3.16.40 \(\int \frac {1}{1-x^{10}} \, dx\) [1540]

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

[Out]

1/5*arctanh(x)+1/20*arctanh(1/2*x*(-5^(1/2)+1)/(x^2+1))*(-5^(1/2)+1)+1/20*arctanh(1/2*x*(5^(1/2)+1)/(x^2+1))*(
5^(1/2)+1)+1/20*arctan(1/2*x*(10-2*5^(1/2))^(1/2)/(-x^2+1))*(10-2*5^(1/2))^(1/2)+1/20*arctan(1/2*x*(10+2*5^(1/
2))^(1/2)/(-x^2+1))*(10+2*5^(1/2))^(1/2)

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Rubi [A]
time = 0.16, 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 = {216, 648, 632, 210, 642, 212} \begin {gather*} -\frac {1}{10} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \text {ArcTan}\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 )} \text {ArcTan}\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 )} \text {ArcTan}\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 )} \text {ArcTan}\left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (4 x+\sqrt {5}+1\right )\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}{40} \left (1+\sqrt {5}\right ) \log \left (x^2+\frac {1}{2} \left (1+\sqrt {5}\right ) x+1\right )+\frac {1}{5} \tanh ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x^10)^(-1),x]

[Out]

-1/10*(Sqrt[(5 + Sqrt[5])/2]*ArcTan[(1 - Sqrt[5] - 4*x)/Sqrt[2*(5 + Sqrt[5])]]) - (Sqrt[(5 - Sqrt[5])/2]*ArcTa
n[(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 + A
rcTanh[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_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 212

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

Rule 216

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*C
os[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n))*Int[1/(r^2 - s^2*x^2), x] + Dis
t[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 632

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 642

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 648

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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.16, size = 289, normalized size = 1.77 \begin {gather*} \frac {1}{40} \left (-2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {1+\sqrt {5}-4 x}{\sqrt {10-2 \sqrt {5}}}\right )+2 \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {1-\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+2 \sqrt {2 \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {-1+\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+2 \sqrt {10-2 \sqrt {5}} \tan ^{-1}\left (\frac {1+\sqrt {5}+4 x}{\sqrt {10-2 \sqrt {5}}}\right )-4 \log (1-x)+4 \log (1+x)-\left (-1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2\right )+\left (-1+\sqrt {5}\right ) \log \left (1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2\right )-\left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right )+\left (1+\sqrt {5}\right ) \log \left (\frac {1}{2} \left (2+x+\sqrt {5} x+2 x^2\right )\right )\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(287\) vs. \(2(121)=242\).
time = 0.20, size = 288, normalized size = 1.77

method	result	size
risch	\(-\frac {\ln \left (x -1\right )}{10}+\frac {\ln \left (x +1\right )}{10}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} +x \right )\right )}{10}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} +x \right )\right )}{10}\)	\(66\)
meijerg	\(-\frac {x \left (\ln \left (1-\left (x^{10}\right )^{\frac {1}{10}}\right )-\ln \left (1+\left (x^{10}\right )^{\frac {1}{10}}\right )+\cos \left (\frac {\pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}+\left (x^{10}\right )^{\frac {1}{5}}\right )-2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}{1-\cos \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}\right )+\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}+\left (x^{10}\right )^{\frac {1}{5}}\right )-2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}{1-\cos \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}\right )-\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}+\left (x^{10}\right )^{\frac {1}{5}}\right )-2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}{1+\cos \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}\right )-\cos \left (\frac {\pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}+\left (x^{10}\right )^{\frac {1}{5}}\right )-2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}{1+\cos \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}\right )\right )}{10 \left (x^{10}\right )^{\frac {1}{10}}}\)	\(254\)
default	\(-\frac {\ln \left (x -1\right )}{10}-\frac {\left (-\sqrt {5}-1\right ) \ln \left (x \sqrt {5}+2 x^{2}+x +2\right )}{40}-\frac {\left (-4-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {1+4 x +\sqrt {5}}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {\left (-\sqrt {5}+1\right ) \ln \left (-x \sqrt {5}+2 x^{2}+x +2\right )}{40}+\frac {\left (4-\frac {\left (-\sqrt {5}+1\right )^{2}}{4}\right ) \arctan \left (\frac {1+4 x -\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {\ln \left (x +1\right )}{10}+\frac {\left (\sqrt {5}-1\right ) \ln \left (x \sqrt {5}+2 x^{2}-x +2\right )}{40}+\frac {\left (4-\frac {\left (\sqrt {5}-1\right )^{2}}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x -1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}-\frac {\left (\sqrt {5}+1\right ) \ln \left (-x \sqrt {5}+2 x^{2}-x +2\right )}{40}-\frac {\left (-4-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x -1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}\)	\(288\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^10+1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 1039 vs. \(2 (113) = 226\).
time = 1.17, size = 1039, normalized size = 6.37 \begin {gather*} \text {Too large to display} \end {gather*}

