Integrand size = 7, antiderivative size = 138 \[ \int \frac {1}{2+x^6} \, dx=\frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}-\frac {\arctan \left (\sqrt {3}-2^{5/6} x\right )}{6\ 2^{5/6}}+\frac {\arctan \left (\sqrt {3}+2^{5/6} x\right )}{6\ 2^{5/6}}-\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{2} \sqrt {3} x+x^2\right )}{4\ 2^{5/6} \sqrt {3}}+\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{2} \sqrt {3} x+x^2\right )}{4\ 2^{5/6} \sqrt {3}} \]
1/6*arctan(1/2*x*2^(5/6))*2^(1/6)+1/12*arctan(x*2^(5/6)-3^(1/2))*2^(1/6)+1 /12*arctan(x*2^(5/6)+3^(1/2))*2^(1/6)-1/24*ln(2^(1/3)+x^2-2^(1/6)*x*3^(1/2 ))*2^(1/6)*3^(1/2)+1/24*ln(2^(1/3)+x^2+2^(1/6)*x*3^(1/2))*2^(1/6)*3^(1/2)
Time = 0.02 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.83 \[ \int \frac {1}{2+x^6} \, dx=\frac {4 \arctan \left (\frac {x}{\sqrt [6]{2}}\right )-2 \arctan \left (\sqrt {3}-2^{5/6} x\right )+2 \arctan \left (\sqrt {3}+2^{5/6} x\right )-\sqrt {3} \log \left (2-2^{5/6} \sqrt {3} x+2^{2/3} x^2\right )+\sqrt {3} \log \left (2+2^{5/6} \sqrt {3} x+2^{2/3} x^2\right )}{12\ 2^{5/6}} \]
(4*ArcTan[x/2^(1/6)] - 2*ArcTan[Sqrt[3] - 2^(5/6)*x] + 2*ArcTan[Sqrt[3] + 2^(5/6)*x] - Sqrt[3]*Log[2 - 2^(5/6)*Sqrt[3]*x + 2^(2/3)*x^2] + Sqrt[3]*Lo g[2 + 2^(5/6)*Sqrt[3]*x + 2^(2/3)*x^2])/(12*2^(5/6))
Time = 0.33 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.286, Rules used = {753, 27, 216, 1142, 25, 27, 1082, 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.
| \(\displaystyle \int \frac {1}{x^6+2} \, dx\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {\int \frac {1}{x^2+\sqrt [3]{2}}dx}{3\ 2^{2/3}}+\frac {\int \frac {2 \sqrt [6]{2}-\sqrt {3} x}{2 \left (x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}\right )}dx}{3\ 2^{5/6}}+\frac {\int \frac {\sqrt {3} x+2 \sqrt [6]{2}}{2 \left (x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}\right )}dx}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {1}{x^2+\sqrt [3]{2}}dx}{3\ 2^{2/3}}+\frac {\int \frac {2 \sqrt [6]{2}-\sqrt {3} x}{x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}+\frac {\int \frac {\sqrt {3} x+2 \sqrt [6]{2}}{x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\int \frac {2 \sqrt [6]{2}-\sqrt {3} x}{x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}+\frac {\int \frac {\sqrt {3} x+2 \sqrt [6]{2}}{x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}+\frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{2^{5/6}}-\frac {1}{2} \sqrt {3} \int -\frac {\sqrt [6]{2} \left (\sqrt {3}-2^{5/6} x\right )}{x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}+\frac {\frac {\int \frac {1}{x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt [6]{2} \left (2^{5/6} x+\sqrt {3}\right )}{x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}+\frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt [6]{2} \left (\sqrt {3}-2^{5/6} x\right )}{x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}+\frac {\frac {\int \frac {1}{x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt [6]{2} \left (2^{5/6} x+\sqrt {3}\right )}{x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}+\frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {\sqrt {3} \int \frac {\sqrt {3}-2^{5/6} x}{x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{2^{5/6}}}{6\ 