Integrand size = 7, antiderivative size = 165 \[ \int \frac {1}{1+x^7} \, dx=\frac {2}{7} \arctan \left (x \sec \left (\frac {\pi }{14}\right )-\tan \left (\frac {\pi }{14}\right )\right ) \cos \left (\frac {\pi }{14}\right )+\frac {2}{7} \arctan \left (x \sec \left (\frac {3 \pi }{14}\right )+\tan \left (\frac {3 \pi }{14}\right )\right ) \cos \left (\frac {3 \pi }{14}\right )+\frac {1}{7} \log (1+x)-\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \log \left (1+x^2-2 x \cos \left (\frac {\pi }{7}\right )\right )-\frac {1}{7} \log \left (1+x^2-2 x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )-\frac {2}{7} \arctan \left (\cot \left (\frac {\pi }{7}\right )-x \csc \left (\frac {\pi }{7}\right )\right ) \sin \left (\frac {\pi }{7}\right )+\frac {1}{7} \log \left (1+x^2+2 x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right ) \]
2/7*arctan(x*sec(1/14*Pi)-tan(1/14*Pi))*cos(1/14*Pi)+2/7*arctan(x*sec(3/14 *Pi)+tan(3/14*Pi))*cos(3/14*Pi)+1/7*ln(1+x)-1/7*cos(1/7*Pi)*ln(1+x^2-2*x*c os(1/7*Pi))-1/7*ln(1+x^2-2*x*sin(1/14*Pi))*sin(1/14*Pi)+2/7*arctan(-cot(1/ 7*Pi)+x*csc(1/7*Pi))*sin(1/7*Pi)+1/7*ln(1+x^2+2*x*sin(3/14*Pi))*sin(3/14*P i)
Time = 0.06 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.01 \[ \int \frac {1}{1+x^7} \, dx=\frac {2}{7} \arctan \left (\sec \left (\frac {\pi }{14}\right ) \left (x-\sin \left (\frac {\pi }{14}\right )\right )\right ) \cos \left (\frac {\pi }{14}\right )+\frac {2}{7} \arctan \left (\sec \left (\frac {3 \pi }{14}\right ) \left (x+\sin \left (\frac {3 \pi }{14}\right )\right )\right ) \cos \left (\frac {3 \pi }{14}\right )+\frac {1}{7} \log (1+x)-\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \log \left (1+x^2-2 x \cos \left (\frac {\pi }{7}\right )\right )-\frac {1}{7} \log \left (1+x^2-2 x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )+\frac {2}{7} \arctan \left (\left (x-\cos \left (\frac {\pi }{7}\right )\right ) \csc \left (\frac {\pi }{7}\right )\right ) \sin \left (\frac {\pi }{7}\right )+\frac {1}{7} \log \left (1+x^2+2 x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right ) \]
(2*ArcTan[Sec[Pi/14]*(x - Sin[Pi/14])]*Cos[Pi/14])/7 + (2*ArcTan[Sec[(3*Pi )/14]*(x + Sin[(3*Pi)/14])]*Cos[(3*Pi)/14])/7 + Log[1 + x]/7 - (Cos[Pi/7]* Log[1 + x^2 - 2*x*Cos[Pi/7]])/7 - (Log[1 + x^2 - 2*x*Sin[Pi/14]]*Sin[Pi/14 ])/7 + (2*ArcTan[(x - Cos[Pi/7])*Csc[Pi/7]]*Sin[Pi/7])/7 + (Log[1 + x^2 + 2*x*Sin[(3*Pi)/14]]*Sin[(3*Pi)/14])/7
Time = 0.38 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {751, 16, 1142, 27, 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.
