Integrand size = 15, antiderivative size = 140 \[ \int (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=-\frac {32 d \sqrt {d x}}{25 a}-\frac {32 (d x)^{5/2}}{125 d}+\frac {16 d^{3/2} \arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{25 a^{5/4}}+\frac {16 d^{3/2} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{25 a^{5/4}}+\frac {8 (d x)^{5/2} \log \left (1-a x^2\right )}{25 d}+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d} \]
-32/125*(d*x)^(5/2)/d+16/25*d^(3/2)*arctan(a^(1/4)*(d*x)^(1/2)/d^(1/2))/a^ (5/4)+16/25*d^(3/2)*arctanh(a^(1/4)*(d*x)^(1/2)/d^(1/2))/a^(5/4)+8/25*(d*x )^(5/2)*ln(-a*x^2+1)/d+2/5*(d*x)^(5/2)*polylog(2,a*x^2)/d-32/25*d*(d*x)^(1 /2)/a
Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.72 \[ \int (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\frac {2 (d x)^{3/2} \left (\frac {40 \arctan \left (\sqrt [4]{a} \sqrt {x}\right )+40 \text {arctanh}\left (\sqrt [4]{a} \sqrt {x}\right )+4 \sqrt [4]{a} \sqrt {x} \left (-20-4 a x^2+5 a x^2 \log \left (1-a x^2\right )\right )}{a^{5/4}}+25 x^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )\right )}{125 x^{3/2}} \]
(2*(d*x)^(3/2)*((40*ArcTan[a^(1/4)*Sqrt[x]] + 40*ArcTanh[a^(1/4)*Sqrt[x]] + 4*a^(1/4)*Sqrt[x]*(-20 - 4*a*x^2 + 5*a*x^2*Log[1 - a*x^2]))/a^(5/4) + 25 *x^(5/2)*PolyLog[2, a*x^2]))/(125*x^(3/2))
Time = 0.34 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.20, 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, 262, 266, 756, 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.
| \(\displaystyle \int (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx\) |
| \(\Big \downarrow \) 7145 |
| \(\displaystyle \frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}-\frac {4}{5} \int -(d x)^{3/2} \log \left (1-a x^2\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{5} \int (d x)^{3/2} \log \left (1-a x^2\right )dx+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\) |
| \(\Big \downarrow \) 2905 |
| \(\displaystyle \frac {4}{5} \left (\frac {4 a \int \frac {x (d x)^{5/2}}{1-a x^2}dx}{5 d}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\) |
| \(\Big \downarrow \) 8 |
| \(\displaystyle \frac {4}{5} \left (\frac {4 a \int \frac {(d x)^{7/2}}{1-a x^2}dx}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {4}{5} \left (\frac {4 a \left (\frac {d^2 \int \frac {(d x)^{3/2}}{1-a x^2}dx}{a}-\frac {2 d (d x)^{5/2}}{5 a}\right )}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {4}{5} \left (\frac {4 a \left (\frac {d^2 \left (\frac {d^2 \int \frac {1}{\sqrt {d x} \left (1-a x^2\right )}dx}{a}-\frac {2 d \sqrt {d x}}{a}\right )}{a}-\frac {2 d (d x)^{5/2}}{5 a}\right )}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {4}{5} \left (\frac {4 a \left (\frac {d^2 \left (\frac {2 d \int \frac {1}{1-a x^2}d\sqrt {d x}}{a}-\frac {2 d \sqrt {d x}}{a}\right )}{a}-\frac {2 d (d x)^{5/2}}{5 a}\right )}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {4}{5} \left (\frac {4 a \left (\frac {d^2 \left (\frac {2 d \left (\frac {1}{2} d \int \frac {1}{d-\sqrt {a} d x}d\sqrt {d x}+\frac {1}{2} d \int \frac {1}{\sqrt {a} x d+d}d\sqrt {d x}\right )}{a}-\frac {2 d \sqrt {d x}}{a}\right )}{a}-\frac {2 d (d x)^{5/2}}{5 