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

Optimal. Leaf size=355 \[ \frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}} \]

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Rubi [A]  time = 0.28, antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1421, 1169, 634, 618, 204, 628} \begin {gather*} \frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\tan ^{-1}\left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x^4)/(1 - x^4 + x^8),x]

[Out]

-ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt[3*(2 - Sqrt[3])]) + ArcTan[(Sqrt[2 + Sqrt[3]] - 2
*x)/Sqrt[2 - Sqrt[3]]]/(4*Sqrt[3*(2 + Sqrt[3])]) + ArcTan[(Sqrt[2 - Sqrt[3]] + 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt
[3*(2 - Sqrt[3])]) - ArcTan[(Sqrt[2 + Sqrt[3]] + 2*x)/Sqrt[2 - Sqrt[3]]]/(4*Sqrt[3*(2 + Sqrt[3])]) + (Sqrt[(2
- Sqrt[3])/3]*Log[1 - Sqrt[2 - Sqrt[3]]*x + x^2])/8 - (Sqrt[(2 - Sqrt[3])/3]*Log[1 + Sqrt[2 - Sqrt[3]]*x + x^2
])/8 - (Sqrt[(2 + Sqrt[3])/3]*Log[1 - Sqrt[2 + Sqrt[3]]*x + x^2])/8 + (Sqrt[(2 + Sqrt[3])/3]*Log[1 + Sqrt[2 +
Sqrt[3]]*x + x^2])/8

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 1421

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

Rubi steps

\begin {align*} \int \frac {1-x^4}{1-x^4+x^8} \, dx &=-\frac {\int \frac {\sqrt {3}+2 x^2}{-1-\sqrt {3} x^2-x^4} \, dx}{2 \sqrt {3}}-\frac {\int \frac {\sqrt {3}-2 x^2}{-1+\sqrt {3} x^2-x^4} \, dx}{2 \sqrt {3}}\\ &=\frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )}-\left (-2+\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 (-2+\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 (2+\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 (2+\sqrt {3}\right ) x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}\\ &=-\left (\frac {1}{8} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \int \frac {-\sqrt {2-\sqrt {3}}+2 x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx+\frac {\left (-2+\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}+2 x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \int \frac {-\sqrt {2+\sqrt {3}}+2 x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \int \frac {\sqrt {2+\sqrt {3}}+2 x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \int \frac {1}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \int \frac {1}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx\\ &=\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {2}{3}-\frac {1}{\sqrt {3}}} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,-\sqrt {2-\sqrt {3}}+2 x\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,\sqrt {2-\sqrt {3}}+2 x\right )\\ &=-\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {2}{3}-\frac {1}{\sqrt {3}}} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 57, normalized size = 0.16 \begin {gather*} -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\&,\frac {\text {$\#$1}^4 \log (x-\text {$\#$1})-\log (x-\text {$\#$1})}{2 \text {$\#$1}^7-\text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^4)/(1 - x^4 + x^8),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^4}{1-x^4+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - x^4)/(1 - x^4 + x^8),x]

[Out]

IntegrateAlgebraic[(1 - x^4)/(1 - x^4 + x^8), x]