| \(\displaystyle \int \frac {1}{x^7+1} \, dx\) |
| \(\Big \downarrow \) 751 |
| \(\displaystyle \frac {2}{7} \int \frac {1-x \sin \left (\frac {\pi }{14}\right )}{x^2-2 \sin \left (\frac {\pi }{14}\right ) x+1}dx+\frac {2}{7} \int \frac {\sin \left (\frac {3 \pi }{14}\right ) x+1}{x^2+2 \sin \left (\frac {3 \pi }{14}\right ) x+1}dx+\frac {2}{7} \int \frac {1-x \cos \left (\frac {\pi }{7}\right )}{x^2-2 \cos \left (\frac {\pi }{7}\right ) x+1}dx+\frac {1}{7} \int \frac {1}{x+1}dx\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2}{7} \int \frac {1-x \sin \left (\frac {\pi }{14}\right )}{x^2-2 \sin \left (\frac {\pi }{14}\right ) x+1}dx+\frac {2}{7} \int \frac {\sin \left (\frac {3 \pi }{14}\right ) x+1}{x^2+2 \sin \left (\frac {3 \pi }{14}\right ) x+1}dx+\frac {2}{7} \int \frac {1-x \cos \left (\frac {\pi }{7}\right )}{x^2-2 \cos \left (\frac {\pi }{7}\right ) x+1}dx+\frac {1}{7} \log (x+1)\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {2}{7} \left (\cos ^2\left (\frac {\pi }{14}\right ) \int \frac {1}{x^2-2 \sin \left (\frac {\pi }{14}\right ) x+1}dx-\frac {1}{2} \sin \left (\frac {\pi }{14}\right ) \int \frac {2 \left (x-\sin \left (\frac {\pi }{14}\right )\right )}{x^2-2 \sin \left (\frac {\pi }{14}\right ) x+1}dx\right )+\frac {2}{7} \left (\frac {1}{2} \sin \left (\frac {3 \pi }{14}\right ) \int \frac {2 \left (x+\sin \left (\frac {3 \pi }{14}\right )\right )}{x^2+2 \sin \left (\frac {3 \pi }{14}\right ) x+1}dx+\cos ^2\left (\frac {3 \pi }{14}\right ) \int \frac {1}{x^2+2 \sin \left (\frac {3 \pi }{14}\right ) x+1}dx\right )+\frac {2}{7} \left (\sin ^2\left (\frac {\pi }{7}\right ) \int \frac {1}{x^2-2 \cos \left (\frac {\pi }{7}\right ) x+1}dx-\frac {1}{2} \cos \left (\frac {\pi }{7}\right ) \int \frac {2 \left (x-\cos \left (\frac {\pi }{7}\right )\right )}{x^2-2 \cos \left (\frac {\pi }{7}\right ) x+1}dx\right )+\frac {1}{7} \log (x+1)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{7} \left (\cos ^2\left (\frac {\pi }{14}\right ) \int \frac {1}{x^2-2 \sin \left (\frac {\pi }{14}\right ) x+1}dx-\sin \left (\frac {\pi }{14}\right ) \int \frac {x-\sin \left (\frac {\pi }{14}\right )}{x^2-2 \sin \left (\frac {\pi }{14}\right ) x+1}dx\right )+\frac {2}{7} \left (\sin \left (\frac {3 \pi }{14}\right ) \int \frac {x+\sin \left (\frac {3 \pi }{14}\right )}{x^2+2 \sin \left (\frac {3 \pi }{14}\right ) x+1}dx+\cos ^2\left (\frac {3 \pi }{14}\right ) \int \frac {1}{x^2+2 \sin \left (\frac {3 \pi }{14}\right ) x+1}dx\right )+\frac {2}{7} \left (\sin ^2\left (\frac {\pi }{7}\right ) \int \frac {1}{x^2-2 \cos \left (\frac {\pi }{7}\right ) x+1}dx-\cos \left (\frac {\pi }{7}\right ) \int \frac {x-\cos \left (\frac {\pi }{7}\right )}{x^2-2 \cos \left (\frac {\pi }{7}\right ) x+1}dx\right )+\frac {1}{7} \log (x+1)\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {2}{7} \left (-\sin \left (\frac {\pi }{14}\right ) \int \frac {x-\sin \left (\frac {\pi }{14}\right )}{x^2-2 \sin \left (\frac {\pi }{14}\right ) x+1}dx-2 \cos ^2\left (\frac {\pi }{14}\right ) \int \frac {1}{-\left (2 x-2 \sin \left (\frac {\pi }{14}\right )\right )^2-4 \cos ^2\left (\frac {\pi }{14}\right )}d\left (2 x-2 \sin \left (\frac {\pi }{14}\right )\right )\right )+\frac {2}{7} \left (\sin \left (\frac {3 \pi }{14}\right ) \int \frac {x+\sin \left (\frac {3 \pi }{14}\right )}{x^2+2 \sin \left (\frac {3 \pi }{14}\right ) x+1}dx-2 \cos ^2\left (\frac {3 \pi }{14}\right ) \int \frac {1}{-\left (2 x+2 \sin \left (\frac {3 \pi }{14}\right )\right )^2-4 \cos ^2\left (\frac {3 \pi }{14}\right )}d\left (2 x+2 \sin \left (\frac {3 \pi }{14}\right )\right )\right )+\frac {2}{7} \left (-\cos \left (\frac {\pi }{7}\right ) \int \frac {x-\cos \left (\frac {\pi }{7}\right )}{x^2-2 \cos \left (\frac {\pi }{7}\right ) x+1}dx-2 \sin ^2\left (\frac {\pi }{7}\right ) \int \frac {1}{-\left (2 x-2 \cos \left (\frac {\pi }{7}\right )\right )^2-4 \sin ^2\left (\frac {\pi }{7}\right )}d\left (2 x-2 \cos \left (\frac {\pi }{7}\right )\right )\right )+\frac {1}{7} \log (x+1)\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2}{7} \left (\sin \left (\frac {\pi }{7}\right ) \arctan \left (\frac {1}{2} \csc \left (\frac {\pi }{7}\right ) \left (2 x-2 \cos \left (\frac {\pi }{7}\right )\right )\right )-\cos \left (\frac {\pi }{7}\right ) \int \frac {x-\cos \left (\frac {\pi }{7}\right )}{x^2-2 \cos \left (\frac {\pi }{7}\right ) x+1}dx\right )+\frac {2}{7} \left (\cos \left (\frac {\pi }{14}\right ) \arctan \left (\frac {1}{2} \sec \left (\frac {\pi }{14}\right ) \left (2 x-2 \sin \left (\frac {\pi }{14}\right )\right )\right )-\sin \left (\frac {\pi }{14}\right ) \int \frac {x-\sin \left (\frac {\pi }{14}\right )}{x^2-2 \sin \left (\frac {\pi }{14}\right ) x+1}dx\right )+\frac {2}{7} \left (\sin \left (\frac {3 \pi }{14}\right ) \int \frac {x+\sin \left (\frac {3 \pi }{14}\right )}{x^2+2 \sin \left (\frac {3 \pi }{14}\right ) x+1}dx+\cos \left (\frac {3 \pi }{14}\right ) \arctan \left (\frac {1}{2} \sec \left (\frac {3 \pi }{14}\right ) \left (2 x+2 \sin \left (\frac {3 \pi }{14}\right )\right )\right )\right )+\frac {1}{7} \log (x+1)\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2}{7} \left (\cos \left (\frac {\pi }{14}\right ) \arctan \left (\frac {1}{2} \sec \left (\frac {\pi }{14}\right ) \left (2 x-2 \sin \left (\frac {\pi }{14}\right )\right )\right )-\frac {1}{2} \sin \left (\frac {\pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac {\pi }{14}\right )+1\right )\right )+\frac {2}{7} \left (\cos \left (\frac {3 \pi }{14}\right ) \arctan \left (\frac {1}{2} \sec \left (\frac {3 \pi }{14}\right ) \left (2 x+2 \sin \left (\frac {3 \pi }{14}\right )\right )\right )+\frac {1}{2} \sin \left (\frac {3 \pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac {3 \pi }{14}\right )+1\right )\right )+\frac {2}{7} \left (\sin \left (\frac {\pi }{7}\right ) \arctan \left (\frac {1}{2} \csc \left (\frac {\pi }{7}\right ) \left (2 x-2 \cos \left (\frac {\pi }{7}\right )\right )\right )-\frac {1}{2} \cos \left (\frac {\pi }{7}\right ) \log \left (x^2-2 x \cos \left (\frac {\pi }{7}\right )+1\right )\right )+\frac {1}{7} \log (x+1)\) |
Log[1 + x]/7 + (2*(ArcTan[(Sec[Pi/14]*(2*x - 2*Sin[Pi/14]))/2]*Cos[Pi/14] - (Log[1 + x^2 - 2*x*Sin[Pi/14]]*Sin[Pi/14])/2))/7 + (2*(-1/2*(Cos[Pi/7]*L og[1 + x^2 - 2*x*Cos[Pi/7]]) + ArcTan[((2*x - 2*Cos[Pi/7])*Csc[Pi/7])/2]*S in[Pi/7]))/7 + (2*(ArcTan[(Sec[(3*Pi)/14]*(2*x + 2*Sin[(3*Pi)/14]))/2]*Cos [(3*Pi)/14] + (Log[1 + x^2 + 2*x*Sin[(3*Pi)/14]]*Sin[(3*Pi)/14])/2))/7
3.15.50.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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_)^(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 - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x]; r/(a*n) Int[1/(r + s*x), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 1)/2}], x]] /; Fre eQ[{a, b}, x] && IGtQ[(n - 3)/2, 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 = 3.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.23
| method | result | size |
