Integrand size = 7, antiderivative size = 138 \[ \int \frac {1}{-2+x^6} \, dx=\frac {\arctan \left (\frac {1}{\sqrt {3}}-\frac {2^{5/6} x}{\sqrt {3}}\right )}{2\ 2^{5/6} \sqrt {3}}-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{5/6} x}{\sqrt {3}}\right )}{2\ 2^{5/6} \sqrt {3}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [6]{2} x+x^2\right )}{12\ 2^{5/6}}-\frac {\log \left (\sqrt [3]{2}+\sqrt [6]{2} x+x^2\right )}{12\ 2^{5/6}} \]
-1/6*arctanh(1/2*x*2^(5/6))*2^(1/6)+1/24*ln(2^(1/3)-2^(1/6)*x+x^2)*2^(1/6) -1/24*ln(2^(1/3)+2^(1/6)*x+x^2)*2^(1/6)-1/12*arctan(-1/3*3^(1/2)+1/3*2^(5/ 6)*x*3^(1/2))*2^(1/6)*3^(1/2)-1/12*arctan(1/3*3^(1/2)+1/3*2^(5/6)*x*3^(1/2 ))*2^(1/6)*3^(1/2)
Time = 0.03 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.88 \[ \int \frac {1}{-2+x^6} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {-1+2^{5/6} x}{\sqrt {3}}\right )+2 \sqrt {3} \arctan \left (\frac {1+2^{5/6} x}{\sqrt {3}}\right )-2 \log \left (2-2^{5/6} x\right )+2 \log \left (2+2^{5/6} x\right )-\log \left (2-2^{5/6} x+2^{2/3} x^2\right )+\log \left (2+2^{5/6} x+2^{2/3} x^2\right )}{12\ 2^{5/6}} \]
-1/12*(2*Sqrt[3]*ArcTan[(-1 + 2^(5/6)*x)/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 2^(5/6)*x)/Sqrt[3]] - 2*Log[2 - 2^(5/6)*x] + 2*Log[2 + 2^(5/6)*x] - Log[2 - 2^(5/6)*x + 2^(2/3)*x^2] + Log[2 + 2^(5/6)*x + 2^(2/3)*x^2])/2^(5/6)
Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.92, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.286, Rules used = {754, 27, 219, 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.
| \(\displaystyle \int \frac {1}{x^6-2} \, dx\) |
| \(\Big \downarrow \) 754 |
| \(\displaystyle -\frac {\int \frac {1}{\sqrt [3]{2}-x^2}dx}{3\ 2^{2/3}}-\frac {\int \frac {2 \sqrt [6]{2}-x}{2 \left (x^2-\sqrt [6]{2} x+\sqrt [3]{2}\right )}dx}{3\ 2^{5/6}}-\frac {\int \frac {x+2 \sqrt [6]{2}}{2 \left (x^2+\sqrt [6]{2} x+\sqrt [3]{2}\right )}dx}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {1}{\sqrt [3]{2}-x^2}dx}{3\ 2^{2/3}}-\frac {\int \frac {2 \sqrt [6]{2}-x}{x^2-\sqrt [6]{2} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}-\frac {\int \frac {x+2 \sqrt [6]{2}}{x^2+\sqrt [6]{2} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\int \frac {2 \sqrt [6]{2}-x}{x^2-\sqrt [6]{2} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}-\frac {\int \frac {x+2 \sqrt [6]{2}}{x^2+\sqrt [6]{2} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {\frac {3 \int \frac {1}{x^2-\sqrt [6]{2} x+\sqrt [3]{2}}dx}{2^{5/6}}-\frac {1}{2} \int -\frac {\sqrt [6]{2} \left (1-2^{5/6} x\right )}{x^2-\sqrt [6]{2} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}-\frac {\frac {3 \int \frac {1}{x^2+\sqrt [6]{2} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {1}{2} \int \frac {\sqrt [6]{2} \left (2^{5/6} x+1\right )}{x^2+\sqrt [6]{2} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {3 \int \frac {1}{x^2-\sqrt [6]{2} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {1}{2} \int \frac {\sqrt [6]{2} \left (1-2^{5/6} x\right )}{x^2-\sqrt [6]{2} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}-\frac {\frac {3 \int \frac {1}{x^2+\sqrt [6]{2} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {1}{2} \int \frac {\sqrt [6]{2} \left (2^{5/6} x+1\right )}{x^2+\sqrt [6]{2} x+\sqrt [3]{2}}dx}{6\ 2^{5/6}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 \int \frac {1}{x^2-\sqrt [6]{2} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {\int \frac {1-2^{5/6} x}{x^2-\sqrt [6]{2} x+\sqrt [3]{2}}dx}{2^{5/6}}}{6\ 2^{5/6}}-\frac {\frac {3 \int \frac {1}{x^2+\sqrt [6]{2} x+\sqrt [3]{2}}dx}{2^{5/6}}+\frac {\int \frac {2^{5/6} x+1}{x^2+\sqrt [6]{2} x+\sqrt [3]{2}}dx}{2^{5/6}}}{6\ 2^{5/6}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {\frac {\int \frac {1-2^{5/6} x}{x^2-\sqrt [6]{2} x+\sqrt [3]{2}}dx}{2^{5/6}}+3 \int \frac {1}{-\left (1-2^{5/6} x\right )^2-3}d\left (1-2^{5/6} x\right )}{6\ 2^{5/6}}-\frac {\frac {\int \frac {2^{5/6} x+1}{x^2+\sqrt [6]{2} x+\sqrt [3]{2}}dx}{2^{5/6}}-3 \int \frac {1}{-\left (2^{5/6} x+1\right )^2-3}d\left (2^{5/6} x+1\right )}{6\ 2^{5/6}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {\int \frac {1-2^{5/6} x}{x^2-\sqrt [6]{2} x+\sqrt [3]{2}}dx}{2^{5/6}}-\sqrt {3} \arctan \left (\frac {1-2^{5/6} x}{\sqrt {3}}\right )}{6\ 2^{5/6}}-\frac {\frac {\int \frac {2^{5/6} x+1}{x^2+\sqrt [6]{2} x+\sqrt [3]{2}}dx}{2^{5/6}}+\sqrt {3} \arctan \left (\frac {2^{5/6} x+1}{\sqrt {3}}\right )}{6\ 2^{5/6}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {-\sqrt {3} \arctan \left (\frac {1-2^{5/6} x}{\sqrt {3}}\right )-\frac {1}{2} \log \left (x^2-\sqrt [6]{2} x+\sqrt [3]{2}\right )}{6\ 2^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {2^{5/6} x+1}{\sqrt {3}}\right )+\frac {1}{2} \log \left (x^2+\sqrt [6]{2} x+\sqrt [3]{2}\right )}{6\ 2^{5/6}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [6]{2}}\right )}{3\ 2^{5/6}}\) |
-1/3*ArcTanh[x/2^(1/6)]/2^(5/6) - (-(Sqrt[3]*ArcTan[(1 - 2^(5/6)*x)/Sqrt[3 ]]) - Log[2^(1/3) - 2^(1/6)*x + x^2]/2)/(6*2^(5/6)) - (Sqrt[3]*ArcTan[(1 + 2^(5/6)*x)/Sqrt[3]] + Log[2^(1/3) + 2^(1/6)*x + x^2]/2)/(6*2^(5/6))
3.1.47.3.1 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[(-(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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[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}, Simp[u = Int[(r - s*Cos[(2*k* Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 *k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*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] && NegQ[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 = 0.18 (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 | \(\frac {\ln \left (2^{\frac {1}{3}}-2^{\frac {1}{6}} x +x^{2}\right ) 2^{\frac {1}{6}}}{24}-\frac {\arctan \left (-\frac {\sqrt {3}}{3}+\frac {2^{\frac {5}{6}} x \sqrt {3}}{3}\right ) 2^{\frac {1}{6}} \sqrt {3}}{12}+\frac {2^{\frac {1}{6}} \ln \left (x -2^{\frac {1}{6}}\right )}{12}-\frac {\ln \left (2^{\frac {1}{3}}+2^{\frac {1}{6}} x +x^{2}\right ) 2^{\frac {1}{6}}}{24}-\frac {\arctan \left (\frac {\sqrt {3}}{3}+\frac {2^{\frac {5}{6}} x \sqrt {3}}{3}\right ) 2^{\frac {1}{6}} \sqrt {3}}{12}-\frac {2^{\frac {1}{6}} \ln \left (x +2^{\frac {1}{6}}\right )}{12}\) | \(111\) |
