\(\int \frac {1}{1+x^7} \, dx\) [1450]

Optimal result

Integrand size = 7, antiderivative size = 165 \[ \int \frac {1}{1+x^7} \, dx=\frac {2}{7} \arctan \left (x \sec \left (\frac {\pi }{14}\right )-\tan \left (\frac {\pi }{14}\right )\right ) \cos \left (\frac {\pi }{14}\right )+\frac {2}{7} \arctan \left (x \sec \left (\frac {3 \pi }{14}\right )+\tan \left (\frac {3 \pi }{14}\right )\right ) \cos \left (\frac {3 \pi }{14}\right )+\frac {1}{7} \log (1+x)-\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \log \left (1+x^2-2 x \cos \left (\frac {\pi }{7}\right )\right )-\frac {1}{7} \log \left (1+x^2-2 x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )-\frac {2}{7} \arctan \left (\cot \left (\frac {\pi }{7}\right )-x \csc \left (\frac {\pi }{7}\right )\right ) \sin \left (\frac {\pi }{7}\right )+\frac {1}{7} \log \left (1+x^2+2 x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right ) \]

[Out]

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*cos(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*Pi)

Rubi [A] (warning: unable to verify)

Time = 0.13 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {207, 648, 632, 210, 642, 31} \[ \int \frac {1}{1+x^7} \, dx=\frac {2}{7} \cos \left (\frac {3 \pi }{14}\right ) \arctan \left (x \sec \left (\frac {3 \pi }{14}\right )+\tan \left (\frac {3 \pi }{14}\right )\right )+\frac {2}{7} \cos \left (\frac {\pi }{14}\right ) \arctan \left (x \sec \left (\frac {\pi }{14}\right )-\tan \left (\frac {\pi }{14}\right )\right )-\frac {2}{7} \sin \left (\frac {\pi }{7}\right ) \arctan \left (\cot \left (\frac {\pi }{7}\right )-x \csc \left (\frac {\pi }{7}\right )\right )+\frac {1}{7} \sin \left (\frac {3 \pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac {3 \pi }{14}\right )+1\right )-\frac {1}{7} \sin \left (\frac {\pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac {\pi }{14}\right )+1\right )-\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \log \left (x^2-2 x \cos \left (\frac {\pi }{7}\right )+1\right )+\frac {1}{7} \log (x+1) \]

[In]

Int[(1 + x^7)^(-1),x]

[Out]

(2*ArcTan[x*Sec[Pi/14] - Tan[Pi/14]]*Cos[Pi/14])/7 + (2*ArcTan[x*Sec[(3*Pi)/14] + Tan[(3*Pi)/14]]*Cos[(3*Pi)/1
4])/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[Cot[Pi/7] - x*Csc[Pi/7]]*Sin[Pi/7])/7 + (Log[1 + x^2 + 2*x*Sin[(3*Pi)/14]]*Sin[(3*Pi)/14])/7

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 207

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] + Dist[2*(r/(a*n)), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[
(n - 3)/2, 0] && PosQ[a/b]

