### 3.56 $$\int \frac{1-x^4}{1-x^4+x^8} \, dx$$

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.216203, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.3, Rules used = {1421, 1169, 634, 618, 204, 628} $\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}$

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 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]

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 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]

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*}

Mathematica [C]  time = 0.0129269, size = 57, normalized size = 0.16 $-\frac{1}{4} \text{RootSum}\left [\text{\#1}^8-\text{\#1}^4+1\& ,\frac{\text{\#1}^4 \log (x-\text{\#1})-\log (x-\text{\#1})}{2 \text{\#1}^7-\text{\#1}^3}\& \right ]$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0., size = 44, normalized size = 0.1 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{ \left ( -{{\it \_R}}^{4}+1 \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}} \end{align*}

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(x-_R),_R=RootOf(_Z^8-_Z^4+1))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Fricas [B]  time = 1.86818, size = 2313, normalized size = 6.52 \begin{align*} \text{result too large to display} \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 1.54208, size = 26, normalized size = 0.07 \begin{align*} - \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{align*}

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)))

________________________________________________________________________________________

Giac [A]  time = 1.12705, size = 342, normalized size = 0.96 \begin{align*} \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{align*}

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)