2^{5/6}}+\frac {\frac {\int \frac {1}{x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {\sqrt {3} \int \frac {2^{5/6} x+\sqrt {3}}{x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{2^{5/6}}}{6\ 2^{5/6}}+\frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\sqrt {3} \int \frac {\sqrt {3}-2^{5/6} x}{x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {\int \frac {1}{-\left (1-\frac {2^{5/6} x}{\sqrt {3}}\right )^2-\frac {1}{3}}d\left (1-\frac {2^{5/6} x}{\sqrt {3}}\right )}{\sqrt {3}}}{6\ 2^{5/6}}+\frac {\frac {\sqrt {3} \int \frac {2^{5/6} x+\sqrt {3}}{x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{2^{5/6}}-\frac {\int \frac {1}{-\left (\frac {2^{5/6} x}{\sqrt {3}}+1\right )^2-\frac {1}{3}}d\left (\frac {2^{5/6} x}{\sqrt {3}}+1\right )}{\sqrt {3}}}{6\ 2^{5/6}}+\frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\sqrt {3} \int \frac {\sqrt {3}-2^{5/6} x}{x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{2^{5/6}}-\arctan \left (\sqrt {3} \left (1-\frac {2^{5/6} x}{\sqrt {3}}\right )\right )}{6\ 2^{5/6}}+\frac {\frac {\sqrt {3} \int \frac {2^{5/6} x+\sqrt {3}}{x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}}dx}{2^{5/6}}+\arctan \left (\sqrt {3} \left (\frac {2^{5/6} x}{\sqrt {3}}+1\right )\right )}{6\ 2^{5/6}}+\frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\arctan \left (\sqrt {3} \left (1-\frac {2^{5/6} x}{\sqrt {3}}\right )\right )-\frac {1}{2} \sqrt {3} \log \left (x^2-\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}\right )}{6\ 2^{5/6}}+\frac {\arctan \left (\sqrt {3} \left (\frac {2^{5/6} x}{\sqrt {3}}+1\right )\right )+\frac {1}{2} \sqrt {3} \log \left (x^2+\sqrt [6]{2} \sqrt {3} x+\sqrt [3]{2}\right )}{6\ 2^{5/6}}+\frac {\arctan \left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
ArcTan[x/2^(1/6)]/(3*2^(5/6)) + (-ArcTan[Sqrt[3]*(1 - (2^(5/6)*x)/Sqrt[3]) ] - (Sqrt[3]*Log[2^(1/3) - 2^(1/6)*Sqrt[3]*x + x^2])/2)/(6*2^(5/6)) + (Arc Tan[Sqrt[3]*(1 + (2^(5/6)*x)/Sqrt[3])] + (Sqrt[3]*Log[2^(1/3) + 2^(1/6)*Sq rt[3]*x + x^2])/2)/(6*2^(5/6))
3.1.48.3.1 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_)^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[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[ (r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2* x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 + s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && PosQ[a /b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.16
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{6}\) | \(22\) |
| default | \(\frac {\arctan \left (\frac {x 2^{\frac {5}{6}}}{2}\right ) 2^{\frac {1}{6}}}{6}+\frac {\arctan \left (x 2^{\frac {5}{6}}-\sqrt {3}\right ) 2^{\frac {1}{6}}}{12}+\frac {\arctan \left (x 2^{\frac {5}{6}}+\sqrt {3}\right ) 2^{\frac {1}{6}}}{12}-\frac {\ln \left (2^{\frac {1}{3}}+x^{2}-2^{\frac {1}{6}} x \sqrt {3}\right ) 2^{\frac {1}{6}} \sqrt {3}}{24}+\frac {\ln \left (2^{\frac {1}{3}}+x^{2}+2^{\frac {1}{6}} x \sqrt {3}\right ) 2^{\frac {1}{6}} \sqrt {3}}{24}\) | \(95\) |