Rule 210

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])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{7} \int \frac {1-x \cos \left (\frac {\pi }{7}\right )}{1+x^2-2 x \cos \left (\frac {\pi }{7}\right )} \, dx+\frac {2}{7} \int \frac {1-x \sin \left (\frac {\pi }{14}\right )}{1+x^2-2 x \sin \left (\frac {\pi }{14}\right )} \, dx+\frac {2}{7} \int \frac {1+x \sin \left (\frac {3 \pi }{14}\right )}{1+x^2+2 x \sin \left (\frac {3 \pi }{14}\right )} \, dx+\frac {1}{7} \int \frac {1}{1+x} \, dx \\ & = \frac {1}{7} \log (1+x)+\frac {1}{7} \left (2 \cos ^2\left (\frac {\pi }{14}\right )\right ) \int \frac {1}{1+x^2-2 x \sin \left (\frac {\pi }{14}\right )} \, dx-\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \int \frac {2 x-2 \cos \left (\frac {\pi }{7}\right )}{1+x^2-2 x \cos \left (\frac {\pi }{7}\right )} \, dx+\frac {1}{7} \left (2 \cos ^2\left (\frac {3 \pi }{14}\right )\right ) \int \frac {1}{1+x^2+2 x \sin \left (\frac {3 \pi }{14}\right )} \, dx-\frac {1}{7} \sin \left (\frac {\pi }{14}\right ) \int \frac {2 x-2 \sin \left (\frac {\pi }{14}\right )}{1+x^2-2 x \sin \left (\frac {\pi }{14}\right )} \, dx+\frac {1}{7} \left (2 \sin ^2\left (\frac {\pi }{7}\right )\right ) \int \frac {1}{1+x^2-2 x \cos \left (\frac {\pi }{7}\right )} \, dx+\frac {1}{7} \sin \left (\frac {3 \pi }{14}\right ) \int \frac {2 x+2 \sin \left (\frac {3 \pi }{14}\right )}{1+x^2+2 x \sin \left (\frac {3 \pi }{14}\right )} \, dx \\ & = \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 {1}{7} \log \left (1+x^2+2 x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right )-\frac {1}{7} \left (4 \cos ^2\left (\frac {\pi }{14}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \cos ^2\left (\frac {\pi }{14}\right )} \, dx,x,2 x-2 \sin \left (\frac {\pi }{14}\right )\right )-\frac {1}{7} \left (4 \cos ^2\left (\frac {3 \pi }{14}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \cos ^2\left (\frac {3 \pi }{14}\right )} \, dx,x,2 x+2 \sin \left (\frac {3 \pi }{14}\right )\right )-\frac {1}{7} \left (4 \sin ^2\left (\frac {\pi }{7}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2-4 \sin ^2\left (\frac {\pi }{7}\right )} \, dx,x,2 x-2 \cos \left (\frac {\pi }{7}\right )\right ) \\ & = \frac {2}{7} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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 ) \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.06 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.01 \[ \int \frac {1}{1+x^7} \, dx=\frac {2}{7} \arctan \left (\sec \left (\frac {\pi }{14}\right ) \left (x-\sin \left (\frac {\pi }{14}\right )\right )\right ) \cos \left (\frac {\pi }{14}\right )+\frac {2}{7} \arctan \left (\sec \left (\frac {3 \pi }{14}\right ) \left (x+\sin \left (\frac {3 \pi }{14}\right )\right )\right ) \cos \left (\frac {3 \pi }{14}\right )+\frac {1}{7} \log (1+x)-\frac {1}{7} \cos \left (\frac {\pi }{7}\right ) \log \left (1+x^2-2 x \cos \left (\frac {\pi }{7}\right )\right )-\frac {1}{7} \log \left (1+x^2-2 x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )+\frac {2}{7} \arctan \left (\left (x-\cos \left (\frac {\pi }{7}\right )\right ) \csc \left (\frac {\pi }{7}\right )\right ) \sin \left (\frac {\pi }{7}\right )+\frac {1}{7} \log \left (1+x^2+2 x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right ) \]

[In]

Integrate[(1 + x^7)^(-1),x]

[Out]

(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*P
i)/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

Maple [C] (verified)

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	\(\frac {x \ln \left (1+\left (x^{7}\right )^{\frac {1}{7}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}-\frac {x \cos \left (\frac {\pi }{7}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}+\left (x^{7}\right )^{\frac {2}{7}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}+\frac {2 x \sin \left (\frac {\pi }{7}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}{1-\cos \left (\frac {\pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}-\frac {x \cos \left (\frac {3 \pi }{7}\right ) \ln \left (1-2 \cos \left (\frac {3 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}+\left (x^{7}\right )^{\frac {2}{7}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}+\frac {2 x \sin \left (\frac {3 \pi }{7}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}{1-\cos \left (\frac {3 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}+\frac {x \cos \left (\frac {2 \pi }{7}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}+\left (x^{7}\right )^{\frac {2}{7}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}+\frac {2 x \sin \left (\frac {2 \pi }{7}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}{1+\cos \left (\frac {2 \pi }{7}\right ) \left (x^{7}\right )^{\frac {1}{7}}}\right )}{7 \left (x^{7}\right )^{\frac {1}{7}}}\)	\(224\)