a}\right )}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {4}{5} \left (\frac {4 a \left (\frac {d^2 \left (\frac {2 d \left (\frac {1}{2} d \int \frac {1}{d-\sqrt {a} d x}d\sqrt {d x}+\frac {\sqrt {d} \arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{a}}\right )}{a}-\frac {2 d \sqrt {d x}}{a}\right )}{a}-\frac {2 d (d x)^{5/2}}{5 a}\right )}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4}{5} \left (\frac {4 a \left (\frac {d^2 \left (\frac {2 d \left (\frac {\sqrt {d} \arctan \left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{a}}+\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {d x}}{\sqrt {d}}\right )}{2 \sqrt [4]{a}}\right )}{a}-\frac {2 d \sqrt {d x}}{a}\right )}{a}-\frac {2 d (d x)^{5/2}}{5 a}\right )}{5 d^2}+\frac {2 (d x)^{5/2} \log \left (1-a x^2\right )}{5 d}\right )+\frac {2 (d x)^{5/2} \operatorname {PolyLog}\left (2,a x^2\right )}{5 d}\) |
(4*((4*a*((-2*d*(d*x)^(5/2))/(5*a) + (d^2*((-2*d*Sqrt[d*x])/a + (2*d*((Sqr t[d]*ArcTan[(a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(2*a^(1/4)) + (Sqrt[d]*ArcTanh[( a^(1/4)*Sqrt[d*x])/Sqrt[d]])/(2*a^(1/4))))/a))/a))/(5*d^2) + (2*(d*x)^(5/2 )*Log[1 - a*x^2])/(5*d)))/5 + (2*(d*x)^(5/2)*PolyLog[2, a*x^2])/(5*d)
3.1.72.3.1 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_) + (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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) 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 = 135, normalized size of antiderivative = 0.96
| method | result | size |
| meijerg | \(-\frac {\left (d x \right )^{\frac {3}{2}} \left (-\frac {4 \sqrt {x}\, \left (-a \right )^{\frac {9}{4}} \left (144 a \,x^{2}+720\right )}{1125 a^{2}}-\frac {16 \sqrt {x}\, \left (-a \right )^{\frac {9}{4}} \left (\ln \left (1-\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (a \,x^{2}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (a \,x^{2}\right )^{\frac {1}{4}}\right )\right )}{25 a^{2} \left (a \,x^{2}\right )^{\frac {1}{4}}}+\frac {16 x^{\frac {5}{2}} \left (-a \right )^{\frac {9}{4}} \ln \left (-a \,x^{2}+1\right )}{25 a}+\frac {4 x^{\frac {5}{2}} \left (-a \right )^{\frac {9}{4}} \operatorname {polylog}\left (2, a \,x^{2}\right )}{5 a}\right )}{2 x^{\frac {3}{2}} \left (-a \right )^{\frac {5}{4}}}\) | \(135\) |
| derivativedivides | \(\frac {\frac {2 \left (d x \right )^{\frac {5}{2}} \operatorname {polylog}\left (2, a \,x^{2}\right )}{5}+\frac {8 \left (d x \right )^{\frac {5}{2}} \ln \left (\frac {-a \,d^{2} x^{2}+d^{2}}{d^{2}}\right )}{25}+\frac {32 a \left (-\frac {\frac {a \left (d x \right )^{\frac {5}{2}}}{5}+d^{2} \sqrt {d x}}{a^{2}}+\frac {d^{2} \left (\frac {d^{2}}{a}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )\right )}{4 a^{2}}\right )}{25}}{d}\) | \(145\) |
| default | \(\frac {\frac {2 \left (d x \right )^{\frac {5}{2}} \operatorname {polylog}\left (2, a \,x^{2}\right )}{5}+\frac {8 \left (d x \right )^{\frac {5}{2}} \ln \left (\frac {-a \,d^{2} x^{2}+d^{2}}{d^{2}}\right )}{25}+\frac {32 a \left (-\frac {\frac {a \left (d x \right )^{\frac {5}{2}}}{5}+d^{2} \sqrt {d x}}{a^{2}}+\frac {d^{2} \left (\frac {d^{2}}{a}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sqrt {d x}+\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}{\sqrt {d x}-\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {d x}}{\left (\frac {d^{2}}{a}\right )^{\frac {1}{4}}}\right )\right )}{4 a^{2}}\right )}{25}}{d}\) | \(145\) |