| risch | \(\frac {\ln \left (1+x \right )}{7}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{5}+\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} +x \right )\right )}{7}\) | \(38\) |
| default | \(\frac {\ln \left (1+x \right )}{7}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{5}+\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\frac {\left (-\textit {\_R}^{5}+2 \textit {\_R}^{4}-3 \textit {\_R}^{3}+4 \textit {\_R}^{2}-5 \textit {\_R} +6\right ) \ln \left (x -\textit {\_R} \right )}{6 \textit {\_R}^{5}-5 \textit {\_R}^{4}+4 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} -1}\right )}{7}\) | \(97\) |
| meijerg | \(\frac {x \ln \left (1+\left (x^{7}\right )^{\frac {1}{7}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}-\frac {x \cos \left (\frac {\pi }{7}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}+\left (x^{7}\right )^{\frac {2}{7}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}+\frac {2 x \sin \left (\frac {\pi }{7}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}{1-\cos \left (\frac {\pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}-\frac {x \cos \left (\frac {3 \pi }{7}\right ) \ln \left (1-2 \cos \left (\frac {3 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}+\left (x^{7}\right )^{\frac {2}{7}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}+\frac {2 x \sin \left (\frac {3 \pi }{7}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}{1-\cos \left (\frac {3 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}+\frac {x \cos \left (\frac {2 \pi }{7}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}+\left (x^{7}\right )^{\frac {2}{7}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}+\frac {2 x \sin \left (\frac {2 \pi }{7}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}{1+\cos \left (\frac {2 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}\) | \(224\) |
Result contains complex when optimal does not.
Time = 3.60 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.39 \[ \int \frac {1}{1+x^7} \, dx=\frac {1}{14} \, {\left (\sqrt {-2.445041867912629? + 0.?e-76 \sqrt {-1}} + 1.246979603717467? + 0.?e-75 \sqrt {-1}\right )} \log \left (2 \, x + \sqrt {-2.445041867912629? + 0.?e-76 \sqrt {-1}} + 1.246979603717467? + 0.?e-75 \sqrt {-1}\right ) - \frac {1}{14} \, {\left (\sqrt {-2.445041867912629? + 0.?e-76 \sqrt {-1}} - 1.246979603717467? + 0.?e-75 \sqrt {-1}\right )} \log \left (2 \, x - \sqrt {-2.445041867912629? + 0.?e-76 \sqrt {-1}} + 1.246979603717467? + 0.?e-75 \sqrt {-1}\right ) + \frac {1}{7} \, \log \left (x + 1\right ) - \left (0.03178870485090206? - 0.1392754160259748? \sqrt {-1}\right ) \, \log \left (x - 0.2225209339563144? + 0.9749279121818236? \sqrt {-1}\right ) - \left (0.03178870485090206? + 0.1392754160259748? \sqrt {-1}\right ) \, \log \left (x - 0.2225209339563144? - 0.9749279121818236? \sqrt {-1}\right ) - \left (0.1287098382717742? - 0.06198339130250831? \sqrt {-1}\right ) \, \log \left (x - 0.9009688679024191? + 0.4338837391175582? \sqrt {-1}\right ) - \left (0.1287098382717742? + 0.06198339130250831? \sqrt {-1}\right ) \, \log \left (x - 0.9009688679024191? - 0.4338837391175582? \sqrt {-1}\right ) \]
