3.222 \(\int \frac{1-x}{1-x^5} \, dx\)

Optimal. Leaf size=109 \[ -\frac{1}{5} (-1)^{2/5} \left (1-(-1)^{2/5}\right ) \log \left ((-1)^{2/5}-x\right )+\frac{1}{5} (-1)^{3/5} \left (1+(-1)^{3/5}\right ) \log \left (-x-(-1)^{3/5}\right )+\frac{1}{5} \sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \log \left (x+\sqrt [5]{-1}\right )-\frac{1}{5} (-1)^{4/5} \left (1-(-1)^{4/5}\right ) \log \left (x-(-1)^{4/5}\right ) \]

[Out]

-((-1)^(2/5)*(1 - (-1)^(2/5))*Log[(-1)^(2/5) - x])/5 + ((-1)^(3/5)*(1 + (-1)^(3/
5))*Log[-(-1)^(3/5) - x])/5 + ((-1)^(1/5)*(1 + (-1)^(1/5))*Log[(-1)^(1/5) + x])/
5 - ((-1)^(4/5)*(1 - (-1)^(4/5))*Log[-(-1)^(4/5) + x])/5

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Rubi [F]  time = 0.0528554, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{1-x}{1-x^5},x\right ) \]

Verification is Not applicable to the result.

[In]  Int[(1 - x)/(1 - x^5),x]

[Out]

-Defer[Int][(-1 - x - x^2 - x^3 - x^4)^(-1), x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} + x^{3} + x^{2} + x + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-x)/(-x**5+1),x)

[Out]

Integral(1/(x**4 + x**3 + x**2 + x + 1), x)

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Mathematica [C]  time = 0.0174208, size = 47, normalized size = 0.43 \[ \text{RootSum}\left [\text{$\#$1}^4+\text{$\#$1}^3+\text{$\#$1}^2+\text{$\#$1}+1\&,\frac{\log (x-\text{$\#$1})}{4 \text{$\#$1}^3+3 \text{$\#$1}^2+2 \text{$\#$1}+1}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x)/(1 - x^5),x]

[Out]

RootSum[1 + #1 + #1^2 + #1^3 + #1^4 & , Log[x - #1]/(1 + 2*#1 + 3*#1^2 + 4*#1^3)
& ]

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Maple [B]  time = 0.024, size = 169, normalized size = 1.6 \[ -{\frac{\sqrt{5}\ln \left ( -x\sqrt{5}+2\,{x}^{2}+x+2 \right ) }{10}}+{\frac{1}{\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{1+4\,x-\sqrt{5}}{\sqrt{10+2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{1+4\,x-\sqrt{5}}{\sqrt{10+2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}\ln \left ( x\sqrt{5}+2\,{x}^{2}+x+2 \right ) }{10}}+{\frac{1}{\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{1+4\,x+\sqrt{5}}{\sqrt{10-2\,\sqrt{5}}}} \right ) }+{\frac{\sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{1+4\,x+\sqrt{5}}{\sqrt{10-2\,\sqrt{5}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-x)/(-x^5+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x - 1}{x^{5} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 1)/(x^5 - 1),x, algorithm="maxima")

[Out]

integrate((x - 1)/(x^5 - 1), x)

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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((x - 1)/(x^5 - 1),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.789137, size = 36, normalized size = 0.33 \[ \operatorname{RootSum}{\left (125 t^{4} + 5 t + 1, \left ( t \mapsto t \log{\left (\frac{375 t^{3}}{11} - \frac{100 t^{2}}{11} + \frac{45 t}{11} + x + \frac{14}{11} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-x)/(-x**5+1),x)

[Out]

RootSum(125*_t**4 + 5*_t + 1, Lambda(_t, _t*log(375*_t**3/11 - 100*_t**2/11 + 45
*_t/11 + x + 14/11)))

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GIAC/XCAS [A]  time = 0.220766, size = 136, normalized size = 1.25 \[ \frac{1}{5} \, \sqrt{-2 \, \sqrt{5} + 5} \arctan \left (\frac{4 \, x - \sqrt{5} + 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) + \frac{1}{5} \, \sqrt{2 \, \sqrt{5} + 5} \arctan \left (\frac{4 \, x + \sqrt{5} + 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) + \frac{1}{10} \, \sqrt{5}{\rm ln}\left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) - \frac{1}{10} \, \sqrt{5}{\rm ln}\left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} - 1\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 1)/(x^5 - 1),x, algorithm="giac")

[Out]

1/5*sqrt(-2*sqrt(5) + 5)*arctan((4*x - sqrt(5) + 1)/sqrt(2*sqrt(5) + 10)) + 1/5*
sqrt(2*sqrt(5) + 5)*arctan((4*x + sqrt(5) + 1)/sqrt(-2*sqrt(5) + 10)) + 1/10*sqr
t(5)*ln(x^2 + 1/2*x*(sqrt(5) + 1) + 1) - 1/10*sqrt(5)*ln(x^2 - 1/2*x*(sqrt(5) -
1) + 1)