3.1447 \(\int \frac{1}{1-x^7} \, dx\)

Optimal. Leaf size=166 \[ -\frac{1}{7} \sin \left (\frac{3 \pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac{3 \pi }{14}\right )+1\right )+\frac{1}{7} \sin \left (\frac{\pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{14}\right )+1\right )+\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{7}\right )+1\right )-\frac{1}{7} \log (1-x)+\frac{2}{7} \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{7}\right ) \left (x+\cos \left (\frac{\pi }{7}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{3 \pi }{14}\right ) \left (x-\sin \left (\frac{3 \pi }{14}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{14}\right ) \left (x+\sin \left (\frac{\pi }{14}\right )\right )\right ) \]

[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

_______________________________________________________________________________________

Rubi [A]  time = 0.272953, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{1}{7} \sin \left (\frac{3 \pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac{3 \pi }{14}\right )+1\right )+\frac{1}{7} \sin \left (\frac{\pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{14}\right )+1\right )+\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{7}\right )+1\right )-\frac{1}{7} \log (1-x)+\frac{2}{7} \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{7}\right ) \left (x+\cos \left (\frac{\pi }{7}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{3 \pi }{14}\right ) \left (x-\sin \left (\frac{3 \pi }{14}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{14}\right ) \left (x+\sin \left (\frac{\pi }{14}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[(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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 75.8887, size = 211, normalized size = 1.27 \[ - \frac{\log{\left (- x + 1 \right )}}{7} + \frac{\log{\left (x^{2} + 2 x \cos{\left (\frac{\pi }{7} \right )} + 1 \right )} \cos{\left (\frac{\pi }{7} \right )}}{7} - \frac{\log{\left (x^{2} - 2 x \cos{\left (\frac{2 \pi }{7} \right )} + 1 \right )} \cos{\left (\frac{2 \pi }{7} \right )}}{7} + \frac{\log{\left (x^{2} + 2 x \cos{\left (\frac{3 \pi }{7} \right )} + 1 \right )} \cos{\left (\frac{3 \pi }{7} \right )}}{7} + \frac{\sqrt{2} \sqrt{- \sin{\left (\frac{3 \pi }{14} \right )} + 1} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \cos{\left (\frac{\pi }{7} \right )}\right )}{\sqrt{- \sin{\left (\frac{3 \pi }{14} \right )} + 1}} \right )}}{7} + \frac{\sqrt{2} \sqrt{\sin{\left (\frac{\pi }{14} \right )} + 1} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \cos{\left (\frac{2 \pi }{7} \right )}\right )}{\sqrt{\sin{\left (\frac{\pi }{14} \right )} + 1}} \right )}}{7} + \frac{\sqrt{2} \sqrt{\sin{\left (\frac{5 \pi }{14} \right )} + 1} \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \cos{\left (\frac{3 \pi }{7} \right )}\right )}{\sqrt{\sin{\left (\frac{5 \pi }{14} \right )} + 1}} \right )}}{7} \]

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 + sqr
t(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(sin(pi/1
4) + 1))/7 + sqrt(2)*sqrt(sin(5*pi/14) + 1)*atan(sqrt(2)*(x + cos(3*pi/7))/sqrt(
sin(5*pi/14) + 1))/7

_______________________________________________________________________________________

Mathematica [A]  time = 0.00700955, size = 166, normalized size = 1. \[ -\frac{1}{7} \sin \left (\frac{3 \pi }{14}\right ) \log \left (x^2-2 x \sin \left (\frac{3 \pi }{14}\right )+1\right )+\frac{1}{7} \sin \left (\frac{\pi }{14}\right ) \log \left (x^2+2 x \sin \left (\frac{\pi }{14}\right )+1\right )+\frac{1}{7} \cos \left (\frac{\pi }{7}\right ) \log \left (x^2+2 x \cos \left (\frac{\pi }{7}\right )+1\right )-\frac{1}{7} \log (1-x)+\frac{2}{7} \sin \left (\frac{\pi }{7}\right ) \tan ^{-1}\left (\csc \left (\frac{\pi }{7}\right ) \left (x+\cos \left (\frac{\pi }{7}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{3 \pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{3 \pi }{14}\right ) \left (x-\sin \left (\frac{3 \pi }{14}\right )\right )\right )+\frac{2}{7} \cos \left (\frac{\pi }{14}\right ) \tan ^{-1}\left (\sec \left (\frac{\pi }{14}\right ) \left (x+\sin \left (\frac{\pi }{14}\right )\right )\right ) \]

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

_______________________________________________________________________________________

Maple [C]  time = 0.014, size = 89, normalized size = 0.5 \[ -{\frac{\ln \left ( -1+x \right ) }{7}}+{\frac{1}{7}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+{{\it \_Z}}^{5}+{{\it \_Z}}^{4}+{{\it \_Z}}^{3}+{{\it \_Z}}^{2}+{\it \_Z}+1 \right ) }{\frac{ \left ({{\it \_R}}^{5}+2\,{{\it \_R}}^{4}+3\,{{\it \_R}}^{3}+4\,{{\it \_R}}^{2}+5\,{\it \_R}+6 \right ) \ln \left ( x-{\it \_R} \right ) }{6\,{{\it \_R}}^{5}+5\,{{\it \_R}}^{4}+4\,{{\it \_R}}^{3}+3\,{{\it \_R}}^{2}+2\,{\it \_R}+1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^7+1),x)

[Out]

-1/7*ln(-1+x)+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))

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

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

_______________________________________________________________________________________

Sympy [A]  time = 0.550645, size = 46, normalized size = 0.28 \[ - \frac{\log{\left (x - 1 \right )}}{7} - \operatorname{RootSum}{\left (117649 t^{6} + 16807 t^{5} + 2401 t^{4} + 343 t^{3} + 49 t^{2} + 7 t + 1, \left ( t \mapsto t \log{\left (- 7 t + x \right )} \right )\right )} \]

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)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218751, size = 171, normalized size = 1.03 \[ \frac{1}{7} \, \cos \left (\frac{3}{7} \, \pi \right ){\rm ln}\left (x^{2} + 2 \, x \cos \left (\frac{3}{7} \, \pi \right ) + 1\right ) - \frac{1}{7} \, \cos \left (\frac{2}{7} \, \pi \right ){\rm ln}\left (x^{2} - 2 \, x \cos \left (\frac{2}{7} \, \pi \right ) + 1\right ) + \frac{1}{7} \, \cos \left (\frac{1}{7} \, \pi \right ){\rm ln}\left (x^{2} + 2 \, x \cos \left (\frac{1}{7} \, \pi \right ) + 1\right ) + \frac{2}{7} \, \arctan \left (\frac{x + \cos \left (\frac{3}{7} \, \pi \right )}{\sin \left (\frac{3}{7} \, \pi \right )}\right ) \sin \left (\frac{3}{7} \, \pi \right ) + \frac{2}{7} \, \arctan \left (\frac{x - \cos \left (\frac{2}{7} \, \pi \right )}{\sin \left (\frac{2}{7} \, \pi \right )}\right ) \sin \left (\frac{2}{7} \, \pi \right ) + \frac{2}{7} \, \arctan \left (\frac{x + \cos \left (\frac{1}{7} \, \pi \right )}{\sin \left (\frac{1}{7} \, \pi \right )}\right ) \sin \left (\frac{1}{7} \, \pi \right ) - \frac{1}{7} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

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*cos
(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(a
bs(x - 1))