[In]

int(1/(x^7+1),x,method=_RETURNVERBOSE)

[Out]

1/7*ln(1+x)+1/7*sum(_R*ln(_R+x),_R=RootOf(_Z^6+_Z^5+_Z^4+_Z^3+_Z^2+_Z+1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.60 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.39 \[ \int \frac {1}{1+x^7} \, dx=\frac {1}{14} \, {\left (\sqrt {-2.445041867912629? + 0.?e-76 \sqrt {-1}} + 1.246979603717467? + 0.?e-75 \sqrt {-1}\right )} \log \left (2 \, x + \sqrt {-2.445041867912629? + 0.?e-76 \sqrt {-1}} + 1.246979603717467? + 0.?e-75 \sqrt {-1}\right ) - \frac {1}{14} \, {\left (\sqrt {-2.445041867912629? + 0.?e-76 \sqrt {-1}} - 1.246979603717467? + 0.?e-75 \sqrt {-1}\right )} \log \left (2 \, x - \sqrt {-2.445041867912629? + 0.?e-76 \sqrt {-1}} + 1.246979603717467? + 0.?e-75 \sqrt {-1}\right ) + \frac {1}{7} \, \log \left (x + 1\right ) - \left (0.03178870485090206? - 0.1392754160259748? \sqrt {-1}\right ) \, \log \left (x - 0.2225209339563144? + 0.9749279121818236? \sqrt {-1}\right ) - \left (0.03178870485090206? + 0.1392754160259748? \sqrt {-1}\right ) \, \log \left (x - 0.2225209339563144? - 0.9749279121818236? \sqrt {-1}\right ) - \left (0.1287098382717742? - 0.06198339130250831? \sqrt {-1}\right ) \, \log \left (x - 0.9009688679024191? + 0.4338837391175582? \sqrt {-1}\right ) - \left (0.1287098382717742? + 0.06198339130250831? \sqrt {-1}\right ) \, \log \left (x - 0.9009688679024191? - 0.4338837391175582? \sqrt {-1}\right ) \]

[In]

integrate(1/(x^7+1),x, algorithm="fricas")

[Out]

1/14*(sqrt(-2.445041867912629? + 0.?e-76*I) + 1.246979603717467? + 0.?e-75*I)*log(2*x + sqrt(-2.44504186791262
9? + 0.?e-76*I) + 1.246979603717467? + 0.?e-75*I) - 1/14*(sqrt(-2.445041867912629? + 0.?e-76*I) - 1.2469796037
17467? + 0.?e-75*I)*log(2*x - sqrt(-2.445041867912629? + 0.?e-76*I) + 1.246979603717467? + 0.?e-75*I) + 1/7*lo
g(x + 1) - (0.03178870485090206? - 0.1392754160259748?*I)*log(x - 0.2225209339563144? + 0.9749279121818236?*I)
- (0.03178870485090206? + 0.1392754160259748?*I)*log(x - 0.2225209339563144? - 0.9749279121818236?*I) - (0.12
87098382717742? - 0.06198339130250831?*I)*log(x - 0.9009688679024191? + 0.4338837391175582?*I) - (0.1287098382
717742? + 0.06198339130250831?*I)*log(x - 0.9009688679024191? - 0.4338837391175582?*I)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.27 \[ \int \frac {1}{1+x^7} \, dx=\frac {\log {\left (x + 1 \right )}}{7} + \operatorname {RootSum} {\left (117649 t^{6} + 16807 t^{5} + 2401 t^{4} + 343 t^{3} + 49 t^{2} + 7 t + 1, \left ( t \mapsto t \log {\left (7 t + x \right )} \right )\right )} \]