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 = 3.33, size = 70, normalized size = 0.43 \begin {gather*} - \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 {gather*}

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 = 1.09, size = 223, normalized size = 1.37 \begin {gather*} \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 {gather*}

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))

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Mupad [B]
time = 1.29, size = 852, normalized size = 5.23 \begin {gather*} -\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{5}+\mathrm {atan}\left (\frac {\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}-\frac {\sqrt {5}}{40}+\frac {1}{40}\right )\,\left (10\,x-\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}+\frac {5\,\sqrt {5}}{2}-\frac {5}{2}\right )\,1{}\mathrm {i}+\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}-\frac {\sqrt {5}}{40}+\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}-\frac {5\,\sqrt {5}}{2}+\frac {5}{2}\right )\,1{}\mathrm {i}}{\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}-\frac {\sqrt {5}}{40}+\frac {1}{40}\right )\,\left (10\,x-\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}+\frac {5\,\sqrt {5}}{2}-\frac {5}{2}\right )-\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}-\frac {\sqrt {5}}{40}+\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}-\frac {5\,\sqrt {5}}{2}+\frac {5}{2}\right )}\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}\,1{}\mathrm {i}}{20}-\frac {\sqrt {5}\,1{}\mathrm {i}}{20}+\frac {1}{20}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}+\frac {\sqrt {5}}{40}-\frac {1}{40}\right )\,\left (10\,x-\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}-\frac {5\,\sqrt {5}}{2}+\frac {5}{2}\right )\,1{}\mathrm {i}+\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}+\frac {\sqrt {5}}{40}-\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}+\frac {5\,\sqrt {5}}{2}-\frac {5}{2}\right )\,1{}\mathrm {i}}{\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}+\frac {\sqrt {5}}{40}-\frac {1}{40}\right )\,\left (10\,x-\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}-\frac {5\,\sqrt {5}}{2}+\frac {5}{2}\right )-\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{40}+\frac {\sqrt {5}}{40}-\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {2}\,\sqrt {-\sqrt {5}-5}}{2}+\frac {5\,\sqrt {5}}{2}-\frac {5}{2}\right )}\right )\,\left (\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-5}\,1{}\mathrm {i}}{20}+\frac {\sqrt {5}\,1{}\mathrm {i}}{20}-\frac {1}{20}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\left (\frac {\sqrt {5}}{40}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (\frac {5\,\sqrt {5}}{2}-10\,x+\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}+\frac {5}{2}\right )\,1{}\mathrm {i}-\left (\frac {\sqrt {5}}{40}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {5}}{2}+\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}+\frac {5}{2}\right )\,1{}\mathrm {i}}{\left (\frac {\sqrt {5}}{40}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (\frac {5\,\sqrt {5}}{2}-10\,x+\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}+\frac {5}{2}\right )+\left (\frac {\sqrt {5}}{40}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {5}}{2}+\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}+\frac {5}{2}\right )}\right )\,\left (\frac {\sqrt {5}\,1{}\mathrm {i}}{20}+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}\,1{}\mathrm {i}}{20}+\frac {1}{20}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\left (\frac {\sqrt {5}}{40}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (10\,x-\frac {5\,\sqrt {5}}{2}+\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}-\frac {5}{2}\right )\,1{}\mathrm {i}+\left (\frac {\sqrt {5}}{40}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {5}}{2}-\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}+\frac {5}{2}\right )\,1{}\mathrm {i}}{\left (\frac {\sqrt {5}}{40}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (10\,x-\frac {5\,\sqrt {5}}{2}+\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}-\frac {5}{2}\right )-\left (\frac {\sqrt {5}}{40}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}}{40}+\frac {1}{40}\right )\,\left (10\,x+\frac {5\,\sqrt {5}}{2}-\frac {5\,\sqrt {2}\,\sqrt {\sqrt {5}-5}}{2}+\frac {5}{2}\right )}\right )\,\left (\frac {\sqrt {5}\,1{}\mathrm {i}}{20}-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-5}\,1{}\mathrm {i}}{20}+\frac {1}{20}{}\mathrm {i}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(x^10 - 1),x)

