3.1540 \(\int \frac{1}{1-x^{10}} \, dx\)

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 = 0.209261, 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, Rules used = {210, 634, 618, 204, 628, 206} \[ -\frac{1}{40} \left (1-\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )+\frac{1}{40} \left (1-\sqrt{5}\right ) \log \left (x^2+\frac{1}{2} \left (1-\sqrt{5}\right ) x+1\right )-\frac{1}{40} \left (1+\sqrt{5}\right ) \log \left (x^2-\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )+\frac{1}{40} \left (1+\sqrt{5}\right ) \log \left (x^2+\frac{1}{2} \left (1+\sqrt{5}\right ) x+1\right )-\frac{1}{10} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{-4 x-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-\frac{1}{10} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (-4 x+\sqrt{5}+1\right )\right )+\frac{1}{10} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 x-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )+\frac{1}{10} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (4 x+\sqrt{5}+1\right )\right )+\frac{1}{5} \tanh ^{-1}(x) \]

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])/10 + (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 + Arc
Tanh[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

Rule 210

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*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r +
s*Cos[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 - s^2*x^2), x])/(a*n) +
Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[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 206

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

Rubi steps

\begin{align*} \int \frac{1}{1-x^{10}} \, dx &=\frac{1}{5} \int \frac{1-\frac{1}{4} \left (-1+\sqrt{5}\right ) x}{1-\frac{1}{2} \left (-1+\sqrt{5}\right ) x+x^2} \, dx+\frac{1}{5} \int \frac{1+\frac{1}{4} \left (-1+\sqrt{5}\right ) x}{1+\frac{1}{2} \left (-1+\sqrt{5}\right ) x+x^2} \, dx+\frac{1}{5} \int \frac{1-\frac{1}{4} \left (1+\sqrt{5}\right ) x}{1-\frac{1}{2} \left (1+\sqrt{5}\right ) x+x^2} \, dx+\frac{1}{5} \int \frac{1+\frac{1}{4} \left (1+\sqrt{5}\right ) x}{1+\frac{1}{2} \left (1+\sqrt{5}\right ) x+x^2} \, dx+\frac{1}{5} \int \frac{1}{1-x^2} \, dx\\ &=\frac{1}{5} \tanh ^{-1}(x)+\frac{1}{40} \left (-1-\sqrt{5}\right ) \int \frac{\frac{1}{2} \left (-1-\sqrt{5}\right )+2 x}{1+\frac{1}{2} \left (-1-\sqrt{5}\right ) x+x^2} \, dx+\frac{1}{40} \left (1-\sqrt{5}\right ) \int \frac{\frac{1}{2} \left (1-\sqrt{5}\right )+2 x}{1+\frac{1}{2} \left (1-\sqrt{5}\right ) x+x^2} \, dx+\frac{1}{40} \left (5-\sqrt{5}\right ) \int \frac{1}{1+\frac{1}{2} \left (-1-\sqrt{5}\right ) x+x^2} \, dx+\frac{1}{40} \left (5-\sqrt{5}\right ) \int \frac{1}{1+\frac{1}{2} \left (1+\sqrt{5}\right ) x+x^2} \, dx+\frac{1}{40} \left (-1+\sqrt{5}\right ) \int \frac{\frac{1}{2} \left (-1+\sqrt{5}\right )+2 x}{1+\frac{1}{2} \left (-1+\sqrt{5}\right ) x+x^2} \, dx+\frac{1}{40} \left (1+\sqrt{5}\right ) \int \frac{\frac{1}{2} \left (1+\sqrt{5}\right )+2 x}{1+\frac{1}{2} \left (1+\sqrt{5}\right ) x+x^2} \, dx+\frac{1}{40} \left (5+\sqrt{5}\right ) \int \frac{1}{1+\frac{1}{2} \left (1-\sqrt{5}\right ) x+x^2} \, dx+\frac{1}{40} \left (5+\sqrt{5}\right ) \int \frac{1}{1+\frac{1}{2} \left (-1+\sqrt{5}\right ) x+x^2} \, dx\\ &=\frac{1}{5} \tanh ^{-1}(x)-\frac{1}{40} \left (1+\sqrt{5}\right ) \log \left (2-x-\sqrt{5} x+2 x^2\right )+\frac{1}{40} \left (1-\sqrt{5}\right ) \log \left (2+x-\sqrt{5} x+2 x^2\right )-\frac{1}{40} \left (1-\sqrt{5}\right ) \log \left (2-x+\sqrt{5} x+2 x^2\right )+\frac{1}{40} \left (1+\sqrt{5}\right ) \log \left (2+x+\sqrt{5} x+2 x^2\right )+\frac{1}{20} \left (-5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \left (-5+\sqrt{5}\right )-x^2} \, dx,x,\frac{1}{2} \left (-1-\sqrt{5}\right )+2 x\right )+\frac{1}{20} \left (-5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \left (-5+\sqrt{5}\right )-x^2} \, dx,x,\frac{1}{2} \left (1+\sqrt{5}\right )+2 x\right )-\frac{1}{20} \left (5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \left (-5-\sqrt{5}\right )-x^2} \, dx,x,\frac{1}{2} \left (1-\sqrt{5}\right )+2 x\right )-\frac{1}{20} \left (5+\sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \left (-5-\sqrt{5}\right )-x^2} \, dx,x,\frac{1}{2} \left (-1+\sqrt{5}\right )+2 x\right )\\ &=-\frac{1}{10} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{1-\sqrt{5}-4 x}{\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 (1+\sqrt{5}-4 x\right )\right )+\frac{1}{10} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{1-\sqrt{5}+4 x}{\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 (1+\sqrt{5}+4 x\right )\right )+\frac{1}{5} \tanh ^{-1}(x)-\frac{1}{40} \left (1+\sqrt{5}\right ) \log \left (2-x-\sqrt{5} x+2 x^2\right )+\frac{1}{40} \left (1-\sqrt{5}\right ) \log \left (2+x-\sqrt{5} x+2 x^2\right )-\frac{1}{40} \left (1-\sqrt{5}\right ) \log \left (2-x+\sqrt{5} x+2 x^2\right )+\frac{1}{40} \left (1+\sqrt{5}\right ) \log \left (2+x+\sqrt{5} x+2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.222981, 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*Sqrt[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 = 0.021, size = 426, normalized size = 2.6 \begin{align*} -{\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 ) } \end{align*}