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fricas [B]  time = 1.63, size = 715, normalized size = 2.01 \begin {gather*} \frac {1}{48} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} - 2 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} \log \left (12 \, x^{2} + 2 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 12\right ) - \frac {1}{48} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} - 2 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} \log \left (12 \, x^{2} - 2 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 12\right ) + \frac {1}{96} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} + 2 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (12 \, x^{2} + \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 12\right ) - \frac {1}{96} \, \sqrt {6} {\left (\sqrt {3} \sqrt {2} + 2 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} \log \left (12 \, x^{2} - \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 12\right ) + \frac {1}{12} \, \sqrt {6} \sqrt {2} \sqrt {\sqrt {3} + 2} \arctan \left (\frac {1}{6} \, \sqrt {6} \sqrt {12 \, x^{2} + 2 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 12} {\left (\sqrt {3} \sqrt {2} - 2 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} + \frac {1}{3} \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} - \sqrt {3} + 2\right ) + \frac {1}{12} \, \sqrt {6} \sqrt {2} \sqrt {\sqrt {3} + 2} \arctan \left (\frac {1}{6} \, \sqrt {6} \sqrt {12 \, x^{2} - 2 \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + 12} {\left (\sqrt {3} \sqrt {2} - 2 \, \sqrt {2}\right )} \sqrt {\sqrt {3} + 2} + \frac {1}{3} \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x - 3 \, \sqrt {2} x\right )} \sqrt {\sqrt {3} + 2} + \sqrt {3} - 2\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {2} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {1}{12} \, \sqrt {6} \sqrt {12 \, x^{2} + \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 12} {\left (\sqrt {3} \sqrt {2} + 2 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} - \frac {1}{6} \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} - \sqrt {3} - 2\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {2} \sqrt {-4 \, \sqrt {3} + 8} \arctan \left (\frac {1}{12} \, \sqrt {6} \sqrt {12 \, x^{2} - \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + 12} {\left (\sqrt {3} \sqrt {2} + 2 \, \sqrt {2}\right )} \sqrt {-4 \, \sqrt {3} + 8} - \frac {1}{6} \, \sqrt {6} {\left (2 \, \sqrt {3} \sqrt {2} x + 3 \, \sqrt {2} x\right )} \sqrt {-4 \, \sqrt {3} + 8} + \sqrt {3} + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*sqrt(6)*(sqrt(3)*sqrt(2) - 2*sqrt(2))*sqrt(sqrt(3) + 2)*log(12*x^2 + 2*sqrt(6)*(2*sqrt(3)*sqrt(2)*x - 3*s
qrt(2)*x)*sqrt(sqrt(3) + 2) + 12) - 1/48*sqrt(6)*(sqrt(3)*sqrt(2) - 2*sqrt(2))*sqrt(sqrt(3) + 2)*log(12*x^2 -
2*sqrt(6)*(2*sqrt(3)*sqrt(2)*x - 3*sqrt(2)*x)*sqrt(sqrt(3) + 2) + 12) + 1/96*sqrt(6)*(sqrt(3)*sqrt(2) + 2*sqrt
(2))*sqrt(-4*sqrt(3) + 8)*log(12*x^2 + sqrt(6)*(2*sqrt(3)*sqrt(2)*x + 3*sqrt(2)*x)*sqrt(-4*sqrt(3) + 8) + 12)
- 1/96*sqrt(6)*(sqrt(3)*sqrt(2) + 2*sqrt(2))*sqrt(-4*sqrt(3) + 8)*log(12*x^2 - sqrt(6)*(2*sqrt(3)*sqrt(2)*x +
3*sqrt(2)*x)*sqrt(-4*sqrt(3) + 8) + 12) + 1/12*sqrt(6)*sqrt(2)*sqrt(sqrt(3) + 2)*arctan(1/6*sqrt(6)*sqrt(12*x^
2 + 2*sqrt(6)*(2*sqrt(3)*sqrt(2)*x - 3*sqrt(2)*x)*sqrt(sqrt(3) + 2) + 12)*(sqrt(3)*sqrt(2) - 2*sqrt(2))*sqrt(s
qrt(3) + 2) + 1/3*sqrt(6)*(2*sqrt(3)*sqrt(2)*x - 3*sqrt(2)*x)*sqrt(sqrt(3) + 2) - sqrt(3) + 2) + 1/12*sqrt(6)*
sqrt(2)*sqrt(sqrt(3) + 2)*arctan(1/6*sqrt(6)*sqrt(12*x^2 - 2*sqrt(6)*(2*sqrt(3)*sqrt(2)*x - 3*sqrt(2)*x)*sqrt(
sqrt(3) + 2) + 12)*(sqrt(3)*sqrt(2) - 2*sqrt(2))*sqrt(sqrt(3) + 2) + 1/3*sqrt(6)*(2*sqrt(3)*sqrt(2)*x - 3*sqrt
(2)*x)*sqrt(sqrt(3) + 2) + sqrt(3) - 2) + 1/24*sqrt(6)*sqrt(2)*sqrt(-4*sqrt(3) + 8)*arctan(1/12*sqrt(6)*sqrt(1
2*x^2 + sqrt(6)*(2*sqrt(3)*sqrt(2)*x + 3*sqrt(2)*x)*sqrt(-4*sqrt(3) + 8) + 12)*(sqrt(3)*sqrt(2) + 2*sqrt(2))*s
qrt(-4*sqrt(3) + 8) - 1/6*sqrt(6)*(2*sqrt(3)*sqrt(2)*x + 3*sqrt(2)*x)*sqrt(-4*sqrt(3) + 8) - sqrt(3) - 2) + 1/
24*sqrt(6)*sqrt(2)*sqrt(-4*sqrt(3) + 8)*arctan(1/12*sqrt(6)*sqrt(12*x^2 - sqrt(6)*(2*sqrt(3)*sqrt(2)*x + 3*sqr
t(2)*x)*sqrt(-4*sqrt(3) + 8) + 12)*(sqrt(3)*sqrt(2) + 2*sqrt(2))*sqrt(-4*sqrt(3) + 8) - 1/6*sqrt(6)*(2*sqrt(3)
*sqrt(2)*x + 3*sqrt(2)*x)*sqrt(-4*sqrt(3) + 8) + sqrt(3) + 2)

