3.51 \(\int \frac {1}{1-x^4+x^8} \, dx\)

Optimal. Leaf size=275 \[ -\frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{4 \sqrt {6}}+\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{4 \sqrt {6}}-\frac {\log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{4 \sqrt {6}}+\frac {\log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{4 \sqrt {6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}} \]

[Out]

-1/12*arctan((-2*x+1/2*6^(1/2)-1/2*2^(1/2))/(1/2*6^(1/2)+1/2*2^(1/2)))*6^(1/2)+1/12*arctan((2*x+1/2*6^(1/2)-1/
2*2^(1/2))/(1/2*6^(1/2)+1/2*2^(1/2)))*6^(1/2)-1/12*arctan((-2*x+1/2*6^(1/2)+1/2*2^(1/2))/(1/2*6^(1/2)-1/2*2^(1
/2)))*6^(1/2)+1/12*arctan((2*x+1/2*6^(1/2)+1/2*2^(1/2))/(1/2*6^(1/2)-1/2*2^(1/2)))*6^(1/2)-1/24*ln(1+x^2-x*(1/
2*6^(1/2)-1/2*2^(1/2)))*6^(1/2)+1/24*ln(1+x^2+x*(1/2*6^(1/2)-1/2*2^(1/2)))*6^(1/2)-1/24*ln(1+x^2-x*(1/2*6^(1/2
)+1/2*2^(1/2)))*6^(1/2)+1/24*ln(1+x^2+x*(1/2*6^(1/2)+1/2*2^(1/2)))*6^(1/2)

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Rubi [A]  time = 0.26, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1346, 1169, 634, 618, 204, 628} \[ -\frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{4 \sqrt {6}}+\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{4 \sqrt {6}}-\frac {\log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{4 \sqrt {6}}+\frac {\log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{4 \sqrt {6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^4 + x^8)^(-1),x]

[Out]

-ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]]/(2*Sqrt[6]) - ArcTan[(Sqrt[2 + Sqrt[3]] - 2*x)/Sqrt[2 - S
qrt[3]]]/(2*Sqrt[6]) + ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]]/(2*Sqrt[6]) + ArcTan[(Sqrt[2 + Sqrt
[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/(2*Sqrt[6]) - Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2]/(4*Sqrt[6]) + Log[1 + Sqrt[2 -
Sqrt[3]]*x + x^2]/(4*Sqrt[6]) - Log[1 - Sqrt[2 + Sqrt[3]]*x + x^2]/(4*Sqrt[6]) + Log[1 + Sqrt[2 + Sqrt[3]]*x
+ x^2]/(4*Sqrt[6])

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 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 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 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 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 1346

Int[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b
/c, 2]}, Dist[1/(2*c*q*r), Int[(r - x^(n/2))/(q - r*x^(n/2) + x^n), x], x] + Dist[1/(2*c*q*r), Int[(r + x^(n/2
))/(q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n/2,
0] && NegQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{1-x^4+x^8} \, dx &=\frac {\int \frac {\sqrt {3}-x^2}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}+\frac {\int \frac {\sqrt {3}+x^2}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}\\ &=\frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )}-\left (-1+\sqrt {3}\right ) x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )}+\left (-1+\sqrt {3}\right ) x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\int \frac {\sqrt {3 \left (2+\sqrt {3}\right )}-\left (1+\sqrt {3}\right ) x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\int \frac {\sqrt {3 \left (2+\sqrt {3}\right )}+\left (1+\sqrt {3}\right ) x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}\\ &=-\frac {\int \frac {-\sqrt {2-\sqrt {3}}+2 x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6}}+\frac {\int \frac {\sqrt {2-\sqrt {3}}+2 x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6}}-\frac {\int \frac {-\sqrt {2+\sqrt {3}}+2 x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6}}+\frac {\int \frac {\sqrt {2+\sqrt {3}}+2 x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6}}+\frac {\int \frac {1}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6 \left (2-\sqrt {3}\right )}}+\frac {\int \frac {1}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6 \left (2-\sqrt {3}\right )}}+\frac {\int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6 \left (2+\sqrt {3}\right )}}+\frac {\int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6 \left (2+\sqrt {3}\right )}}\\ &=-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,-\sqrt {2-\sqrt {3}}+2 x\right )}{2 \sqrt {6 \left (2-\sqrt {3}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,\sqrt {2-\sqrt {3}}+2 x\right )}{2 \sqrt {6 \left (2-\sqrt {3}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )}{2 \sqrt {6 \left (2+\sqrt {3}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )}{2 \sqrt {6 \left (2+\sqrt {3}\right )}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 42, normalized size = 0.15 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {\log (x-\text {$\#$1})}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^4 + x^8)^(-1),x]

