Integrand size = 15, antiderivative size = 103 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{(d x)^{3/2}} \, dx=-\frac {16 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {16 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {8 \log \left (1-a x^2\right )}{d \sqrt {d x}}-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{d \sqrt {d x}} \]
-16*a^(1/4)*arctan(a^(1/4)*(d*x)^(1/2)/d^(1/2))/d^(3/2)+16*a^(1/4)*arctanh (a^(1/4)*(d*x)^(1/2)/d^(1/2))/d^(3/2)+8*ln(-a*x^2+1)/d/(d*x)^(1/2)-2*polyl og(2,a*x^2)/d/(d*x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.60 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{(d x)^{3/2}} \, dx=\frac {x \operatorname {Gamma}\left (\frac {3}{4}\right ) \left (16 a x^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},a x^2\right )+12 \log \left (1-a x^2\right )-3 \operatorname {PolyLog}\left (2,a x^2\right )\right )}{2 (d x)^{3/2} \operatorname {Gamma}\left (\frac {7}{4}\right )} \]
(x*Gamma[3/4]*(16*a*x^2*Hypergeometric2F1[3/4, 1, 7/4, a*x^2] + 12*Log[1 - a*x^2] - 3*PolyLog[2, a*x^2]))/(2*(d*x)^(3/2)*Gamma[7/4])
Time = 0.31 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {7145, 25, 2905, 8, 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.
| \(\displaystyle \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{(d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 7145 |
| \(\displaystyle 4 \int -\frac {\log \left (1-a x^2\right )}{(d x)^{3/2}}dx-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \frac {\log \left (1-a x^2\right )}{(d x)^{3/2}}dx-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 2905 |
| \(\displaystyle -4 \left (-\frac {4 a \int \frac {x}{\sqrt {d x} \left (1-a x^2\right )}dx}{d}-\frac {2 \log \left (1-a x^2\right )}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 8 |
| \(\displaystyle -4 \left (-\frac {4 a \int \frac {\sqrt {d x}}{1-a x^2}dx}{d^2}-\frac {2 \log \left (1-a x^2\right )}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -4 \left (-\frac {8 a \int \frac {d^3 x}{d^2-a d^2 x^2}d\sqrt {d x}}{d^3}-\frac {2 \log \left (1-a x^2\right )}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -4 \left (-\frac {8 a \int \frac {d x}{d^2-a d^2 x^2}d\sqrt {d x}}{d}-\frac {2 \log \left (1-a x^2\right )}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -4 \left (-\frac {8 a \left (\frac {\int \frac {1}{d-\sqrt {a} d x}d\sqrt {d x}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a} x d+d}d\sqrt {d x}}{2 \sqrt {a}}\right )}{d}-\frac {2 \log \left (1-a x^2\right )}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -4 \left (-\frac {8 a \left (\frac {\int \frac {1}{d-\sqrt {a} d x}d\sqrt {d x}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 a^{3/4} \sqrt {d}}\right )}{d}-\frac {2 \log \left (1-a x^2\right )}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{d \sqrt {d x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -4 \left (-\frac {8 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 a^{3/4} \sqrt {d}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 a^{3/4} \sqrt {d}}\right )}{d}-\frac {2 \log \left (1-a x^2\right )}{d \sqrt {d x}}\right )-\frac {2 \operatorname {PolyLog}\left (2,a x^2\right )}{d \sqrt {d x}}\) |
-4*((-8*a*(-1/2*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]]/(a^(3/4)*Sqrt[d]) + Ar cTanh[(a^(1/4)*Sqrt[d*x])/Sqrt[d]]/(2*a^(3/4)*Sqrt[d])))/d - (2*Log[1 - a* x^2])/(d*Sqrt[d*x])) - (2*PolyLog[2, a*x^2])/(d*Sqrt[d*x])
3.1.75.3.1 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] :> 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 = 0.47 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.08
| method | result | size |