| meijerg | \(\frac {2^{\frac {1}{6}} x \left (\ln \left (1-\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}\right )-\ln \left (1+\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}\right )+\frac {\ln \left (1-\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}+\frac {2^{\frac {2}{3}} \left (x^{6}\right )^{\frac {1}{3}}}{2}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{4-2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\frac {2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{2}+\frac {2^{\frac {2}{3}} \left (x^{6}\right )^{\frac {1}{3}}}{2}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, 2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}{4+2^{\frac {5}{6}} \left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{12 \left (x^{6}\right )^{\frac {1}{6}}}\) | \(157\) |
Time = 0.24 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.92 \[ \int \frac {1}{-2+x^6} \, dx=-\frac {1}{384} \cdot 32^{\frac {5}{6}} {\left (\sqrt {-3} + 1\right )} \log \left (32^{\frac {5}{6}} {\left (\sqrt {-3} + 1\right )} + 32 \, x\right ) + \frac {1}{384} \cdot 32^{\frac {5}{6}} {\left (\sqrt {-3} + 1\right )} \log \left (-32^{\frac {5}{6}} {\left (\sqrt {-3} + 1\right )} + 32 \, x\right ) - \frac {1}{384} \cdot 32^{\frac {5}{6}} {\left (\sqrt {-3} - 1\right )} \log \left (32^{\frac {5}{6}} {\left (\sqrt {-3} - 1\right )} + 32 \, x\right ) + \frac {1}{384} \cdot 32^{\frac {5}{6}} {\left (\sqrt {-3} - 1\right )} \log \left (-32^{\frac {5}{6}} {\left (\sqrt {-3} - 1\right )} + 32 \, x\right ) - \frac {1}{192} \cdot 32^{\frac {5}{6}} \log \left (16 \, x + 32^{\frac {5}{6}}\right ) + \frac {1}{192} \cdot 32^{\frac {5}{6}} \log \left (16 \, x - 32^{\frac {5}{6}}\right ) \]
-1/384*32^(5/6)*(sqrt(-3) + 1)*log(32^(5/6)*(sqrt(-3) + 1) + 32*x) + 1/384 *32^(5/6)*(sqrt(-3) + 1)*log(-32^(5/6)*(sqrt(-3) + 1) + 32*x) - 1/384*32^( 5/6)*(sqrt(-3) - 1)*log(32^(5/6)*(sqrt(-3) - 1) + 32*x) + 1/384*32^(5/6)*( sqrt(-3) - 1)*log(-32^(5/6)*(sqrt(-3) - 1) + 32*x) - 1/192*32^(5/6)*log(16 *x + 32^(5/6)) + 1/192*32^(5/6)*log(16*x - 32^(5/6))
Time = 0.32 (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 = 112, normalized size of antiderivative = 0.81 \[ \int \frac {1}{-2+x^6} \, dx=-\frac {1}{12} \, \sqrt {3} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {5}{6}} {\left (2 \, x + 2^{\frac {1}{6}}\right )}\right ) - \frac {1}{12} \, \sqrt {3} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {5}{6}} {\left (2 \, x - 2^{\frac {1}{6}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {1}{6}} \log \left (x^{2} + 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) + \frac {1}{24} \cdot 2^{\frac {1}{6}} \log \left (x^{2} - 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) - \frac {1}{12} \cdot 2^{\frac {1}{6}} \log \left (x + 2^{\frac {1}{6}}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{6}} \log \left (x - 2^{\frac {1}{6}}\right ) \]
