3.1538 \(\int \frac{1}{1-x^{10}} \, dx\)

Optimal. Leaf size=163 \[ \frac{1}{20} \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{\sqrt{10-2 \sqrt{5}} x}{2 \left (1-x^2\right )}\right )+\frac{1}{20} \sqrt{10+2 \sqrt{5}} \tan ^{-1}\left (\frac{\sqrt{10+2 \sqrt{5}} x}{2 \left (1-x^2\right )}\right )+\frac{1}{20} \left (1-\sqrt{5}\right ) \tanh ^{-1}\left (\frac{\left (1-\sqrt{5}\right ) x}{2 \left (x^2+1\right )}\right )+\frac{1}{20} \left (1+\sqrt{5}\right ) \tanh ^{-1}\left (\frac{\left (1+\sqrt{5}\right ) x}{2 \left (x^2+1\right )}\right )+\frac{1}{5} \tanh ^{-1}(x) \]

[Out]

(Sqrt[10 - 2*Sqrt[5]]*ArcTan[(Sqrt[10 - 2*Sqrt[5]]*x)/(2*(1 - x^2))])/20 + (Sqrt
[10 + 2*Sqrt[5]]*ArcTan[(Sqrt[10 + 2*Sqrt[5]]*x)/(2*(1 - x^2))])/20 + ArcTanh[x]
/5 + ((1 - Sqrt[5])*ArcTanh[((1 - Sqrt[5])*x)/(2*(1 + x^2))])/20 + ((1 + Sqrt[5]
)*ArcTanh[((1 + Sqrt[5])*x)/(2*(1 + x^2))])/20

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Rubi [A]  time = 0.469714, antiderivative size = 325, normalized size of antiderivative = 1.99, number of steps used = 10, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667 \[ -\frac{1}{40} \left (1-\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )+\frac{1}{40} \left (1-\sqrt{5}\right ) \log \left (x^2+\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )-\frac{1}{40} \left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )+\frac{1}{40} \left (1+\sqrt{5}\right ) \log \left (x^2+\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )-\frac{1}{10} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{-4 x-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-\frac{1}{10} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (-4 x+\sqrt{5}+1\right )\right )+\frac{1}{10} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 x-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )+\frac{1}{10} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (4 x+\sqrt{5}+1\right )\right )+\frac{1}{5} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^10)^(-1),x]

[Out]

-(Sqrt[(5 + Sqrt[5])/2]*ArcTan[(1 - Sqrt[5] - 4*x)/Sqrt[2*(5 + Sqrt[5])]])/10 -
(Sqrt[(5 - Sqrt[5])/2]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(1 + Sqrt[5] - 4*x))/2])/1
0 + (Sqrt[(5 + Sqrt[5])/2]*ArcTan[(1 - Sqrt[5] + 4*x)/Sqrt[2*(5 + Sqrt[5])]])/10
+ (Sqrt[(5 - Sqrt[5])/2]*ArcTan[(Sqrt[(5 + Sqrt[5])/10]*(1 + Sqrt[5] + 4*x))/2]
)/10 + ArcTanh[x]/5 - ((1 - Sqrt[5])*Log[1 - ((1 - Sqrt[5])*x)/2 + x^2])/40 + ((
1 - Sqrt[5])*Log[1 + ((1 - Sqrt[5])*x)/2 + x^2])/40 - ((1 + Sqrt[5])*Log[1 - ((1
+ Sqrt[5])*x)/2 + x^2])/40 + ((1 + Sqrt[5])*Log[1 + ((1 + Sqrt[5])*x)/2 + x^2])
/40