| meijerg | \(\frac {2^{\frac {1}{6}} \left (-\frac {x \sqrt {3}\, \ln \left (1-\frac {\sqrt {3}\, 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}+\frac {2^{\frac {2}{3}} \left (x^{6}\right )^{\frac {1}{3}}}{2}\right )}{2 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \arctan \left (\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{4-\sqrt {3}\, 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}\right )}{\left (x^{6}\right )^{\frac {1}{6}}}+\frac {2 x \arctan \left (\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}\right )}{\left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \sqrt {3}\, \ln \left (1+\frac {\sqrt {3}\, 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}+\frac {2^{\frac {2}{3}} \left (x^{6}\right )^{\frac {1}{3}}}{2}\right )}{2 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \arctan \left (\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{4+\sqrt {3}\, 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}\right )}{\left (x^{6}\right )^{\frac {1}{6}}}\right )}{12}\) | \(170\) |
Time = 0.26 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.19 \[ \int \frac {1}{2+x^6} \, dx=\frac {1}{384} \cdot 32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} + 1\right )} \log \left (32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} + 1\right )} + 32 \, x\right ) - \frac {1}{384} \cdot 32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} + 1\right )} \log \left (-32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} + 1\right )} + 32 \, x\right ) + \frac {1}{384} \cdot 32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} - 1\right )} \log \left (32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} - 1\right )} + 32 \, x\right ) - \frac {1}{384} \cdot 32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} - 1\right )} \log \left (-32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} {\left (\sqrt {-3} - 1\right )} + 32 \, x\right ) + \frac {1}{192} \cdot 32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} \log \left (32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} + 16 \, x\right ) - \frac {1}{192} \cdot 32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} \log \left (-32^{\frac {5}{6}} \left (-1\right )^{\frac {1}{6}} + 16 \, x\right ) \]
1/384*32^(5/6)*(-1)^(1/6)*(sqrt(-3) + 1)*log(32^(5/6)*(-1)^(1/6)*(sqrt(-3) + 1) + 32*x) - 1/384*32^(5/6)*(-1)^(1/6)*(sqrt(-3) + 1)*log(-32^(5/6)*(-1 )^(1/6)*(sqrt(-3) + 1) + 32*x) + 1/384*32^(5/6)*(-1)^(1/6)*(sqrt(-3) - 1)* log(32^(5/6)*(-1)^(1/6)*(sqrt(-3) - 1) + 32*x) - 1/384*32^(5/6)*(-1)^(1/6) *(sqrt(-3) - 1)*log(-32^(5/6)*(-1)^(1/6)*(sqrt(-3) - 1) + 32*x) + 1/192*32 ^(5/6)*(-1)^(1/6)*log(32^(5/6)*(-1)^(1/6) + 16*x) - 1/192*32^(5/6)*(-1)^(1 /6)*log(-32^(5/6)*(-1)^(1/6) + 16*x)
Time = 0.14 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.10 \[ \int \frac {1}{2+x^6} \, dx=\operatorname {RootSum} {\left (1492992 t^{6} + 1, \left ( t \mapsto t \log {\left (12 t + x \right )} \right )\right )} \]
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78 \[ \int \frac {1}{2+x^6} \, dx=\frac {1}{24} \, \sqrt {3} 2^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) - \frac {1}{24} \, \sqrt {3} 2^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{6}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {5}{6}} {\left (2 \, x + \sqrt {3} 2^{\frac {1}{6}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{6}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {5}{6}} {\left (2 \, x - \sqrt {3} 2^{\frac {1}{6}}\right )}\right ) + \frac {1}{6} \cdot 2^{\frac {1}{6}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {5}{6}} x\right ) \]