-1/12*sqrt(3)*2^(1/6)*arctan(1/6*sqrt(3)*2^(5/6)*(2*x + 2^(1/6))) - 1/12*s qrt(3)*2^(1/6)*arctan(1/6*sqrt(3)*2^(5/6)*(2*x - 2^(1/6))) - 1/24*2^(1/6)* log(x^2 + 2^(1/6)*x + 2^(1/3)) + 1/24*2^(1/6)*log(x^2 - 2^(1/6)*x + 2^(1/3 )) - 1/12*2^(1/6)*log(x + 2^(1/6)) + 1/12*2^(1/6)*log(x - 2^(1/6))
Time = 0.30 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.83 \[ \int \frac {1}{-2+x^6} \, dx=-\frac {1}{12} \, \sqrt {3} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {5}{6}} {\left (2 \, x + 2^{\frac {1}{6}}\right )}\right ) - \frac {1}{12} \, \sqrt {3} 2^{\frac {1}{6}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {5}{6}} {\left (2 \, x - 2^{\frac {1}{6}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {1}{6}} \log \left (x^{2} + 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) + \frac {1}{24} \cdot 2^{\frac {1}{6}} \log \left (x^{2} - 2^{\frac {1}{6}} x + 2^{\frac {1}{3}}\right ) - \frac {1}{12} \cdot 2^{\frac {1}{6}} \log \left ({\left | x + 2^{\frac {1}{6}} \right |}\right ) + \frac {1}{12} \cdot 2^{\frac {1}{6}} \log \left ({\left | x - 2^{\frac {1}{6}} \right |}\right ) \]
-1/12*sqrt(3)*2^(1/6)*arctan(1/6*sqrt(3)*2^(5/6)*(2*x + 2^(1/6))) - 1/12*s qrt(3)*2^(1/6)*arctan(1/6*sqrt(3)*2^(5/6)*(2*x - 2^(1/6))) - 1/24*2^(1/6)* log(x^2 + 2^(1/6)*x + 2^(1/3)) + 1/24*2^(1/6)*log(x^2 - 2^(1/6)*x + 2^(1/3 )) - 1/12*2^(1/6)*log(abs(x + 2^(1/6))) + 1/12*2^(1/6)*log(abs(x - 2^(1/6) ))
Time = 0.21 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.01 \[ \int \frac {1}{-2+x^6} \, dx=-\frac {2^{1/6}\,\mathrm {atanh}\left (\frac {2^{5/6}\,x}{2}\right )}{6}+\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{1/6}\,x\,1{}\mathrm {i}}{2\,\left (-\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}-\frac {2^{1/6}\,\sqrt {3}\,x}{2\,\left (-\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{12}+\frac {2^{1/6}\,\mathrm {atan}\left (\frac {2^{1/6}\,x\,1{}\mathrm {i}}{2\,\left (\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}+\frac {2^{1/6}\,\sqrt {3}\,x}{2\,\left (\frac {2^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{12} \]
(2^(1/6)*atan((2^(1/6)*x*1i)/(2*((2^(1/3)*3^(1/2)*1i)/2 - 2^(1/3)/2)) - (2 ^(1/6)*3^(1/2)*x)/(2*((2^(1/3)*3^(1/2)*1i)/2 - 2^(1/3)/2)))*(3^(1/2)*1i + 1)*1i)/12 - (2^(1/6)*atanh((2^(5/6)*x)/2))/6 + (2^(1/6)*atan((2^(1/6)*x*1i )/(2*((2^(1/3)*3^(1/2)*1i)/2 + 2^(1/3)/2)) + (2^(1/6)*3^(1/2)*x)/(2*((2^(1 /3)*3^(1/2)*1i)/2 + 2^(1/3)/2)))*(3^(1/2)*1i - 1)*1i)/12
Time = 0.00 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.69 \[ \int \frac {1}{-2+x^6} \, dx=\frac {2^{\frac {1}{6}} \left (2 \sqrt {3}\, \mathit {atan} \left (\frac {\left (2^{\frac {1}{6}}-2 x \right ) 2^{\frac {5}{6}}}{2 \sqrt {3}}\right )-2 \sqrt {3}\, \mathit {atan} \left (\frac {\left (2^{\frac {1}{6}}+2 x \right ) 2^{\frac {5}{6}}}{2 \sqrt {3}}\right )-2 \,\mathrm {log}\left (2^{\frac {1}{6}}+x \right )+2 \,\mathrm {log}\left (-2^{\frac {1}{6}}+x \right )+\mathrm {log}\left (-2^{\frac {1}{6}} x +2^{\frac {1}{3}}+x^{2}\right )-\mathrm {log}\left (2^{\frac {1}{6}} x +2^{\frac {1}{3}}+x^{2}\right )\right )}{24} \]