}\, \left (-\frac {64 x^{\frac {3}{2}} \left (-a \right )^{\frac {7}{4}}}{27 a}-\frac {16 x^{\frac {3}{2}} \left (-a \right )^{\frac {7}{4}} \left (\ln \left (1-\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (a \,x^{2}\right )^{\frac {1}{4}}\right )+2 \arctan \left (\left (a \,x^{2}\right )^{\frac {1}{4}}\right )\right )}{9 a \left (a \,x^{2}\right )^{\frac {3}{4}}}+\frac {16 x^{\frac {3}{2}} \left (-a \right )^{\frac {7}{4}} \ln \left (-a \,x^{2}+1\right )}{9 a}+\frac {4 x^{\frac {3}{2}} \left (-a \right )^{\frac {7}{4}} \operatorname {polylog}\left (2, a \,x^{2}\right )}{3 a}\right )}{2 \sqrt {x}\, \left (-a \right )^{\frac {3}{4}}}\) | \(127\) |
| derivativedivides | \(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} \operatorname {polylog}\left (2, a \,x^{2}\right )}{3}+\frac {8 \left (d x \right )^{\frac {3}{2}} \ln \left (\frac {-a \,d^{2} x^{2}+d^{2}}{d^{2}}\right )}{9}+\frac {32 a \left (-\frac {\left (d x \right )^{\frac {3}{2}}}{3 a}-\frac {d^{2} \left (2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )\right )}{4 a^{2} \left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )}{9}}{d}\) | \(134\) |
| default | \(\frac {\frac {2 \left (d x \right )^{\frac {3}{2}} \operatorname {polylog}\left (2, a \,x^{2}\right )}{3}+\frac {8 \left (d x \right )^{\frac {3}{2}} \ln \left (\frac {-a \,d^{2} x^{2}+d^{2}}{d^{2}}\right )}{9}+\frac {32 a \left (-\frac {\left (d x \right )^{\frac {3}{2}}}{3 a}-\frac {d^{2} \left (2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )\right )}{4 a^{2} \left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )}{9}}{d}\) | \(134\) |
-1/2*(d*x)^(1/2)/x^(1/2)/(-a)^(3/4)*(-64/27*x^(3/2)*(-a)^(7/4)/a-16/9*x^(3 /2)*(-a)^(7/4)/a/(a*x^2)^(3/4)*(ln(1-(a*x^2)^(1/4))-ln(1+(a*x^2)^(1/4))+2* arctan((a*x^2)^(1/4)))+16/9*x^(3/2)*(-a)^(7/4)*ln(-a*x^2+1)/a+4/3*x^(3/2)* (-a)^(7/4)/a*polylog(2,a*x^2))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.38 \[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\frac {2}{27} \, \sqrt {d x} {\left (9 \, x {\rm Li}_2\left (a x^{2}\right ) + 12 \, x \log \left (-a x^{2} + 1\right ) - 16 \, x\right )} + \frac {8}{9} \, \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}} \log \left (512 \, a^{2} \left (\frac {d^{2}}{a^{3}}\right )^{\frac {3}{4}} + 512 \, \sqrt {d x} d\right ) - \frac {8}{9} i \, \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}} \log \left (512 i \, a^{2} \left (\frac {d^{2}}{a^{3}}\right )^{\frac {3}{4}} + 512 \, \sqrt {d x} d\right ) + \frac {8}{9} i \, \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}} \log \left (-512 i \, a^{2} \left (\frac {d^{2}}{a^{3}}\right )^{\frac {3}{4}} + 512 \, \sqrt {d x} d\right ) - \frac {8}{9} \, \left (\frac {d^{2}}{a^{3}}\right )^{\frac {1}{4}} \log \left (-512 \, a^{2} \left (\frac {d^{2}}{a^{3}}\right )^{\frac {3}{4}} + 512 \, \sqrt {d x} d\right ) \]
2/27*sqrt(d*x)*(9*x*dilog(a*x^2) + 12*x*log(-a*x^2 + 1) - 16*x) + 8/9*(d^2 /a^3)^(1/4)*log(512*a^2*(d^2/a^3)^(3/4) + 512*sqrt(d*x)*d) - 8/9*I*(d^2/a^ 3)^(1/4)*log(512*I*a^2*(d^2/a^3)^(3/4) + 512*sqrt(d*x)*d) + 8/9*I*(d^2/a^3 )^(1/4)*log(-512*I*a^2*(d^2/a^3)^(3/4) + 512*sqrt(d*x)*d) - 8/9*(d^2/a^3)^ (1/4)*log(-512*a^2*(d^2/a^3)^(3/4) + 512*sqrt(d*x)*d)
\[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\int \sqrt {d x} \operatorname {Li}_{2}\left (a x^{2}\right )\, dx \]
Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.11 \[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=-\frac {2 \, {\left (12 \, d^{2} {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {a}}{\sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} \sqrt {a}} + \frac {\log \left (\frac {\sqrt {d x} \sqrt {a} - \sqrt {\sqrt {a} d}}{\sqrt {d x} \sqrt {a} + \sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} \sqrt {a}}\right )} + 8 \, \left (d x\right )^{\frac {3}{2}} {\left (3 \, \log \left (d\right ) + 2\right )} - 9 \, \left (d x\right )^{\frac {3}{2}} {\rm Li}_2\left (a x^{2}\right ) - 12 \, \left (d x\right )^{\frac {3}{2}} \log \left (-a d^{2} x^{2} + d^{2}\right )\right )}}{27 \, d} \]
-2/27*(12*d^2*(2*arctan(sqrt(d*x)*sqrt(a)/sqrt(sqrt(a)*d))/(sqrt(sqrt(a)*d )*sqrt(a)) + log((sqrt(d*x)*sqrt(a) - sqrt(sqrt(a)*d))/(sqrt(d*x)*sqrt(a) + sqrt(sqrt(a)*d)))/(sqrt(sqrt(a)*d)*sqrt(a))) + 8*(d*x)^(3/2)*(3*log(d) + 2) - 9*(d*x)^(3/2)*dilog(a*x^2) - 12*(d*x)^(3/2)*log(-a*d^2*x^2 + d^2))/d
\[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\int { \sqrt {d x} {\rm Li}_2\left (a x^{2}\right ) \,d x } \]
Timed out. \[ \int \sqrt {d x} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\int \mathrm {polylog}\left (2,a\,x^2\right )\,\sqrt {d\,x} \,d x \]