Integrand size = 14, antiderivative size = 319 \[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=-\frac {19}{54} i (1-i x)^{5/6} \sqrt [6]{1+i x}-\frac {1}{18} i (1-i x)^{5/6} (1+i x)^{7/6}+\frac {1}{3} (1-i x)^{5/6} (1+i x)^{7/6} x+\frac {19}{162} i \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19}{162} i \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19}{81} i \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {19 i \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{108 \sqrt {3}}+\frac {19 i \log \left (1+\frac {\sqrt [3]{1-i x}}{\sqrt [3]{1+i x}}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )}{108 \sqrt {3}} \]
-19/54*I*(1-I*x)^(5/6)*(1+I*x)^(1/6)-1/18*I*(1-I*x)^(5/6)*(1+I*x)^(7/6)+1/ 3*(1-I*x)^(5/6)*(1+I*x)^(7/6)*x-19/81*I*arctan((1-I*x)^(1/6)/(1+I*x)^(1/6) )-19/162*I*arctan(2*(1-I*x)^(1/6)/(1+I*x)^(1/6)-3^(1/2))-19/162*I*arctan(2 *(1-I*x)^(1/6)/(1+I*x)^(1/6)+3^(1/2))-19/324*I*ln(1+(1-I*x)^(1/3)/(1+I*x)^ (1/3)-(1-I*x)^(1/6)*3^(1/2)/(1+I*x)^(1/6))*3^(1/2)+19/324*I*ln(1+(1-I*x)^( 1/3)/(1+I*x)^(1/3)+(1-I*x)^(1/6)*3^(1/2)/(1+I*x)^(1/6))*3^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.23 \[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=\frac {1}{90} (1-i x)^{5/6} \left (5 \sqrt [6]{1+i x} \left (-i+7 x+6 i x^2\right )-38 i \sqrt [6]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {5}{6},\frac {11}{6},\frac {1}{2}-\frac {i x}{2}\right )\right ) \]
((1 - I*x)^(5/6)*(5*(1 + I*x)^(1/6)*(-I + 7*x + (6*I)*x^2) - (38*I)*2^(1/6 )*Hypergeometric2F1[-1/6, 5/6, 11/6, 1/2 - (I/2)*x]))/90
Time = 0.41 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.95, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {5585, 101, 27, 90, 60, 73, 854, 824, 27, 216, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int x^2 e^{\frac {1}{3} i \arctan (x)} \, dx\) |
| \(\Big \downarrow \) 5585 |
| \(\displaystyle \int \frac {\sqrt [6]{1+i x} x^2}{\sqrt [6]{1-i x}}dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {1}{3} \int -\frac {\sqrt [6]{i x+1} (i x+3)}{3 \sqrt [6]{1-i x}}dx+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} (1-i x)^{5/6} (1+i x)^{7/6} x-\frac {1}{9} \int \frac {\sqrt [6]{i x+1} (i x+3)}{\sqrt [6]{1-i x}}dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{9} \left (-\frac {19}{6} \int \frac {\sqrt [6]{i x+1}}{\sqrt [6]{1-i x}}dx-\frac {1}{2} i (1-i x)^{5/6} (1+i x)^{7/6}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{9} \left (-\frac {19}{6} \left (\frac {1}{3} \int \frac {1}{\sqrt [6]{1-i x} (i x+1)^{5/6}}dx+i (1-i x)^{5/6} \sqrt [6]{1+i x}\right )-\frac {1}{2} i (1-i x)^{5/6} (1+i x)^{7/6}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{9} \left (-\frac {19}{6} \left (2 i \int \frac {(1-i x)^{2/3}}{(i x+1)^{5/6}}d\sqrt [6]{1-i x}+i (1-i x)^{5/6} \sqrt [6]{1+i x}\right )-\frac {1}{2} i (1-i x)^{5/6} (1+i x)^{7/6}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {1}{9} \left (-\frac {19}{6} \left (2 i \int \frac {(1-i x)^{2/3}}{2-i x}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+i (1-i x)^{5/6} \sqrt [6]{1+i x}\right )-\frac {1}{2} i (1-i x)^{5/6} (1+i x)^{7/6}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {1}{9} \left (-\frac {19}{6} \left (2 i \left (\frac {1}{3} \int \frac {1}{\sqrt [3]{1-i x}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\frac {1}{3} \int -\frac {1-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{2 \left (\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1\right )}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\frac {1}{3} \int -\frac {\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}{2 \left (\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1\right )}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+i (1-i x)^{5/6} \sqrt [6]{1+i x}\right )-\frac {1}{2} i (1-i x)^{5/6} (1+i x)^{7/6}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (-\frac {19}{6} \left (2 i \left (\frac {1}{3} \int \frac {1}{\sqrt [3]{1-i x}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+i (1-i x)^{5/6} \sqrt [6]{1+i x}\right )-\frac {1}{2} i (1-i x)^{5/6} (1+i x)^{7/6}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{9} \left (-\frac {19}{6} \left (2 i \left (-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\right )+i (1-i x)^{5/6} \sqrt [6]{1+i x}\right )-\frac {1}{2} i (1-i x)^{5/6} (1+i x)^{7/6}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{9} \left (-\frac {19}{6} \left (2 i \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\sqrt {3}}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\right )+i (1-i x)^{5/6} \sqrt [6]{1+i x}\right )-\frac {1}{2} i (1-i x)^{5/6} (1+i x)^{7/6}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{9} \left (-\frac {19}{6} \left (2 i \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\sqrt {3}}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\right )+i (1-i x)^{5/6} \sqrt [6]{1+i x}\right )-\frac {1}{2} i (1-i x)^{5/6} (1+i x)^{7/6}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{9} \left (-\frac {19}{6} \left (2 i \left (\frac {1}{6} \left (-\int \frac {1}{-\sqrt [3]{1-i x}-1}d\left (\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{6} \left (-\int \frac {1}{-\sqrt [3]{1-i x}-1}d\left (\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\sqrt {3}}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\right )+i (1-i x)^{5/6} \sqrt [6]{1+i x}\right )-\frac {1}{2} i (1-i x)^{5/6} (1+i x)^{7/6}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{9} \left (-\frac {19}{6} \left (2 i \left (\frac {1}{6} \left (-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}}{\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\right )+\frac {1}{6} \left (\arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+\sqrt {3}}{\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}+1}d\frac {\sqrt [6]{1-i x}}{\sqrt [6]{i x+1}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\right )+i (1-i x)^{5/6} \sqrt [6]{1+i x}\right )-\frac {1}{2} i (1-i x)^{5/6} (1+i x)^{7/6}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{9} \left (-\frac {19}{6} \left (2 i \left (\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\sqrt [3]{1-i x}-\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )\right )+\frac {1}{6} \left (\arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}\right )-\frac {1}{2} \sqrt {3} \log \left (\sqrt [3]{1-i x}+\frac {\sqrt {3} \sqrt [6]{1-i x}}{\sqrt [6]{1+i x}}+1\right )\right )\right )+i (1-i x)^{5/6} \sqrt [6]{1+i x}\right )-\frac {1}{2} i (1-i x)^{5/6} (1+i x)^{7/6}\right )+\frac {1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}\) |
((1 - I*x)^(5/6)*(1 + I*x)^(7/6)*x)/3 + ((-1/2*I)*(1 - I*x)^(5/6)*(1 + I*x )^(7/6) - (19*(I*(1 - I*x)^(5/6)*(1 + I*x)^(1/6) + (2*I)*(ArcTan[(1 - I*x) ^(1/6)/(1 + I*x)^(1/6)]/3 + (-ArcTan[Sqrt[3] - (2*(1 - I*x)^(1/6))/(1 + I* x)^(1/6)] + (Sqrt[3]*Log[1 + (1 - I*x)^(1/3) - (Sqrt[3]*(1 - I*x)^(1/6))/( 1 + I*x)^(1/6)])/2)/6 + (ArcTan[Sqrt[3] + (2*(1 - I*x)^(1/6))/(1 + I*x)^(1 /6)] - (Sqrt[3]*Log[1 + (1 - I*x)^(1/3) + (Sqrt[3]*(1 - I*x)^(1/6))/(1 + I *x)^(1/6)])/2)/6)))/6)/9
3.2.15.3.1 Defintions of rubi rules used
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_))^(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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*((1 - I*a *x)^(I*(n/2))/(1 + I*a*x)^(I*(n/2))), x] /; FreeQ[{a, m, n}, x] && !Intege rQ[(I*n - 1)/2]
\[\int {\left (\frac {i x +1}{\sqrt {x^{2}+1}}\right )}^{\frac {1}{3}} x^{2}d x\]
Time = 0.26 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.65 \[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=-\frac {19}{324} \, {\left (-i \, \sqrt {3} + 1\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) - \frac {19}{324} \, {\left (-i \, \sqrt {3} - 1\right )} \log \left (\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) - \frac {19}{324} \, {\left (i \, \sqrt {3} + 1\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + \frac {1}{2} i\right ) - \frac {19}{324} \, {\left (i \, \sqrt {3} - 1\right )} \log \left (-\frac {1}{2} \, \sqrt {3} + \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {1}{2} i\right ) + \frac {1}{54} \, {\left (18 \, x^{3} - 3 i \, x^{2} - x - 22 i\right )} \left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - \frac {19}{162} \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} + i\right ) + \frac {19}{162} \, \log \left (\left (\frac {i \, \sqrt {x^{2} + 1}}{x + i}\right )^{\frac {1}{3}} - i\right ) \]
-19/324*(-I*sqrt(3) + 1)*log(1/2*sqrt(3) + (I*sqrt(x^2 + 1)/(x + I))^(1/3) + 1/2*I) - 19/324*(-I*sqrt(3) - 1)*log(1/2*sqrt(3) + (I*sqrt(x^2 + 1)/(x + I))^(1/3) - 1/2*I) - 19/324*(I*sqrt(3) + 1)*log(-1/2*sqrt(3) + (I*sqrt(x ^2 + 1)/(x + I))^(1/3) + 1/2*I) - 19/324*(I*sqrt(3) - 1)*log(-1/2*sqrt(3) + (I*sqrt(x^2 + 1)/(x + I))^(1/3) - 1/2*I) + 1/54*(18*x^3 - 3*I*x^2 - x - 22*I)*(I*sqrt(x^2 + 1)/(x + I))^(1/3) - 19/162*log((I*sqrt(x^2 + 1)/(x + I ))^(1/3) + I) + 19/162*log((I*sqrt(x^2 + 1)/(x + I))^(1/3) - I)
\[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=\int x^{2} \sqrt [3]{\frac {i \left (x - i\right )}{\sqrt {x^{2} + 1}}}\, dx \]
\[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=\int { x^{2} \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}} \,d x } \]
\[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=\int { x^{2} \left (\frac {i \, x + 1}{\sqrt {x^{2} + 1}}\right )^{\frac {1}{3}} \,d x } \]
Timed out. \[ \int e^{\frac {1}{3} i \arctan (x)} x^2 \, dx=\int x^2\,{\left (\frac {1+x\,1{}\mathrm {i}}{\sqrt {x^2+1}}\right )}^{1/3} \,d x \]