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Rubi in Sympy [A]  time = 38.7375, size = 291, normalized size = 1.79 \[ - \left (- \frac{\sqrt{5}}{40} + \frac{1}{40}\right ) \log{\left (x^{2} + x \left (- \frac{1}{2} + \frac{\sqrt{5}}{2}\right ) + 1 \right )} + \left (\frac{1}{40} + \frac{\sqrt{5}}{40}\right ) \log{\left (x^{2} + x \left (\frac{1}{2} + \frac{\sqrt{5}}{2}\right ) + 1 \right )} - \left (\frac{1}{40} + \frac{\sqrt{5}}{40}\right ) \log{\left (x^{2} + x \left (- \frac{\sqrt{5}}{2} - \frac{1}{2}\right ) + 1 \right )} + \left (- \frac{\sqrt{5}}{40} + \frac{1}{40}\right ) \log{\left (x^{2} + x \left (- \frac{\sqrt{5}}{2} + \frac{1}{2}\right ) + 1 \right )} + \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \operatorname{atan}{\left (\frac{x + \frac{1}{4} + \frac{\sqrt{5}}{4}}{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}} \right )}}{5} + \frac{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}} \operatorname{atan}{\left (\frac{x - \frac{\sqrt{5}}{4} - \frac{1}{4}}{\sqrt{- \frac{\sqrt{5}}{8} + \frac{5}{8}}} \right )}}{5} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \operatorname{atan}{\left (\frac{x - \frac{1}{4} + \frac{\sqrt{5}}{4}}{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}} \right )}}{5} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \operatorname{atan}{\left (\frac{x - \frac{\sqrt{5}}{4} + \frac{1}{4}}{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}} \right )}}{5} + \frac{\operatorname{atanh}{\left (x \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**10+1),x)

[Out]

-(-sqrt(5)/40 + 1/40)*log(x**2 + x*(-1/2 + sqrt(5)/2) + 1) + (1/40 + sqrt(5)/40)
*log(x**2 + x*(1/2 + sqrt(5)/2) + 1) - (1/40 + sqrt(5)/40)*log(x**2 + x*(-sqrt(5
)/2 - 1/2) + 1) + (-sqrt(5)/40 + 1/40)*log(x**2 + x*(-sqrt(5)/2 + 1/2) + 1) + sq
rt(-sqrt(5)/8 + 5/8)*atan((x + 1/4 + sqrt(5)/4)/sqrt(-sqrt(5)/8 + 5/8))/5 + sqrt
(-sqrt(5)/8 + 5/8)*atan((x - sqrt(5)/4 - 1/4)/sqrt(-sqrt(5)/8 + 5/8))/5 + sqrt(s
qrt(5)/8 + 5/8)*atan((x - 1/4 + sqrt(5)/4)/sqrt(sqrt(5)/8 + 5/8))/5 + sqrt(sqrt(
5)/8 + 5/8)*atan((x - sqrt(5)/4 + 1/4)/sqrt(sqrt(5)/8 + 5/8))/5 + atanh(x)/5

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Mathematica [A]  time = 0.68479, size = 289, normalized size = 1.77 \[ \frac{1}{40} \left (-\left (\sqrt{5}-1\right ) \log \left (x^2-\frac{1}{2} \left (\sqrt{5}-1\right ) x+1\right )+\left (\sqrt{5}-1\right ) \log \left (x^2+\frac{1}{2} \left (\sqrt{5}-1\right ) x+1\right )-\left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )+\left (1+\sqrt{5}\right ) \log \left (\frac{1}{2} \left (2 x^2+\sqrt{5} x+x+2\right )\right )-4 \log (1-x)+4 \log (x+1)-2 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{-4 x+\sqrt{5}+1}{\sqrt{10-2 \sqrt{5}}}\right )+2 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 x-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )+2 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 x+\sqrt{5}-1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )+2 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{4 x+\sqrt{5}+1}{\sqrt{10-2 \sqrt{5}}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^10)^(-1),x]

[Out]

(-2*Sqrt[10 - 2*Sqrt[5]]*ArcTan[(1 + Sqrt[5] - 4*x)/Sqrt[10 - 2*Sqrt[5]]] + 2*Sq
rt[2*(5 + Sqrt[5])]*ArcTan[(1 - Sqrt[5] + 4*x)/Sqrt[2*(5 + Sqrt[5])]] + 2*Sqrt[2
*(5 + Sqrt[5])]*ArcTan[(-1 + Sqrt[5] + 4*x)/Sqrt[2*(5 + Sqrt[5])]] + 2*Sqrt[10 -
2*Sqrt[5]]*ArcTan[(1 + Sqrt[5] + 4*x)/Sqrt[10 - 2*Sqrt[5]]] - 4*Log[1 - x] + 4*
Log[1 + x] - (-1 + Sqrt[5])*Log[1 - ((-1 + Sqrt[5])*x)/2 + x^2] + (-1 + Sqrt[5])
*Log[1 + ((-1 + Sqrt[5])*x)/2 + x^2] - (1 + Sqrt[5])*Log[1 - ((1 + Sqrt[5])*x)/2
+ x^2] + (1 + Sqrt[5])*Log[(2 + x + Sqrt[5]*x + 2*x^2)/2])/40