1/14*(sqrt(-2.445041867912629? + 0.?e-76*I) + 1.246979603717467? + 0.?e-75 *I)*log(2*x + sqrt(-2.445041867912629? + 0.?e-76*I) + 1.246979603717467? + 0.?e-75*I) - 1/14*(sqrt(-2.445041867912629? + 0.?e-76*I) - 1.246979603717 467? + 0.?e-75*I)*log(2*x - sqrt(-2.445041867912629? + 0.?e-76*I) + 1.2469 79603717467? + 0.?e-75*I) + 1/7*log(x + 1) - (0.03178870485090206? - 0.139 2754160259748?*I)*log(x - 0.2225209339563144? + 0.9749279121818236?*I) - ( 0.03178870485090206? + 0.1392754160259748?*I)*log(x - 0.2225209339563144? - 0.9749279121818236?*I) - (0.1287098382717742? - 0.06198339130250831?*I)* log(x - 0.9009688679024191? + 0.4338837391175582?*I) - (0.1287098382717742 ? + 0.06198339130250831?*I)*log(x - 0.9009688679024191? - 0.43388373911755 82?*I)
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.27 \[ \int \frac {1}{1+x^7} \, dx=\frac {\log {\left (x + 1 \right )}}{7} + \operatorname {RootSum} {\left (117649 t^{6} + 16807 t^{5} + 2401 t^{4} + 343 t^{3} + 49 t^{2} + 7 t + 1, \left ( t \mapsto t \log {\left (7 t + x \right )} \right )\right )} \]
log(x + 1)/7 + RootSum(117649*_t**6 + 16807*_t**5 + 2401*_t**4 + 343*_t**3 + 49*_t**2 + 7*_t + 1, Lambda(_t, _t*log(7*_t + x)))
\[ \int \frac {1}{1+x^7} \, dx=\int { \frac {1}{x^{7} + 1} \,d x } \]
-1/7*integrate((x^5 - 2*x^4 + 3*x^3 - 4*x^2 + 5*x - 6)/(x^6 - x^5 + x^4 - x^3 + x^2 - x + 1), x) + 1/7*log(x + 1)
Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.78 \[ \int \frac {1}{1+x^7} \, dx=-\frac {1}{7} \, \cos \left (\frac {3}{7} \, \pi \right ) \log \left (x^{2} - 2 \, x \cos \left (\frac {3}{7} \, \pi \right ) + 1\right ) + \frac {1}{7} \, \cos \left (\frac {2}{7} \, \pi \right ) \log \left (x^{2} + 2 \, x \cos \left (\frac {2}{7} \, \pi \right ) + 1\right ) - \frac {1}{7} \, \cos \left (\frac {1}{7} \, \pi \right ) \log \left (x^{2} - 2 \, x \cos \left (\frac {1}{7} \, \pi \right ) + 1\right ) + \frac {2}{7} \, \arctan \left (\frac {x - \cos \left (\frac {3}{7} \, \pi \right )}{\sin \left (\frac {3}{7} \, \pi \right )}\right ) \sin \left (\frac {3}{7} \, \pi \right ) + \frac {2}{7} \, \arctan \left (\frac {x + \cos \left (\frac {2}{7} \, \pi \right )}{\sin \left (\frac {2}{7} \, \pi \right )}\right ) \sin \left (\frac {2}{7} \, \pi \right ) + \frac {2}{7} \, \arctan \left (\frac {x - \cos \left (\frac {1}{7} \, \pi \right )}{\sin \left (\frac {1}{7} \, \pi \right )}\right ) \sin \left (\frac {1}{7} \, \pi \right ) + \frac {1}{7} \, \log \left ({\left | x + 1 \right |}\right ) \]
-1/7*cos(3/7*pi)*log(x^2 - 2*x*cos(3/7*pi) + 1) + 1/7*cos(2/7*pi)*log(x^2 + 2*x*cos(2/7*pi) + 1) - 1/7*cos(1/7*pi)*log(x^2 - 2*x*cos(1/7*pi) + 1) + 2/7*arctan((x - cos(3/7*pi))/sin(3/7*pi))*sin(3/7*pi) + 2/7*arctan((x + co s(2/7*pi))/sin(2/7*pi))*sin(2/7*pi) + 2/7*arctan((x - cos(1/7*pi))/sin(1/7 *pi))*sin(1/7*pi) + 1/7*log(abs(x + 1))
Time = 6.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.62 \[ \int \frac {1}{1+x^7} \, dx=\frac {\ln \left (x+1\right )}{7}-\frac {{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\,\ln \left (x-{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\right )}{7}-\frac {{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\,\ln \left (x-{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\right )}{7}-\frac {{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\,\ln \left (x-{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\right )}{7}+\frac {\ln \left (x+{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}\right )\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}}{7}+\frac {\ln \left (x+{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}\right )\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}}{7}+\frac {\ln \left (x+{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}\right )\,{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}}{7} \]