[Out]

RootSum[1 - #1^4 + #1^8 & , Log[x - #1]/(-#1^3 + 2*#1^7) & ]/4

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fricas [A]  time = 0.45, size = 215, normalized size = 0.78 \[ -\frac {1}{6} \, \sqrt {3} \sqrt {2} \arctan \left (-\frac {\sqrt {3} \sqrt {2} {\left (x^{3} - x\right )} + x^{2} - \sqrt {x^{4} + \sqrt {3} \sqrt {2} {\left (x^{3} + x\right )} + 3 \, x^{2} + 1} {\left (\sqrt {3} \sqrt {2} x - 2\right )}}{3 \, x^{2} - 2}\right ) - \frac {1}{6} \, \sqrt {3} \sqrt {2} \arctan \left (-\frac {\sqrt {3} \sqrt {2} {\left (x^{3} - x\right )} - x^{2} - \sqrt {x^{4} - \sqrt {3} \sqrt {2} {\left (x^{3} + x\right )} + 3 \, x^{2} + 1} {\left (\sqrt {3} \sqrt {2} x + 2\right )}}{3 \, x^{2} - 2}\right ) + \frac {1}{24} \, \sqrt {3} \sqrt {2} \log \left (x^{4} + \sqrt {3} \sqrt {2} {\left (x^{3} + x\right )} + 3 \, x^{2} + 1\right ) - \frac {1}{24} \, \sqrt {3} \sqrt {2} \log \left (x^{4} - \sqrt {3} \sqrt {2} {\left (x^{3} + x\right )} + 3 \, x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^8-x^4+1),x, algorithm="fricas")

[Out]

-1/6*sqrt(3)*sqrt(2)*arctan(-(sqrt(3)*sqrt(2)*(x^3 - x) + x^2 - sqrt(x^4 + sqrt(3)*sqrt(2)*(x^3 + x) + 3*x^2 +
1)*(sqrt(3)*sqrt(2)*x - 2))/(3*x^2 - 2)) - 1/6*sqrt(3)*sqrt(2)*arctan(-(sqrt(3)*sqrt(2)*(x^3 - x) - x^2 - sqr
t(x^4 - sqrt(3)*sqrt(2)*(x^3 + x) + 3*x^2 + 1)*(sqrt(3)*sqrt(2)*x + 2))/(3*x^2 - 2)) + 1/24*sqrt(3)*sqrt(2)*lo
g(x^4 + sqrt(3)*sqrt(2)*(x^3 + x) + 3*x^2 + 1) - 1/24*sqrt(3)*sqrt(2)*log(x^4 - sqrt(3)*sqrt(2)*(x^3 + x) + 3*
x^2 + 1)

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giac [A]  time = 1.06, size = 205, normalized size = 0.75 \[ \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{24} \, \sqrt {6} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{24} \, \sqrt {6} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{24} \, \sqrt {6} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{24} \, \sqrt {6} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^8-x^4+1),x, algorithm="giac")

[Out]

