Integrand size = 7, antiderivative size = 80 \[ \int \frac {1}{1+x^6} \, dx=-\frac {1}{6} \arctan \left (\sqrt {3}-2 x\right )+\frac {\arctan (x)}{3}+\frac {1}{6} \arctan \left (\sqrt {3}+2 x\right )-\frac {\log \left (1-\sqrt {3} x+x^2\right )}{4 \sqrt {3}}+\frac {\log \left (1+\sqrt {3} x+x^2\right )}{4 \sqrt {3}} \]
1/3*arctan(x)+1/6*arctan(2*x-3^(1/2))+1/6*arctan(2*x+3^(1/2))-1/12*ln(1+x^ 2-x*3^(1/2))*3^(1/2)+1/12*ln(1+x^2+x*3^(1/2))*3^(1/2)
Time = 0.01 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int \frac {1}{1+x^6} \, dx=\frac {1}{12} \left (-2 \arctan \left (\sqrt {3}-2 x\right )+4 \arctan (x)+2 \arctan \left (\sqrt {3}+2 x\right )-\sqrt {3} \log \left (1-\sqrt {3} x+x^2\right )+\sqrt {3} \log \left (1+\sqrt {3} x+x^2\right )\right ) \]
(-2*ArcTan[Sqrt[3] - 2*x] + 4*ArcTan[x] + 2*ArcTan[Sqrt[3] + 2*x] - Sqrt[3 ]*Log[1 - Sqrt[3]*x + x^2] + Sqrt[3]*Log[1 + Sqrt[3]*x + x^2])/12
Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {753, 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.
\(\displaystyle \int \frac {1}{x^6+1} \, dx\) |
\(\Big \downarrow \) 753 |
\(\displaystyle \frac {1}{3} \int \frac {1}{x^2+1}dx+\frac {1}{3} \int \frac {2-\sqrt {3} x}{2 \left (x^2-\sqrt {3} x+1\right )}dx+\frac {1}{3} \int \frac {\sqrt {3} x+2}{2 \left (x^2+\sqrt {3} x+1\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {1}{x^2+1}dx+\frac {1}{6} \int \frac {2-\sqrt {3} x}{x^2-\sqrt {3} x+1}dx+\frac {1}{6} \int \frac {\sqrt {3} x+2}{x^2+\sqrt {3} x+1}dx\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{6} \int \frac {2-\sqrt {3} x}{x^2-\sqrt {3} x+1}dx+\frac {1}{6} \int \frac {\sqrt {3} x+2}{x^2+\sqrt {3} x+1}dx+\frac {\arctan (x)}{3}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {1}{x^2-\sqrt {3} x+1}dx-\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-2 x}{x^2-\sqrt {3} x+1}dx\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{x^2+\sqrt {3} x+1}dx+\frac {1}{2} \sqrt {3} \int \frac {2 x+\sqrt {3}}{x^2+\sqrt {3} x+1}dx\right )+\frac {\arctan (x)}{3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {1}{x^2-\sqrt {3} x+1}dx+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 x}{x^2-\sqrt {3} x+1}dx\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{x^2+\sqrt {3} x+1}dx+\frac {1}{2} \sqrt {3} \int \frac {2 x+\sqrt {3}}{x^2+\sqrt {3} x+1}dx\right )+\frac {\arctan (x)}{3}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 x}{x^2-\sqrt {3} x+1}dx-\int \frac {1}{-\left (2 x-\sqrt {3}\right )^2-1}d\left (2 x-\sqrt {3}\right )\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {2 x+\sqrt {3}}{x^2+\sqrt {3} x+1}dx-\int \frac {1}{-\left (2 x+\sqrt {3}\right )^2-1}d\left (2 x+\sqrt {3}\right )\right )+\frac {\arctan (x)}{3}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 x}{x^2-\sqrt {3} x+1}dx-\arctan \left (\sqrt {3}-2 x\right )\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {2 x+\sqrt {3}}{x^2+\sqrt {3} x+1}dx+\arctan \left (2 x+\sqrt {3}\right )\right )+\frac {\arctan (x)}{3}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{6} \left (-\arctan \left (\sqrt {3}-2 x\right )-\frac {1}{2} \sqrt {3} \log \left (x^2-\sqrt {3} x+1\right )\right )+\frac {1}{6} \left (\arctan \left (2 x+\sqrt {3}\right )+\frac {1}{2} \sqrt {3} \log \left (x^2+\sqrt {3} x+1\right )\right )+\frac {\arctan (x)}{3}\) |
ArcTan[x]/3 + (-ArcTan[Sqrt[3] - 2*x] - (Sqrt[3]*Log[1 - Sqrt[3]*x + x^2]) /2)/6 + (ArcTan[Sqrt[3] + 2*x] + (Sqrt[3]*Log[1 + Sqrt[3]*x + x^2])/2)/6
3.14.69.3.1 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] :> 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.46 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.35
method | result | size |
risch | \(\frac {\arctan \left (x \right )}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} +x \right )\right )}{6}\) | \(28\) |
