3.1450 \(\int \frac{1}{1+x^7} \, dx\)

Optimal. Leaf size=165 \[ \frac{1}{7} \sin \left (\frac{3 \pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac{3 \pi }{14}\right )+1\right )-\frac{1}{7} \sin \left (\frac{\pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{14}\right )+1\right )-\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{7}\right )+1\right )+\frac{1}{7} \log (x+1)+\frac{2}{7} \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (x \sec \left (\frac{3 \pi }{14}\right )+\tan \left (\frac{3 \pi }{14}\right )\right )+\frac{2}{7} \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (x \sec \left (\frac{\pi }{14}\right )-\tan \left (\frac{\pi }{14}\right )\right )-\frac{2}{7} \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{7}\right )-x \csc \left (\frac{\pi }{7}\right )\right ) \]

[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

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Rubi [A]  time = 0.139025, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {201, 634, 618, 204, 628, 31} \[ \frac{1}{7} \sin \left (\frac{3 \pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac{3 \pi }{14}\right )+1\right )-\frac{1}{7} \sin \left (\frac{\pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{14}\right )+1\right )-\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{7}\right )+1\right )+\frac{1}{7} \log (x+1)+\frac{2}{7} \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (x \sec \left (\frac{3 \pi }{14}\right )+\tan \left (\frac{3 \pi }{14}\right )\right )+\frac{2}{7} \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (x \sec \left (\frac{\pi }{14}\right )-\tan \left (\frac{\pi }{14}\right )\right )-\frac{2}{7} \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{7}\right )-x \csc \left (\frac{\pi }{7}\right )\right ) \]

Antiderivative was successfully verified.

[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 201

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*
Int[1/(r + s*x), x])/(a*n) + 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 634

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]

Rule 618

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

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 31

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

Rubi steps

\begin{align*} \int \frac{1}{1+x^7} \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.0039307, size = 166, normalized size = 1.01 \[ \frac{1}{7} \sin \left (\frac{3 \pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac{3 \pi }{14}\right )+1\right )-\frac{1}{7} \sin \left (\frac{\pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{14}\right )+1\right )-\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{7}\right )+1\right )+\frac{1}{7} \log (x+1)+\frac{2}{7} \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{7}\right ) \left (x-\cos \left (\frac{\pi }{7}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{3 \pi }{14}\right ) \left (x+\sin \left (\frac{3 \pi }{14}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{14}\right ) \left (x-\sin \left (\frac{\pi }{14}\right )\right )\right ) \]

Antiderivative was successfully verified.

[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

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Maple [C]  time = 0.008, size = 97, normalized size = 0.6 \begin{align*}{\frac{1}{7}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{5}+{{\it \_Z}}^{4}-{{\it \_Z}}^{3}+{{\it \_Z}}^{2}-{\it \_Z}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{5}+2\,{{\it \_R}}^{4}-3\,{{\it \_R}}^{3}+4\,{{\it \_R}}^{2}-5\,{\it \_R}+6 \right ) \ln \left ( x-{\it \_R} \right ) }{6\,{{\it \_R}}^{5}-5\,{{\it \_R}}^{4}+4\,{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+2\,{\it \_R}-1}}}+{\frac{\ln \left ( 1+x \right ) }{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{7} \, \int \frac{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}\,{d x} + \frac{1}{7} \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [C]  time = 24.3663, size = 1010, normalized size = 6.12 \begin{align*} \frac{1}{14} \,{\left (\sqrt{-2.445041867912629? + 0.?e-37 \sqrt{-1}} + 1.246979603717467? + 0.?e-36 \sqrt{-1}\right )} \log \left (2 \, x + \sqrt{-2.445041867912629? + 0.?e-37 \sqrt{-1}} + 1.246979603717467? + 0.?e-36 \sqrt{-1}\right ) - \frac{1}{14} \,{\left (\sqrt{-2.445041867912629? + 0.?e-37 \sqrt{-1}} - 1.246979603717467? + 0.?e-36 \sqrt{-1}\right )} \log \left (2 \, x - \sqrt{-2.445041867912629? + 0.?e-37 \sqrt{-1}} + 1.246979603717467? + 0.?e-36 \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? + 0.4338837391175581? \sqrt{-1}\right ) - \left (0.1287098382717742? + 0.06198339130250831? \sqrt{-1}\right ) \, \log \left (x - 0.9009688679024191? - 0.4338837391175582? \sqrt{-1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*(sqrt(-2.445041867912629? + 0.?e-37*I) + 1.246979603717467? + 0.?e-36*I)*log(2*x + sqrt(-2.44504186791262
9? + 0.?e-37*I) + 1.246979603717467? + 0.?e-36*I) - 1/14*(sqrt(-2.445041867912629? + 0.?e-37*I) - 1.2469796037
17467? + 0.?e-36*I)*log(2*x - sqrt(-2.445041867912629? + 0.?e-37*I) + 1.246979603717467? + 0.?e-36*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? + 0.4338837391175581?*I) - (0.1287098382
717742? + 0.06198339130250831?*I)*log(x - 0.9009688679024191? - 0.4338837391175582?*I)

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Sympy [A]  time = 0.174362, size = 44, normalized size = 0.27 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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Giac [A]  time = 1.14935, size = 174, normalized size = 1.05 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))