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)*ar
ctan((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. \begin{align*} \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 ) \end{align*}

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*integrate((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 [B]  time = 9.7963, size = 3841, normalized size = 23.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^10+1),x, algorithm="fricas")

[Out]

1/40*(sqrt(5) + 2*sqrt(-3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 1/8*(2*sqrt(1/2)*sqrt(sqrt(5) -
5) + sqrt(5) - 3)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(
5) - 1)^2 + sqrt(1/2)*sqrt(sqrt(5) - 5) + 1/2*sqrt(5) - 5/2) - 1)*log(2*x + 1/2*sqrt(5) + sqrt(-3/16*(2*sqrt(1
/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 1/8*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(2*sqrt(1/2)*sqrt(s
qrt(5) - 5) - sqrt(5) - 1) - 3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + sqrt(1/2)*sqrt(sqrt(5) - 5
) + 1/2*sqrt(5) - 5/2) - 1/2) + 1/40*(sqrt(5) - 2*sqrt(-3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 +
1/8*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) - 3)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 3/16*(2*sqr
t(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)^2 + sqrt(1/2)*sqrt(sqrt(5) - 5) + 1/2*sqrt(5) - 5/2) - 1)*log(2*x + 1/
2*sqrt(5) - sqrt(-3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)^2 + 1/8*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) +
sqrt(5) - 3)*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1) - 3/16*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1
)^2 + sqrt(1/2)*sqrt(sqrt(5) - 5) + 1/2*sqrt(5) - 5/2) - 1/2) + 1/40*(sqrt(5) + 2*sqrt(-3/16*(sqrt(5) + 20*sqr
t(-1/200*sqrt(5) - 1/40) - 1)^2 - 1/8*(sqrt(5) + 20*sqrt(-1/200*sqrt(5) - 1/40) + 3)*(sqrt(5) - 20*sqrt(-1/200
*sqrt(5) - 1/40) - 1) - 3/16*(sqrt(5) - 20*sqrt(-1/200*sqrt(5) - 1/40) - 1)^2 - 1/2*sqrt(5) - 10*sqrt(-1/200*s
qrt(5) - 1/40) - 5/2) + 1)*log(2*x + 1/2*sqrt(5) + sqrt(-3/16*(sqrt(5) + 20*sqrt(-1/200*sqrt(5) - 1/40) - 1)^2
- 1/8*(sqrt(5) + 20*sqrt(-1/200*sqrt(5) - 1/40) + 3)*(sqrt(5) - 20*sqrt(-1/200*sqrt(5) - 1/40) - 1) - 3/16*(s
qrt(5) - 20*sqrt(-1/200*sqrt(5) - 1/40) - 1)^2 - 1/2*sqrt(5) - 10*sqrt(-1/200*sqrt(5) - 1/40) - 5/2) + 1/2) +
1/40*(sqrt(5) - 2*sqrt(-3/16*(sqrt(5) + 20*sqrt(-1/200*sqrt(5) - 1/40) - 1)^2 - 1/8*(sqrt(5) + 20*sqrt(-1/200*
sqrt(5) - 1/40) + 3)*(sqrt(5) - 20*sqrt(-1/200*sqrt(5) - 1/40) - 1) - 3/16*(sqrt(5) - 20*sqrt(-1/200*sqrt(5) -
1/40) - 1)^2 - 1/2*sqrt(5) - 10*sqrt(-1/200*sqrt(5) - 1/40) - 5/2) + 1)*log(2*x + 1/2*sqrt(5) - sqrt(-3/16*(s
qrt(5) + 20*sqrt(-1/200*sqrt(5) - 1/40) - 1)^2 - 1/8*(sqrt(5) + 20*sqrt(-1/200*sqrt(5) - 1/40) + 3)*(sqrt(5) -
20*sqrt(-1/200*sqrt(5) - 1/40) - 1) - 3/16*(sqrt(5) - 20*sqrt(-1/200*sqrt(5) - 1/40) - 1)^2 - 1/2*sqrt(5) - 1
0*sqrt(-1/200*sqrt(5) - 1/40) - 5/2) + 1/2) + 1/40*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) - sqrt(5) - 1)*log(x + 1/2*s
qrt(1/2)*sqrt(sqrt(5) - 5) - 1/4*sqrt(5) - 1/4) - 1/40*(2*sqrt(1/2)*sqrt(sqrt(5) - 5) + sqrt(5) + 1)*log(x - 1
/2*sqrt(1/2)*sqrt(sqrt(5) - 5) - 1/4*sqrt(5) - 1/4) - 1/40*(sqrt(5) - 20*sqrt(-1/200*sqrt(5) - 1/40) - 1)*log(
x - 1/4*sqrt(5) + 5*sqrt(-1/200*sqrt(5) - 1/40) + 1/4) - 1/40*(sqrt(5) + 20*sqrt(-1/200*sqrt(5) - 1/40) - 1)*l
og(x - 1/4*sqrt(5) - 5*sqrt(-1/200*sqrt(5) - 1/40) + 1/4) + 1/10*log(x + 1) - 1/10*log(x - 1)

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Sympy [A]  time = 2.9148, size = 70, normalized size = 0.43 \begin{align*} - \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 )} \end{align*}

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 [A]  time = 1.17854, size = 301, normalized size = 1.85 \begin{align*} \frac{1}{40} \,{\left (\sqrt{5} + 1\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) - \frac{1}{40} \,{\left (\sqrt{5} + 1\right )} \log \left (x^{2} - \frac{1}{2} \, x{\left (\sqrt{5} + 1\right )} + 1\right ) + \frac{1}{40} \,{\left (\sqrt{5} - 1\right )} \log \left (x^{2} + \frac{1}{2} \, x{\left (\sqrt{5} - 1\right )} + 1\right ) - \frac{1}{40} \,{\left (\sqrt{5} - 1\right )} \log \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} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{10} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

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)*log(x^2 + 1/2*x*(sqrt(5) + 1) + 1) - 1/40*(sqrt(5) + 1)*log(x^2 - 1/2*x*(sqrt(5) + 1) + 1)
+ 1/40*(sqrt(5) - 1)*log(x^2 + 1/2*x*(sqrt(5) - 1) + 1) - 1/40*(sqrt(5) - 1)*log(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) + 10)*arc
tan((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*log(abs(x
+ 1)) - 1/10*log(abs(x - 1))