| meijerg | \(-\frac {x^{\frac {3}{2}} \left (-a \right )^{\frac {1}{4}} \left (-\frac {16 x^{\frac {3}{2}} \left (-a \right )^{\frac {3}{4}} \left (\ln \left (1-\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (a \,x^{2}\right )^{\frac {1}{4}}\right )+2 \arctan \left (\left (a \,x^{2}\right )^{\frac {1}{4}}\right )\right )}{\left (a \,x^{2}\right )^{\frac {3}{4}}}+\frac {16 \left (-a \right )^{\frac {3}{4}} \ln \left (-a \,x^{2}+1\right )}{\sqrt {x}\, a}-\frac {4 \left (-a \right )^{\frac {3}{4}} \operatorname {polylog}\left (2, a \,x^{2}\right )}{\sqrt {x}\, a}\right )}{2 \left (d x \right )^{\frac {3}{2}}}\) | \(111\) |
| derivativedivides | \(\frac {-\frac {2 \operatorname {polylog}\left (2, a \,x^{2}\right )}{\sqrt {d x}}+\frac {8 \ln \left (\frac {-a \,d^{2} x^{2}+d^{2}}{d^{2}}\right )}{\sqrt {d x}}-\frac {8 \left (2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )\right )}{\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}}{d}\) | \(114\) |
| default | \(\frac {-\frac {2 \operatorname {polylog}\left (2, a \,x^{2}\right )}{\sqrt {d x}}+\frac {8 \ln \left (\frac {-a \,d^{2} x^{2}+d^{2}}{d^{2}}\right )}{\sqrt {d x}}-\frac {8 \left (2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )\right )}{\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}}{d}\) | \(114\) |
-1/2/(d*x)^(3/2)*x^(3/2)*(-a)^(1/4)*(-16*x^(3/2)*(-a)^(3/4)/(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/x^(1/ 2)*(-a)^(3/4)*ln(-a*x^2+1)/a-4/x^(1/2)*(-a)^(3/4)/a*polylog(2,a*x^2))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.69 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{(d x)^{3/2}} \, dx=\frac {2 \, {\left (4 \, d^{2} x \left (\frac {a}{d^{6}}\right )^{\frac {1}{4}} \log \left (512 \, d^{5} \left (\frac {a}{d^{6}}\right )^{\frac {3}{4}} + 512 \, \sqrt {d x} a\right ) - 4 i \, d^{2} x \left (\frac {a}{d^{6}}\right )^{\frac {1}{4}} \log \left (512 i \, d^{5} \left (\frac {a}{d^{6}}\right )^{\frac {3}{4}} + 512 \, \sqrt {d x} a\right ) + 4 i \, d^{2} x \left (\frac {a}{d^{6}}\right )^{\frac {1}{4}} \log \left (-512 i \, d^{5} \left (\frac {a}{d^{6}}\right )^{\frac {3}{4}} + 512 \, \sqrt {d x} a\right ) - 4 \, d^{2} x \left (\frac {a}{d^{6}}\right )^{\frac {1}{4}} \log \left (-512 \, d^{5} \left (\frac {a}{d^{6}}\right )^{\frac {3}{4}} + 512 \, \sqrt {d x} a\right ) - \sqrt {d x} {\left ({\rm Li}_2\left (a x^{2}\right ) - 4 \, \log \left (-a x^{2} + 1\right )\right )}\right )}}{d^{2} x} \]
2*(4*d^2*x*(a/d^6)^(1/4)*log(512*d^5*(a/d^6)^(3/4) + 512*sqrt(d*x)*a) - 4* I*d^2*x*(a/d^6)^(1/4)*log(512*I*d^5*(a/d^6)^(3/4) + 512*sqrt(d*x)*a) + 4*I *d^2*x*(a/d^6)^(1/4)*log(-512*I*d^5*(a/d^6)^(3/4) + 512*sqrt(d*x)*a) - 4*d ^2*x*(a/d^6)^(1/4)*log(-512*d^5*(a/d^6)^(3/4) + 512*sqrt(d*x)*a) - sqrt(d* x)*(dilog(a*x^2) - 4*log(-a*x^2 + 1)))/(d^2*x)
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{(d x)^{3/2}} \, dx=\int \frac {\operatorname {Li}_{2}\left (a x^{2}\right )}{\left (d x\right )^{\frac {3}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.19 \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{(d x)^{3/2}} \, dx=-\frac {2 \, {\left (4 \, a {\left (\frac {2 \, \arctan \left (\frac {\sqrt {d x} \sqrt {a}}{\sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} \sqrt {a}} + \frac {\log \left (\frac {\sqrt {d x} \sqrt {a} - \sqrt {\sqrt {a} d}}{\sqrt {d x} \sqrt {a} + \sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d} \sqrt {a}}\right )} + \frac {{\rm Li}_2\left (a x^{2}\right ) - 4 \, \log \left (-a d^{2} x^{2} + d^{2}\right ) + 8 \, \log \left (d\right )}{\sqrt {d x}}\right )}}{d} \]
-2*(4*a*(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))) + (dilog(a*x^2) - 4*log(-a*d^2*x^ 2 + d^2) + 8*log(d))/sqrt(d*x))/d
\[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{(d x)^{3/2}} \, dx=\int { \frac {{\rm Li}_2\left (a x^{2}\right )}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {PolyLog}\left (2,a x^2\right )}{(d x)^{3/2}} \, dx=\int \frac {\mathrm {polylog}\left (2,a\,x^2\right )}{{\left (d\,x\right )}^{3/2}} \,d x \]