[Out]

atan((((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 - 5^(1/2)/40 + 1/40)*(10*x - (5*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 + (
5*5^(1/2))/2 - 5/2)*1i + ((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 - 5^(1/2)/40 + 1/40)*(10*x + (5*2^(1/2)*(- 5^(1/2
) - 5)^(1/2))/2 - (5*5^(1/2))/2 + 5/2)*1i)/(((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 - 5^(1/2)/40 + 1/40)*(10*x - (
5*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 + (5*5^(1/2))/2 - 5/2) - ((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 - 5^(1/2)/40 +
1/40)*(10*x + (5*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 - (5*5^(1/2))/2 + 5/2)))*((2^(1/2)*(- 5^(1/2) - 5)^(1/2)*1i
)/20 - (5^(1/2)*1i)/20 + 1i/20) - (atan(x*1i)*1i)/5 + atan((((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 + 5^(1/2)/40 -
1/40)*(10*x - (5*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 - (5*5^(1/2))/2 + 5/2)*1i + ((2^(1/2)*(- 5^(1/2) - 5)^(1/2)
)/40 + 5^(1/2)/40 - 1/40)*(10*x + (5*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 + (5*5^(1/2))/2 - 5/2)*1i)/(((2^(1/2)*(-
5^(1/2) - 5)^(1/2))/40 + 5^(1/2)/40 - 1/40)*(10*x - (5*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 - (5*5^(1/2))/2 + 5/2
) - ((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 + 5^(1/2)/40 - 1/40)*(10*x + (5*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 + (5*
5^(1/2))/2 - 5/2)))*((2^(1/2)*(- 5^(1/2) - 5)^(1/2)*1i)/20 + (5^(1/2)*1i)/20 - 1i/20) + atan(((5^(1/2)/40 + (2
^(1/2)*(5^(1/2) - 5)^(1/2))/40 + 1/40)*((5*5^(1/2))/2 - 10*x + (5*2^(1/2)*(5^(1/2) - 5)^(1/2))/2 + 5/2)*1i - (
5^(1/2)/40 + (2^(1/2)*(5^(1/2) - 5)^(1/2))/40 + 1/40)*(10*x + (5*5^(1/2))/2 + (5*2^(1/2)*(5^(1/2) - 5)^(1/2))/
2 + 5/2)*1i)/((5^(1/2)/40 + (2^(1/2)*(5^(1/2) - 5)^(1/2))/40 + 1/40)*((5*5^(1/2))/2 - 10*x + (5*2^(1/2)*(5^(1/
2) - 5)^(1/2))/2 + 5/2) + (5^(1/2)/40 + (2^(1/2)*(5^(1/2) - 5)^(1/2))/40 + 1/40)*(10*x + (5*5^(1/2))/2 + (5*2^
(1/2)*(5^(1/2) - 5)^(1/2))/2 + 5/2)))*((5^(1/2)*1i)/20 + (2^(1/2)*(5^(1/2) - 5)^(1/2)*1i)/20 + 1i/20) + atan((
(5^(1/2)/40 - (2^(1/2)*(5^(1/2) - 5)^(1/2))/40 + 1/40)*(10*x - (5*5^(1/2))/2 + (5*2^(1/2)*(5^(1/2) - 5)^(1/2))
/2 - 5/2)*1i + (5^(1/2)/40 - (2^(1/2)*(5^(1/2) - 5)^(1/2))/40 + 1/40)*(10*x + (5*5^(1/2))/2 - (5*2^(1/2)*(5^(1
/2) - 5)^(1/2))/2 + 5/2)*1i)/((5^(1/2)/40 - (2^(1/2)*(5^(1/2) - 5)^(1/2))/40 + 1/40)*(10*x - (5*5^(1/2))/2 + (
5*2^(1/2)*(5^(1/2) - 5)^(1/2))/2 - 5/2) - (5^(1/2)/40 - (2^(1/2)*(5^(1/2) - 5)^(1/2))/40 + 1/40)*(10*x + (5*5^
(1/2))/2 - (5*2^(1/2)*(5^(1/2) - 5)^(1/2))/2 + 5/2)))*((5^(1/2)*1i)/20 - (2^(1/2)*(5^(1/2) - 5)^(1/2)*1i)/20 +
1i/20)

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