### 3.58 $$\int \frac{1-x^4}{x^2 (1-x^4+x^8)} \, dx$$

Optimal. Leaf size=280 $-\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}}$

[Out]

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

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Rubi [A]  time = 0.207993, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 7, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.304, Rules used = {1504, 1372, 1169, 634, 618, 204, 628} $-\frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{4 \sqrt{6}}-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{6}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{6}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 - x^4)/(x^2*(1 - x^4 + x^8)),x]

[Out]

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

Rule 1504

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Simp[(d*(f*x)^(m + 1)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*f*(m + 1)), x] + Dist[1/(a*f^n*(m + 1)), Int[(f*x)^
(m + n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n*(p + 1) + 1)*x^n,
x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -
1] && IntegerQ[p]

Rule 1372

Int[(x_)^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[
2*q - b/c, 2]}, -Dist[1/(2*c*r), Int[(x^(m - 3*(n/2))*(q - r*x^(n/2)))/(q - r*x^(n/2) + x^n), x], x] + Dist[1/
(2*c*r), Int[(x^(m - 3*(n/2))*(q + r*x^(n/2)))/(q + r*x^(n/2) + x^n), x], x]]] /; FreeQ[{a, b, c}, x] && EqQ[n
2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n/2, 0] && IGtQ[m, 0] && GeQ[m, (3*n)/2] && LtQ[m, 2*n] && NegQ[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 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}{x^2 \left (1-x^4+x^8\right )} \, dx &=-\frac{1}{x}-\int \frac{x^6}{1-x^4+x^8} \, dx\\ &=-\frac{1}{x}+\frac{\int \frac{1-\sqrt{3} x^2}{1-\sqrt{3} x^2+x^4} \, dx}{2 \sqrt{3}}-\frac{\int \frac{1+\sqrt{3} x^2}{1+\sqrt{3} x^2+x^4} \, dx}{2 \sqrt{3}}\\ &=-\frac{1}{x}-\frac{\int \frac{\sqrt{2-\sqrt{3}}-\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}}+\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}}-\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}}+\left (1+\sqrt{3}\right ) x}{1+\sqrt{2+\sqrt{3}} x+x^2} \, dx}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}\\ &=-\frac{1}{x}-\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{1}{x}-\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{1}{x}+\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*}

Mathematica [C]  time = 0.015771, size = 47, normalized size = 0.17 $-\frac{1}{4} \text{RootSum}\left [\text{\#1}^8-\text{\#1}^4+1\& ,\frac{\text{\#1}^3 \log (x-\text{\#1})}{2 \text{\#1}^4-1}\& \right ]-\frac{1}{x}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(1 - x^4)/(x^2*(1 - x^4 + x^8)),x]

[Out]

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

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Maple [C]  time = 0.007, size = 38, normalized size = 0.1 \begin{align*} -{\frac{\sum _{{\it \_R}={\it RootOf} \left ( 9\,{{\it \_Z}}^{4}+1 \right ) }{\it \_R}\,\ln \left ( 9\,{{\it \_R}}^{3}x-3\,{{\it \_R}}^{2}+{x}^{2} \right ) }{4}}-{x}^{-1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4+1)/x^2/(x^8-x^4+1),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{x} - \int \frac{x^{6}}{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^2/(x^8-x^4+1),x, algorithm="maxima")

[Out]

-1/x - integrate(x^6/(x^8 - x^4 + 1), x)

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Fricas [A]  time = 1.62903, size = 617, normalized size = 2.2 \begin{align*} \frac{4 \, \sqrt{3} \sqrt{2} x \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 ) + 4 \, \sqrt{3} \sqrt{2} x \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 ) + \sqrt{3} \sqrt{2} x \log \left (x^{4} + \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1\right ) - \sqrt{3} \sqrt{2} x \log \left (x^{4} - \sqrt{3} \sqrt{2}{\left (x^{3} + x\right )} + 3 \, x^{2} + 1\right ) - 24}{24 \, x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+1)/x^2/(x^8-x^4+1),x, algorithm="fricas")

[Out]

1/24*(4*sqrt(3)*sqrt(2)*x*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)) + 4*sqrt(3)*sqrt(2)*x*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)) + sqrt(3)*sqrt(2)*x*
log(x^4 + sqrt(3)*sqrt(2)*(x^3 + x) + 3*x^2 + 1) - sqrt(3)*sqrt(2)*x*log(x^4 - sqrt(3)*sqrt(2)*(x^3 + x) + 3*x
^2 + 1) - 24)/x

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Sympy [A]  time = 0.305509, size = 168, normalized size = 0.6 \begin{align*} - \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} - \frac{1}{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**4+1)/x**2/(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(sq
rt(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 - 1/x

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Giac [A]  time = 1.13547, size = 284, normalized size = 1.01 \begin{align*} -\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 ) - \frac{1}{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+1)/x^2/(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) + sqr
t(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) - 1/x