1/12*sqrt(6)*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2))) + 1/12*sqrt(6)*arctan((4*x - sqrt(6) + sqrt
(2))/(sqrt(6) + sqrt(2))) + 1/12*sqrt(6)*arctan((4*x + sqrt(6) + sqrt(2))/(sqrt(6) - sqrt(2))) + 1/12*sqrt(6)*
arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2))) + 1/24*sqrt(6)*log(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1)
- 1/24*sqrt(6)*log(x^2 - 1/2*x*(sqrt(6) + sqrt(2)) + 1) + 1/24*sqrt(6)*log(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1
) - 1/24*sqrt(6)*log(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1)

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maple [C]  time = 0.01, size = 30, normalized size = 0.11 \[ \frac {\RootOf \left (9 \textit {\_Z}^{4}+1\right ) \ln \left (3 \RootOf \left (9 \textit {\_Z}^{4}+1\right )^{2}+3 \RootOf \left (9 \textit {\_Z}^{4}+1\right ) x +x^{2}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8-x^4+1),x)

[Out]

1/4*sum(_R*ln(3*_R^2+3*_R*x+x^2),_R=RootOf(9*_Z^4+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{8} - x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^8-x^4+1),x, algorithm="maxima")

[Out]

integrate(1/(x^8 - x^4 + 1), x)

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mupad [B]  time = 0.10, size = 53, normalized size = 0.19 \[ \sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (\frac {1}{3}+\frac {1}{3}{}\mathrm {i}\right )}{\frac {2\,x^2}{3}-\frac {2}{3}{}\mathrm {i}}\right )\,\left (-\frac {1}{12}-\frac {1}{12}{}\mathrm {i}\right )+\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (\frac {1}{3}-\frac {1}{3}{}\mathrm {i}\right )}{\frac {2\,x^2}{3}+\frac {2}{3}{}\mathrm {i}}\right )\,\left (-\frac {1}{12}+\frac {1}{12}{}\mathrm {i}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8 - x^4 + 1),x)

[Out]

- 6^(1/2)*atan((6^(1/2)*x*(1/3 + 1i/3))/((2*x^2)/3 - 2i/3))*(1/12 + 1i/12) - 6^(1/2)*atan((6^(1/2)*x*(1/3 - 1i
/3))/((2*x^2)/3 + 2i/3))*(1/12 - 1i/12)

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sympy [A]  time = 0.22, size = 165, normalized size = 0.60 \[ \frac {\sqrt {6} \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{3} - \frac {1}{3} \right )} + 2 \operatorname {atan}{\left (\sqrt {6} x^{3} - 4 x^{2} + 2 \sqrt {6} x - 3 \right )}\right )}{24} + \frac {\sqrt {6} \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{3} + \frac {1}{3} \right )} + 2 \operatorname {atan}{\left (\sqrt {6} x^{3} + 4 x^{2} + 2 \sqrt {6} x + 3 \right )}\right )}{24} - \frac {\sqrt {6} \log {\left (x^{4} - \sqrt {6} x^{3} + 3 x^{2} - \sqrt {6} x + 1 \right )}}{24} + \frac {\sqrt {6} \log {\left (x^{4} + \sqrt {6} x^{3} + 3 x^{2} + \sqrt {6} x + 1 \right )}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**8-x**4+1),x)

[Out]

sqrt(6)*(2*atan(sqrt(6)*x/3 - 1/3) + 2*atan(sqrt(6)*x**3 - 4*x**2 + 2*sqrt(6)*x - 3))/24 + sqrt(6)*(2*atan(sqr
t(6)*x/3 + 1/3) + 2*atan(sqrt(6)*x**3 + 4*x**2 + 2*sqrt(6)*x + 3))/24 - sqrt(6)*log(x**4 - sqrt(6)*x**3 + 3*x*
*2 - sqrt(6)*x + 1)/24 + sqrt(6)*log(x**4 + sqrt(6)*x**3 + 3*x**2 + sqrt(6)*x + 1)/24

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