[In]

integrate(1/(x**7+1),x)

[Out]

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)))

Maxima [F]

\[ \int \frac {1}{1+x^7} \, dx=\int { \frac {1}{x^{7} + 1} \,d x } \]

[In]

integrate(1/(x^7+1),x, algorithm="maxima")

[Out]

-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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.78 \[ \int \frac {1}{1+x^7} \, dx=-\frac {1}{7} \, \cos \left (\frac {3}{7} \, \pi \right ) \log \left (x^{2} - 2 \, x \cos \left (\frac {3}{7} \, \pi \right ) + 1\right ) + \frac {1}{7} \, \cos \left (\frac {2}{7} \, \pi \right ) \log \left (x^{2} + 2 \, x \cos \left (\frac {2}{7} \, \pi \right ) + 1\right ) - \frac {1}{7} \, \cos \left (\frac {1}{7} \, \pi \right ) \log \left (x^{2} - 2 \, x \cos \left (\frac {1}{7} \, \pi \right ) + 1\right ) + \frac {2}{7} \, \arctan \left (\frac {x - \cos \left (\frac {3}{7} \, \pi \right )}{\sin \left (\frac {3}{7} \, \pi \right )}\right ) \sin \left (\frac {3}{7} \, \pi \right ) + \frac {2}{7} \, \arctan \left (\frac {x + \cos \left (\frac {2}{7} \, \pi \right )}{\sin \left (\frac {2}{7} \, \pi \right )}\right ) \sin \left (\frac {2}{7} \, \pi \right ) + \frac {2}{7} \, \arctan \left (\frac {x - \cos \left (\frac {1}{7} \, \pi \right )}{\sin \left (\frac {1}{7} \, \pi \right )}\right ) \sin \left (\frac {1}{7} \, \pi \right ) + \frac {1}{7} \, \log \left ({\left | x + 1 \right |}\right ) \]

[In]

integrate(1/(x^7+1),x, algorithm="giac")

[Out]

-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 +
cos(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))

Mupad [B] (verification not implemented)

Time = 6.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.62 \[ \int \frac {1}{1+x^7} \, dx=\frac {\ln \left (x+1\right )}{7}-\frac {{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\,\ln \left (x-{\mathrm {e}}^{\frac {\pi \,1{}\mathrm {i}}{7}}\right )}{7}-\frac {{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\,\ln \left (x-{\mathrm {e}}^{\frac {\pi \,3{}\mathrm {i}}{7}}\right )}{7}-\frac {{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\,\ln \left (x-{\mathrm {e}}^{\frac {\pi \,5{}\mathrm {i}}{7}}\right )}{7}+\frac {\ln \left (x+{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}\right )\,{\mathrm {e}}^{\frac {\pi \,2{}\mathrm {i}}{7}}}{7}+\frac {\ln \left (x+{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}\right )\,{\mathrm {e}}^{\frac {\pi \,4{}\mathrm {i}}{7}}}{7}+\frac {\ln \left (x+{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}\right )\,{\mathrm {e}}^{\frac {\pi \,6{}\mathrm {i}}{7}}}{7} \]

[In]

int(1/(x^7 + 1),x)

[Out]

log(x + 1)/7 - (exp((pi*1i)/7)*log(x - exp((pi*1i)/7)))/7 - (exp((pi*3i)/7)*log(x - exp((pi*3i)/7)))/7 - (exp(
(pi*5i)/7)*log(x - exp((pi*5i)/7)))/7 + (log(x + exp((pi*2i)/7))*exp((pi*2i)/7))/7 + (log(x + exp((pi*4i)/7))*
exp((pi*4i)/7))/7 + (log(x + exp((pi*6i)/7))*exp((pi*6i)/7))/7