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Maple [B]  time = 0.04, size = 426, normalized size = 2.6 \[ -{\frac{\ln \left ( -1+x \right ) }{10}}+{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}+x+2 \right ) \sqrt{5}}{40}}+{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}+x+2 \right ) }{40}}+{\frac{1}{2\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{1+4\,x+\sqrt{5}}{\sqrt{10-2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{10\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{1+4\,x+\sqrt{5}}{\sqrt{10-2\,\sqrt{5}}}} \right ) }-{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}+x+2 \right ) \sqrt{5}}{40}}+{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}+x+2 \right ) }{40}}+{\frac{1}{2\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{1+4\,x-\sqrt{5}}{\sqrt{10+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{10\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{1+4\,x-\sqrt{5}}{\sqrt{10+2\,\sqrt{5}}}} \right ) }+{\frac{\ln \left ( 1+x \right ) }{10}}+{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) \sqrt{5}}{40}}-{\frac{\ln \left ( x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{40}}+{\frac{1}{2\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{10\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{\sqrt{5}+4\,x-1}{\sqrt{10+2\,\sqrt{5}}}} \right ) }-{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) \sqrt{5}}{40}}-{\frac{\ln \left ( -x\sqrt{5}+2\,{x}^{2}-x+2 \right ) }{40}}+{\frac{1}{2\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{10\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{-\sqrt{5}+4\,x-1}{\sqrt{10-2\,\sqrt{5}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^10+1),x)

[Out]

-1/10*ln(-1+x)+1/40*ln(x*5^(1/2)+2*x^2+x+2)*5^(1/2)+1/40*ln(x*5^(1/2)+2*x^2+x+2)
+1/2/(10-2*5^(1/2))^(1/2)*arctan((1+4*x+5^(1/2))/(10-2*5^(1/2))^(1/2))-1/10/(10-
2*5^(1/2))^(1/2)*arctan((1+4*x+5^(1/2))/(10-2*5^(1/2))^(1/2))*5^(1/2)-1/40*ln(-x
*5^(1/2)+2*x^2+x+2)*5^(1/2)+1/40*ln(-x*5^(1/2)+2*x^2+x+2)+1/2/(10+2*5^(1/2))^(1/
2)*arctan((1+4*x-5^(1/2))/(10+2*5^(1/2))^(1/2))+1/10/(10+2*5^(1/2))^(1/2)*arctan
((1+4*x-5^(1/2))/(10+2*5^(1/2))^(1/2))*5^(1/2)+1/10*ln(1+x)+1/40*ln(x*5^(1/2)+2*
x^2-x+2)*5^(1/2)-1/40*ln(x*5^(1/2)+2*x^2-x+2)+1/2/(10+2*5^(1/2))^(1/2)*arctan((5
^(1/2)+4*x-1)/(10+2*5^(1/2))^(1/2))+1/10/(10+2*5^(1/2))^(1/2)*arctan((5^(1/2)+4*
x-1)/(10+2*5^(1/2))^(1/2))*5^(1/2)-1/40*ln(-x*5^(1/2)+2*x^2-x+2)*5^(1/2)-1/40*ln
(-x*5^(1/2)+2*x^2-x+2)+1/2/(10-2*5^(1/2))^(1/2)*arctan((-5^(1/2)+4*x-1)/(10-2*5^
(1/2))^(1/2))-1/10/(10-2*5^(1/2))^(1/2)*arctan((-5^(1/2)+4*x-1)/(10-2*5^(1/2))^(
1/2))*5^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{10} \, \int \frac{x^{3} + 2 \, x^{2} + 3 \, x + 4}{x^{4} + x^{3} + x^{2} + x + 1}\,{d x} - \frac{1}{10} \, \int \frac{x^{3} - 2 \, x^{2} + 3 \, x - 4}{x^{4} - x^{3} + x^{2} - x + 1}\,{d x} + \frac{1}{10} \, \log \left (x + 1\right ) - \frac{1}{10} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(x^10 - 1),x, algorithm="maxima")

