3.1448 \(\int \frac{1}{1+x^7} \, dx\)

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)/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[Cot[P
i/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 = 0.294185, 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 \[ \frac{1}{7} \sin \left (\frac{3 \pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac{3 \pi }{14}\right )+1\right )-\frac{1}{7} \sin \left (\frac{\pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac{\pi }{14}\right )+1\right )-\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \log \left (x^2-2 x \cos \left (\frac{\pi }{7}\right )+1\right )+\frac{1}{7} \log (x+1)+\frac{2}{7} \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (x \sec \left (\frac{3 \pi }{14}\right )+\tan \left (\frac{3 \pi }{14}\right )\right )+\frac{2}{7} \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (x \sec \left (\frac{\pi }{14}\right )-\tan \left (\frac{\pi }{14}\right )\right )-\frac{2}{7} \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{7}\right )-x \csc \left (\frac{\pi }{7}\right )\right ) \]

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)/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[Cot[P
i/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 in Sympy [A]  time = 72.2724, size = 209, normalized size = 1.27 \[ \frac{\log{\left (x + 1 \right )}}{7} - \frac{\log{\left (x^{2} - 2 x \cos{\left (\frac{\pi }{7} \right )} + 1 \right )} \cos{\left (\frac{\pi }{7} \right )}}{7} + \frac{\log{\left (x^{2} + 2 x \cos{\left (\frac{2 \pi }{7} \right )} + 1 \right )} \cos{\left (\frac{2 \pi }{7} \right )}}{7} - \frac{\log{\left (x^{2} - 2 x \cos{\left (\frac{3 \pi }{7} \right )} + 1 \right )} \cos{\left (\frac{3 \pi }{7} \right )}}{7} + \frac{\sqrt{2} \sqrt{\sin{\left (\frac{5 \pi }{14} \right )} + 1} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \sin{\left (\frac{\pi }{14} \right )}\right )}{\sqrt{\sin{\left (\frac{5 \pi }{14} \right )} + 1}} \right )}}{7} + \frac{\sqrt{2} \sqrt{- \sin{\left (\frac{3 \pi }{14} \right )} + 1} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \cos{\left (\frac{\pi }{7} \right )}\right )}{\sqrt{- \sin{\left (\frac{3 \pi }{14} \right )} + 1}} \right )}}{7} + \frac{\sqrt{2} \sqrt{\sin{\left (\frac{\pi }{14} \right )} + 1} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \cos{\left (\frac{2 \pi }{7} \right )}\right )}{\sqrt{\sin{\left (\frac{\pi }{14} \right )} + 1}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**7+1),x)

[Out]

log(x + 1)/7 - log(x**2 - 2*x*cos(pi/7) + 1)*cos(pi/7)/7 + log(x**2 + 2*x*cos(2*
pi/7) + 1)*cos(2*pi/7)/7 - log(x**2 - 2*x*cos(3*pi/7) + 1)*cos(3*pi/7)/7 + sqrt(
2)*sqrt(sin(5*pi/14) + 1)*atan(sqrt(2)*(x - sin(pi/14))/sqrt(sin(5*pi/14) + 1))/
7 + sqrt(2)*sqrt(-sin(3*pi/14) + 1)*atan(sqrt(2)*(x - cos(pi/7))/sqrt(-sin(3*pi/
14) + 1))/7 + sqrt(2)*sqrt(sin(pi/14) + 1)*atan(sqrt(2)*(x + cos(2*pi/7))/sqrt(s
in(pi/14) + 1))/7

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Mathematica [A]  time = 0.00675196, 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*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

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Maple [C]  time = 0.014, size = 97, normalized size = 0.6 \[{\frac{1}{7}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}-{{\it \_Z}}^{5}+{{\it \_Z}}^{4}-{{\it \_Z}}^{3}+{{\it \_Z}}^{2}-{\it \_Z}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{5}+2\,{{\it \_R}}^{4}-3\,{{\it \_R}}^{3}+4\,{{\it \_R}}^{2}-5\,{\it \_R}+6 \right ) \ln \left ( x-{\it \_R} \right ) }{6\,{{\it \_R}}^{5}-5\,{{\it \_R}}^{4}+4\,{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+2\,{\it \_R}-1}}}+{\frac{\ln \left ( 1+x \right ) }{7}} \]

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. \[ -\frac{1}{7} \, \int \frac{x^{5} - 2 \, x^{4} + 3 \, x^{3} - 4 \, x^{2} + 5 \, x - 6}{x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1}\,{d x} + \frac{1}{7} \, \log \left (x + 1\right ) \]

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.546635, size = 44, normalized size = 0.27 \[ \frac{\log{\left (x + 1 \right )}}{7} + \operatorname{RootSum}{\left (117649 t^{6} + 16807 t^{5} + 2401 t^{4} + 343 t^{3} + 49 t^{2} + 7 t + 1, \left ( t \mapsto t \log{\left (7 t + x \right )} \right )\right )} \]

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/XCAS [A]  time = 0.220261, size = 174, normalized size = 1.05 \[ -\frac{1}{7} \, \cos \left (\frac{3}{7} \, \pi \right ){\rm ln}\left (x^{2} - 2 \, x \cos \left (\frac{3}{7} \, \pi \right ) + 1\right ) + \frac{1}{7} \, \cos \left (\frac{2}{7} \, \pi \right ){\rm ln}\left (x^{2} + 2 \, x \cos \left (\frac{2}{7} \, \pi \right ) + 1\right ) - \frac{1}{7} \, \cos \left (\frac{1}{7} \, \pi \right ){\rm ln}\left (x^{2} - 2 \, x \cos \left (\frac{1}{7} \, \pi \right ) + 1\right ) + \frac{2}{7} \, \arctan \left (\frac{x - \cos \left (\frac{3}{7} \, \pi \right )}{\sin \left (\frac{3}{7} \, \pi \right )}\right ) \sin \left (\frac{3}{7} \, \pi \right ) + \frac{2}{7} \, \arctan \left (\frac{x + \cos \left (\frac{2}{7} \, \pi \right )}{\sin \left (\frac{2}{7} \, \pi \right )}\right ) \sin \left (\frac{2}{7} \, \pi \right ) + \frac{2}{7} \, \arctan \left (\frac{x - \cos \left (\frac{1}{7} \, \pi \right )}{\sin \left (\frac{1}{7} \, \pi \right )}\right ) \sin \left (\frac{1}{7} \, \pi \right ) + \frac{1}{7} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

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)*ln(x^2 - 2*x*cos(3/7*pi) + 1) + 1/7*cos(2/7*pi)*ln(x^2 + 2*x*co
s(2/7*pi) + 1) - 1/7*cos(1/7*pi)*ln(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*ln(
abs(x + 1))