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

Optimal. Leaf size=360 \[ \frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

[Out]

-x^(-1) + ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt[3*(2 - Sqrt[3])]) - ArcTan[(Sqrt[2 + Sqr
t[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 = 0.238585, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1368, 1506, 1279, 1169, 634, 618, 204, 628} \[ \frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )-\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )+\frac{1}{8} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )-\frac{1}{x}+\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}-\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )}{4 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right )}{4 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(1 - x^4 + x^8)),x]

[Out]

-x^(-1) + ArcTan[(Sqrt[2 - Sqrt[3]] - 2*x)/Sqrt[2 + Sqrt[3]]]/(4*Sqrt[3*(2 - Sqrt[3])]) - ArcTan[(Sqrt[2 + Sqr
t[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 1368

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

Rule 1506

Int[(((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Wit
h[{q = Rt[a*c, 2]}, With[{r = Rt[2*c*q - b*c, 2]}, Dist[c/(2*q*r), Int[((f*x)^m*Simp[d*r - (c*d - e*q)*x^(n/2)
, x])/(q - r*x^(n/2) + c*x^n), x], x] + Dist[c/(2*q*r), Int[((f*x)^m*Simp[d*r + (c*d - e*q)*x^(n/2), x])/(q +
r*x^(n/2) + c*x^n), x], x]] /;  !LtQ[2*c*q - b*c, 0]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[n2, 2*n] && LtQ[b
^2 - 4*a*c, 0] && IntegersQ[m, n/2] && LtQ[0, m, n] && PosQ[a*c]

Rule 1279

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(e*f
*(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 3)), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m -
2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[
{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (Inte
gerQ[p] || IntegerQ[m])

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^2 \left (1-x^4+x^8\right )} \, dx &=-\frac{1}{x}+\int \frac{x^2 \left (1-x^4\right )}{1-x^4+x^8} \, dx\\ &=-\frac{1}{x}+\frac{\int \frac{x^2 \left (\sqrt{3}-2 x^2\right )}{1-\sqrt{3} x^2+x^4} \, dx}{2 \sqrt{3}}+\frac{\int \frac{x^2 \left (\sqrt{3}+2 x^2\right )}{1+\sqrt{3} x^2+x^4} \, dx}{2 \sqrt{3}}\\ &=-\frac{1}{x}-\frac{\int \frac{-2+\sqrt{3} x^2}{1-\sqrt{3} x^2+x^4} \, dx}{2 \sqrt{3}}-\frac{\int \frac{2+\sqrt{3} x^2}{1+\sqrt{3} x^2+x^4} \, dx}{2 \sqrt{3}}\\ &=-\frac{1}{x}-\frac{\int \frac{2 \sqrt{2-\sqrt{3}}-\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{2 \sqrt{2-\sqrt{3}}+\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{-2 \sqrt{2+\sqrt{3}}-\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{-2 \sqrt{2+\sqrt{3}}+\left (-2-\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{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 )} \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}{x}+\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}{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}{x}+\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{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}{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 = 0.015709, size = 61, normalized size = 0.17 \[ -\frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\& ,\frac{\text{$\#$1}^4 \log (x-\text{$\#$1})-\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}}\& \right ]-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(1 - x^4 + x^8)),x]

[Out]

-x^(-1) - RootSum[1 - #1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-#1 + 2*#1^5) & ]/4

________________________________________________________________________________________

Maple [C]  time = 0.009, size = 52, 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}}^{6}-{{\it \_R}}^{2} \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}-{{\it \_R}}^{3}}}}-{x}^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sum((_R^6-_R^2)/(2*_R^7-_R^3)*ln(x-_R),_R=RootOf(_Z^8-_Z^4+1))-1/x

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{x} - \int \frac{x^{6} - x^{2}}{x^{8} - x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x - integrate((x^6 - x^2)/(x^8 - x^4 + 1), x)

________________________________________________________________________________________

Fricas [B]  time = 1.82402, size = 2327, normalized size = 6.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.29061, size = 29, normalized size = 0.08 \begin{align*} \operatorname{RootSum}{\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log{\left (- 442368 t^{7} + 384 t^{3} + x \right )} \right )\right )} - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(x**8-x**4+1),x)

[Out]

RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(-442368*_t**7 + 384*_t**3 + x))) - 1/x

________________________________________________________________________________________

Giac [A]  time = 1.20009, size = 348, normalized size = 0.97 \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 ) - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(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) + sq
rt(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*(s
qrt(6) - 3*sqrt(2))*log(x^2 - 1/2*x*(sqrt(6) - sqrt(2)) + 1) - 1/x