-1/2*(d*x)^(3/2)/x^(3/2)/(-a)^(5/4)*(-4/1125*x^(1/2)*(-a)^(9/4)*(144*a*x^2 +720)/a^2-16/25*x^(1/2)*(-a)^(9/4)/a^2/(a*x^2)^(1/4)*(ln(1-(a*x^2)^(1/4))- ln(1+(a*x^2)^(1/4))-2*arctan((a*x^2)^(1/4)))+16/25*x^(5/2)*(-a)^(9/4)*ln(- a*x^2+1)/a+4/5*x^(5/2)*(-a)^(9/4)/a*polylog(2,a*x^2))
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.34 \[ \int (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\frac {2 \, {\left (20 \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {d x} d + 8 \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}}\right ) + 20 i \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {d x} d + 8 i \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}}\right ) - 20 i \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {d x} d - 8 i \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}}\right ) - 20 \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {d x} d - 8 \, a \left (\frac {d^{6}}{a^{5}}\right )^{\frac {1}{4}}\right ) + {\left (25 \, a d x^{2} {\rm Li}_2\left (a x^{2}\right ) + 20 \, a d x^{2} \log \left (-a x^{2} + 1\right ) - 16 \, a d x^{2} - 80 \, d\right )} \sqrt {d x}\right )}}{125 \, a} \]
2/125*(20*a*(d^6/a^5)^(1/4)*log(8*sqrt(d*x)*d + 8*a*(d^6/a^5)^(1/4)) + 20* I*a*(d^6/a^5)^(1/4)*log(8*sqrt(d*x)*d + 8*I*a*(d^6/a^5)^(1/4)) - 20*I*a*(d ^6/a^5)^(1/4)*log(8*sqrt(d*x)*d - 8*I*a*(d^6/a^5)^(1/4)) - 20*a*(d^6/a^5)^ (1/4)*log(8*sqrt(d*x)*d - 8*a*(d^6/a^5)^(1/4)) + (25*a*d*x^2*dilog(a*x^2) + 20*a*d*x^2*log(-a*x^2 + 1) - 16*a*d*x^2 - 80*d)*sqrt(d*x))/a
Timed out. \[ \int (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.14 \[ \int (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\frac {2 \, {\left (\frac {25 \, \left (d x\right )^{\frac {5}{2}} a {\rm Li}_2\left (a x^{2}\right ) + 20 \, \left (d x\right )^{\frac {5}{2}} a \log \left (-a d^{2} x^{2} + d^{2}\right ) - 8 \, \left (d x\right )^{\frac {5}{2}} {\left (5 \, a \log \left (d\right ) + 2 \, a\right )} - 80 \, \sqrt {d x} d^{2}}{a} + \frac {20 \, {\left (\frac {2 \, d^{3} \arctan \left (\frac {\sqrt {d x} \sqrt {a}}{\sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d}} - \frac {d^{3} \log \left (\frac {\sqrt {d x} \sqrt {a} - \sqrt {\sqrt {a} d}}{\sqrt {d x} \sqrt {a} + \sqrt {\sqrt {a} d}}\right )}{\sqrt {\sqrt {a} d}}\right )}}{a}\right )}}{125 \, d} \]
2/125*((25*(d*x)^(5/2)*a*dilog(a*x^2) + 20*(d*x)^(5/2)*a*log(-a*d^2*x^2 + d^2) - 8*(d*x)^(5/2)*(5*a*log(d) + 2*a) - 80*sqrt(d*x)*d^2)/a + 20*(2*d^3* arctan(sqrt(d*x)*sqrt(a)/sqrt(sqrt(a)*d))/sqrt(sqrt(a)*d) - d^3*log((sqrt( d*x)*sqrt(a) - sqrt(sqrt(a)*d))/(sqrt(d*x)*sqrt(a) + sqrt(sqrt(a)*d)))/sqr t(sqrt(a)*d))/a)/d
\[ \int (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\int { \left (d x\right )^{\frac {3}{2}} {\rm Li}_2\left (a x^{2}\right ) \,d x } \]
Timed out. \[ \int (d x)^{3/2} \operatorname {PolyLog}\left (2,a x^2\right ) \, dx=\int \mathrm {polylog}\left (2,a\,x^2\right )\,{\left (d\,x\right )}^{3/2} \,d x \]