1/24*sqrt(3)*2^(1/6)*log(x^2 + sqrt(3)*2^(1/6)*x + 2^(1/3)) - 1/24*sqrt(3) *2^(1/6)*log(x^2 - sqrt(3)*2^(1/6)*x + 2^(1/3)) + 1/12*2^(1/6)*arctan(1/2* 2^(5/6)*(2*x + sqrt(3)*2^(1/6))) + 1/12*2^(1/6)*arctan(1/2*2^(5/6)*(2*x - sqrt(3)*2^(1/6))) + 1/6*2^(1/6)*arctan(1/2*2^(5/6)*x)
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.78 \[ \int \frac {1}{2+x^6} \, dx=\frac {1}{24} \, \sqrt {3} 2^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) - \frac {1}{24} \, \sqrt {3} 2^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{6}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {5}{6}} {\left (2 \, x + \sqrt {3} 2^{\frac {1}{6}}\right )}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{6}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {5}{6}} {\left (2 \, x - \sqrt {3} 2^{\frac {1}{6}}\right )}\right ) + \frac {1}{6} \cdot 2^{\frac {1}{6}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {5}{6}} x\right ) \]
1/24*sqrt(3)*2^(1/6)*log(x^2 + sqrt(3)*2^(1/6)*x + 2^(1/3)) - 1/24*sqrt(3) *2^(1/6)*log(x^2 - sqrt(3)*2^(1/6)*x + 2^(1/3)) + 1/12*2^(1/6)*arctan(1/2* 2^(5/6)*(2*x + sqrt(3)*2^(1/6))) + 1/12*2^(1/6)*arctan(1/2*2^(5/6)*(2*x - sqrt(3)*2^(1/6))) + 1/6*2^(1/6)*arctan(1/2*2^(5/6)*x)
Time = 0.13 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.98 \[ \int \frac {1}{2+x^6} \, dx=\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{5/6}\,x}{2}\right )}{6}+\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{1/6}\,x}{2\,\left (-\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}+\frac {2^{1/6}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,\left (-\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}\right )\,\left (\sqrt {3}-\mathrm {i}\right )\,1{}\mathrm {i}}{12}+\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{1/6}\,x}{2\,\left (\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}-\frac {2^{1/6}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,\left (\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}\right )\,\left (\sqrt {3}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{12} \]
(2^(1/6)*atan((2^(5/6)*x)/2))/6 + (2^(1/6)*atan((2^(1/6)*x)/(2*((2^(1/3)*3 ^(1/2)*1i)/2 - 2^(1/3)/2)) + (2^(1/6)*3^(1/2)*x*1i)/(2*((2^(1/3)*3^(1/2)*1 i)/2 - 2^(1/3)/2)))*(3^(1/2) - 1i)*1i)/12 + (2^(1/6)*atan((2^(1/6)*x)/(2*( (2^(1/3)*3^(1/2)*1i)/2 + 2^(1/3)/2)) - (2^(1/6)*3^(1/2)*x*1i)/(2*((2^(1/3) *3^(1/2)*1i)/2 + 2^(1/3)/2)))*(3^(1/2) + 1i)*1i)/12
Time = 0.00 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.64 \[ \int \frac {1}{2+x^6} \, dx=\frac {2^{\frac {1}{6}} \left (-2 \mathit {atan} \left (\frac {\left (2^{\frac {1}{6}} \sqrt {3}-2 x \right ) 2^{\frac {5}{6}}}{2}\right )+2 \mathit {atan} \left (\frac {\left (2^{\frac {1}{6}} \sqrt {3}+2 x \right ) 2^{\frac {5}{6}}}{2}\right )+4 \mathit {atan} \left (\frac {x 2^{\frac {5}{6}}}{2}\right )-\sqrt {3}\, \mathrm {log}\left (-2^{\frac {1}{6}} \sqrt {3}\, x +2^{\frac {1}{3}}+x^{2}\right )+\sqrt {3}\, \mathrm {log}\left (2^{\frac {1}{6}} \sqrt {3}\, x +2^{\frac {1}{3}}+x^{2}\right )\right )}{24} \]