[Out]

1/10*integrate((x^3 + 2*x^2 + 3*x + 4)/(x^4 + x^3 + x^2 + x + 1), x) - 1/10*inte
grate((x^3 - 2*x^2 + 3*x - 4)/(x^4 - x^3 + x^2 - x + 1), x) + 1/10*log(x + 1) -
1/10*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^10 - 1),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 12.5254, size = 70, normalized size = 0.43 \[ - \frac{\log{\left (x - 1 \right )}}{10} + \frac{\log{\left (x + 1 \right )}}{10} - \operatorname{RootSum}{\left (10000 t^{4} - 1000 t^{3} + 100 t^{2} - 10 t + 1, \left ( t \mapsto t \log{\left (- 10 t + x \right )} \right )\right )} - \operatorname{RootSum}{\left (10000 t^{4} + 1000 t^{3} + 100 t^{2} + 10 t + 1, \left ( t \mapsto t \log{\left (- 10 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**10+1),x)

[Out]

-log(x - 1)/10 + log(x + 1)/10 - RootSum(10000*_t**4 - 1000*_t**3 + 100*_t**2 -
10*_t + 1, Lambda(_t, _t*log(-10*_t + x))) - RootSum(10000*_t**4 + 1000*_t**3 +
100*_t**2 + 10*_t + 1, Lambda(_t, _t*log(-10*_t + x)))

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GIAC/XCAS [A]  time = 0.230918, size = 301, normalized size = 1.85 \[ \frac{1}{40} \,{\left (\sqrt{5} + 1\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) - \frac{1}{40} \,{\left (\sqrt{5} + 1\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) + \frac{1}{40} \,{\left (\sqrt{5} - 1\right )}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} - 1\right )} + 1\right ) - \frac{1}{40} \,{\left (\sqrt{5} - 1\right )}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} - 1\right )} + 1\right ) + \frac{1}{20} \, \sqrt{2 \, \sqrt{5} + 10} \arctan \left (\frac{4 \, x + \sqrt{5} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) + \frac{1}{20} \, \sqrt{2 \, \sqrt{5} + 10} \arctan \left (\frac{4 \, x - \sqrt{5} + 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) + \frac{1}{20} \, \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{4 \, x + \sqrt{5} + 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) + \frac{1}{20} \, \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{4 \, x - \sqrt{5} - 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) + \frac{1}{10} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{10} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(x^10 - 1),x, algorithm="giac")

[Out]

1/40*(sqrt(5) + 1)*ln(x^2 + 1/2*x*(sqrt(5) + 1) + 1) - 1/40*(sqrt(5) + 1)*ln(x^2
- 1/2*x*(sqrt(5) + 1) + 1) + 1/40*(sqrt(5) - 1)*ln(x^2 + 1/2*x*(sqrt(5) - 1) +
1) - 1/40*(sqrt(5) - 1)*ln(x^2 - 1/2*x*(sqrt(5) - 1) + 1) + 1/20*sqrt(2*sqrt(5)
+ 10)*arctan((4*x + sqrt(5) - 1)/sqrt(2*sqrt(5) + 10)) + 1/20*sqrt(2*sqrt(5) + 1
0)*arctan((4*x - sqrt(5) + 1)/sqrt(2*sqrt(5) + 10)) + 1/20*sqrt(-2*sqrt(5) + 10)
*arctan((4*x + sqrt(5) + 1)/sqrt(-2*sqrt(5) + 10)) + 1/20*sqrt(-2*sqrt(5) + 10)*
arctan((4*x - sqrt(5) - 1)/sqrt(-2*sqrt(5) + 10)) + 1/10*ln(abs(x + 1)) - 1/10*l
n(abs(x - 1))