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giac [A]  time = 0.46, size = 253, normalized size = 0.71 \begin {gather*} \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(sqrt(6) + 3*sqrt(2))*arctan((4*x + sqrt(6) - sqrt(2))/(sqrt(6) + sqrt(2))) + 1/24*(sqrt(6) + 3*sqrt(2))*
arctan((4*x - sqrt(6) + sqrt(2))/(sqrt(6) + sqrt(2))) + 1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x + sqrt(6) + sqr
t(2))/(sqrt(6) - sqrt(2))) + 1/24*(sqrt(6) - 3*sqrt(2))*arctan((4*x - sqrt(6) - sqrt(2))/(sqrt(6) - sqrt(2)))
+ 1/48*(sqrt(6) + 3*sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) + sqrt(2)) + 1) - 1/48*(sqrt(6) + 3*sqrt(2))*log(x^2 - 1
/2*x*(sqrt(6) + sqrt(2)) + 1) + 1/48*(sqrt(6) - 3*sqrt(2))*log(x^2 + 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/48*(sq
rt(6) - 3*sqrt(2))*log(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1)

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maple [C]  time = 0.01, size = 44, normalized size = 0.12 \begin {gather*} \frac {\left (-\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{4}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )+x \right )}{8 \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{7}-4 \RootOf \left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sum((-_R^4+1)/(2*_R^7-_R^3)*ln(-_R+x),_R=RootOf(_Z^8-_Z^4+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.67, size = 208, normalized size = 0.59 \begin {gather*} -\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}-\frac {\sqrt {3}\,x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {3}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4 - 1)/(x^8 - x^4 + 1),x)

[Out]

(2^(3/4)*3^(1/2)*atan((2^(1/4)*x)/(2*(3^(1/2)*1i + 1)^(1/4)) - (2^(1/4)*3^(1/2)*x*1i)/(2*(3^(1/2)*1i + 1)^(1/4
)))*(3^(1/2)*1i + 1)^(1/4)*1i)/12 - (3^(1/2)*atan((x*1i)/(8 - 3^(1/2)*8i)^(1/4) - (3^(1/2)*x)/(8 - 3^(1/2)*8i)
^(1/4))*(8 - 3^(1/2)*8i)^(1/4))/12 - (3^(1/2)*atan(x/(8 - 3^(1/2)*8i)^(1/4) + (3^(1/2)*x*1i)/(8 - 3^(1/2)*8i)^
(1/4))*(8 - 3^(1/2)*8i)^(1/4)*1i)/12 + (2^(3/4)*3^(1/2)*atan((2^(1/4)*x*1i)/(2*(3^(1/2)*1i + 1)^(1/4)) + (2^(1
/4)*3^(1/2)*x)/(2*(3^(1/2)*1i + 1)^(1/4)))*(3^(1/2)*1i + 1)^(1/4))/12

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sympy [A]  time = 3.10, size = 26, normalized size = 0.07 \begin {gather*} - \operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left (t \mapsto t \log {\left (9216 t^{5} - 8 t + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(9216*_t**5 - 8*_t + x)))

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