default | \(\frac {\arctan \left (x \right )}{3}+\frac {\arctan \left (2 x -\sqrt {3}\right )}{6}+\frac {\arctan \left (2 x +\sqrt {3}\right )}{6}-\frac {\ln \left (1+x^{2}-\sqrt {3}\, x \right ) \sqrt {3}}{12}+\frac {\ln \left (1+x^{2}+\sqrt {3}\, x \right ) \sqrt {3}}{12}\) | \(61\) |
meijerg | \(-\frac {x \sqrt {3}\, \ln \left (1-\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{12 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \arctan \left (\frac {\left (x^{6}\right )^{\frac {1}{6}}}{2-\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \arctan \left (\left (x^{6}\right )^{\frac {1}{6}}\right )}{3 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \sqrt {3}\, \ln \left (1+\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{12 \left (x^{6}\right )^{\frac {1}{6}}}+\frac {x \arctan \left (\frac {\left (x^{6}\right )^{\frac {1}{6}}}{2+\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\) | \(133\) |
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.69 \[ \int \frac {1}{1+x^6} \, dx=\frac {1}{12} \, \sqrt {2} \sqrt {i \, \sqrt {3} + 1} \log \left (2 \, x + \sqrt {2} \sqrt {i \, \sqrt {3} + 1}\right ) - \frac {1}{12} \, \sqrt {2} \sqrt {i \, \sqrt {3} + 1} \log \left (2 \, x - \sqrt {2} \sqrt {i \, \sqrt {3} + 1}\right ) + \frac {1}{12} \, \sqrt {2} \sqrt {-i \, \sqrt {3} + 1} \log \left (2 \, x + \sqrt {2} \sqrt {-i \, \sqrt {3} + 1}\right ) - \frac {1}{12} \, \sqrt {2} \sqrt {-i \, \sqrt {3} + 1} \log \left (2 \, x - \sqrt {2} \sqrt {-i \, \sqrt {3} + 1}\right ) + \frac {1}{3} \, \arctan \left (x\right ) \]
1/12*sqrt(2)*sqrt(I*sqrt(3) + 1)*log(2*x + sqrt(2)*sqrt(I*sqrt(3) + 1)) - 1/12*sqrt(2)*sqrt(I*sqrt(3) + 1)*log(2*x - sqrt(2)*sqrt(I*sqrt(3) + 1)) + 1/12*sqrt(2)*sqrt(-I*sqrt(3) + 1)*log(2*x + sqrt(2)*sqrt(-I*sqrt(3) + 1)) - 1/12*sqrt(2)*sqrt(-I*sqrt(3) + 1)*log(2*x - sqrt(2)*sqrt(-I*sqrt(3) + 1) ) + 1/3*arctan(x)
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.85 \[ \int \frac {1}{1+x^6} \, dx=- \frac {\sqrt {3} \log {\left (x^{2} - \sqrt {3} x + 1 \right )}}{12} + \frac {\sqrt {3} \log {\left (x^{2} + \sqrt {3} x + 1 \right )}}{12} + \frac {\operatorname {atan}{\left (x \right )}}{3} + \frac {\operatorname {atan}{\left (2 x - \sqrt {3} \right )}}{6} + \frac {\operatorname {atan}{\left (2 x + \sqrt {3} \right )}}{6} \]
-sqrt(3)*log(x**2 - sqrt(3)*x + 1)/12 + sqrt(3)*log(x**2 + sqrt(3)*x + 1)/ 12 + atan(x)/3 + atan(2*x - sqrt(3))/6 + atan(2*x + sqrt(3))/6
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {1}{1+x^6} \, dx=\frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) - \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + \frac {1}{6} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{6} \, \arctan \left (2 \, x - \sqrt {3}\right ) + \frac {1}{3} \, \arctan \left (x\right ) \]
1/12*sqrt(3)*log(x^2 + sqrt(3)*x + 1) - 1/12*sqrt(3)*log(x^2 - sqrt(3)*x + 1) + 1/6*arctan(2*x + sqrt(3)) + 1/6*arctan(2*x - sqrt(3)) + 1/3*arctan(x )
Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {1}{1+x^6} \, dx=\frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) - \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + \frac {1}{6} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{6} \, \arctan \left (2 \, x - \sqrt {3}\right ) + \frac {1}{3} \, \arctan \left (x\right ) \]
1/12*sqrt(3)*log(x^2 + sqrt(3)*x + 1) - 1/12*sqrt(3)*log(x^2 - sqrt(3)*x + 1) + 1/6*arctan(2*x + sqrt(3)) + 1/6*arctan(2*x - sqrt(3)) + 1/3*arctan(x )
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06 \[ \int \frac {1}{1+x^6} \, dx=\frac {\mathrm {atan}\left (x\right )}{3}+\mathrm {atan}\left (\frac {x}{-1+\sqrt {3}\,1{}\mathrm {i}}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\mathrm {atan}\left (\frac {x}{1+\sqrt {3}\,1{}\mathrm {i}}-\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]