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

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

[Out]

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

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Rubi [A]  time = 0.199669, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {1373, 1094, 634, 618, 204, 628} \[ \frac{\log \left (x^2-\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\log \left (x^2+\sqrt{2-\sqrt{3}} x+1\right )}{8 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\log \left (x^2-\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{\log \left (x^2+\sqrt{2+\sqrt{3}} x+1\right )}{8 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{3}}-2 x}{\sqrt{2+\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2+\sqrt{3}}-2 x}{\sqrt{2-\sqrt{3}}}\right )-\frac{1}{4} \sqrt{\frac{1}{3} \left (2-\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{3}}}{\sqrt{2+\sqrt{3}}}\right )+\frac{1}{4} \sqrt{\frac{1}{3} \left (2+\sqrt{3}\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2+\sqrt{3}}}{\sqrt{2-\sqrt{3}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/(1 - x^4 + x^8),x]

[Out]

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

Rule 1373

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 - n/2)/(q - r*x^(n/2) + x^n), x], x] - Dist[1/(2*c*r), Int[x^(m - 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
] && IGtQ[m, 0] && GeQ[m, n/2] && LtQ[m, (3*n)/2] && NegQ[b^2 - 4*a*c]

Rule 1094

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-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)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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{x^2}{1-x^4+x^8} \, dx &=\frac{\int \frac{1}{1-\sqrt{3} x^2+x^4} \, dx}{2 \sqrt{3}}-\frac{\int \frac{1}{1+\sqrt{3} x^2+x^4} \, dx}{2 \sqrt{3}}\\ &=-\frac{\int \frac{\sqrt{2-\sqrt{3}}-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}}+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}}-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}}+x}{1+\sqrt{2+\sqrt{3}} x+x^2} \, dx}{4 \sqrt{3 \left (2+\sqrt{3}\right )}}\\ &=-\frac{\int \frac{1}{1-\sqrt{2-\sqrt{3}} x+x^2} \, dx}{8 \sqrt{3}}-\frac{\int \frac{1}{1+\sqrt{2-\sqrt{3}} x+x^2} \, dx}{8 \sqrt{3}}+\frac{\int \frac{1}{1-\sqrt{2+\sqrt{3}} x+x^2} \, dx}{8 \sqrt{3}}+\frac{\int \frac{1}{1+\sqrt{2+\sqrt{3}} x+x^2} \, dx}{8 \sqrt{3}}+\frac{\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{\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{\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{\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{\log \left (1-\sqrt{2-\sqrt{3}} x+x^2\right )}{8 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\log \left (1+\sqrt{2-\sqrt{3}} x+x^2\right )}{8 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\log \left (1-\sqrt{2+\sqrt{3}} x+x^2\right )}{8 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{\log \left (1+\sqrt{2+\sqrt{3}} x+x^2\right )}{8 \sqrt{3 \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 )}{4 \sqrt{3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{3}-x^2} \, dx,x,\sqrt{2-\sqrt{3}}+2 x\right )}{4 \sqrt{3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{3}-x^2} \, dx,x,-\sqrt{2+\sqrt{3}}+2 x\right )}{4 \sqrt{3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-2+\sqrt{3}-x^2} \, dx,x,\sqrt{2+\sqrt{3}}+2 x\right )}{4 \sqrt{3}}\\ &=\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{\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{\log \left (1-\sqrt{2-\sqrt{3}} x+x^2\right )}{8 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\log \left (1+\sqrt{2-\sqrt{3}} x+x^2\right )}{8 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\log \left (1-\sqrt{2+\sqrt{3}} x+x^2\right )}{8 \sqrt{3 \left (2+\sqrt{3}\right )}}+\frac{\log \left (1+\sqrt{2+\sqrt{3}} x+x^2\right )}{8 \sqrt{3 \left (2+\sqrt{3}\right )}}\\ \end{align*}

Mathematica [C]  time = 0.011243, size = 40, normalized size = 0.11 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^4+1\& ,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5-\text{$\#$1}}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(1 - x^4 + x^8),x]

[Out]

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

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Maple [C]  time = 0.007, size = 40, normalized size = 0.1 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}-{{\it \_Z}}^{4}+1 \right ) }{\frac{{{\it \_R}}^{2}\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^2/(x^8-x^4+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/(x^8 - x^4 + 1), x)

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Fricas [B]  time = 1.80795, size = 1914, normalized size = 5.39 \begin{align*} -\frac{1}{48} \, \sqrt{6}{\left (\sqrt{3} \sqrt{2} - 2 \, \sqrt{2}\right )} \sqrt{\sqrt{3} + 2} \log \left (2 \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{\sqrt{3} + 2} + 12 \, x^{2} + 12\right ) + \frac{1}{48} \, \sqrt{6}{\left (\sqrt{3} \sqrt{2} - 2 \, \sqrt{2}\right )} \sqrt{\sqrt{3} + 2} \log \left (-2 \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{\sqrt{3} + 2} + 12 \, x^{2} + 12\right ) - \frac{1}{96} \, \sqrt{6}{\left (\sqrt{3} \sqrt{2} + 2 \, \sqrt{2}\right )} \sqrt{-4 \, \sqrt{3} + 8} \log \left (\sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{-4 \, \sqrt{3} + 8} + 12 \, x^{2} + 12\right ) + \frac{1}{96} \, \sqrt{6}{\left (\sqrt{3} \sqrt{2} + 2 \, \sqrt{2}\right )} \sqrt{-4 \, \sqrt{3} + 8} \log \left (-\sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{-4 \, \sqrt{3} + 8} + 12 \, x^{2} + 12\right ) - \frac{1}{12} \, \sqrt{6} \sqrt{2} \sqrt{\sqrt{3} + 2} \arctan \left (-\frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{\sqrt{3} + 2} + \frac{1}{6} \, \sqrt{6} \sqrt{2} \sqrt{2 \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{\sqrt{3} + 2} + 12 \, x^{2} + 12} \sqrt{\sqrt{3} + 2} - \sqrt{3} - 2\right ) - \frac{1}{12} \, \sqrt{6} \sqrt{2} \sqrt{\sqrt{3} + 2} \arctan \left (-\frac{1}{3} \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{\sqrt{3} + 2} + \frac{1}{6} \, \sqrt{6} \sqrt{2} \sqrt{-2 \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{\sqrt{3} + 2} + 12 \, x^{2} + 12} \sqrt{\sqrt{3} + 2} + \sqrt{3} + 2\right ) + \frac{1}{24} \, \sqrt{6} \sqrt{2} \sqrt{-4 \, \sqrt{3} + 8} \arctan \left (-\frac{1}{6} \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{-4 \, \sqrt{3} + 8} + \frac{1}{12} \, \sqrt{6} \sqrt{2} \sqrt{\sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{-4 \, \sqrt{3} + 8} + 12 \, x^{2} + 12} \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}{6} \, \sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{-4 \, \sqrt{3} + 8} + \frac{1}{12} \, \sqrt{6} \sqrt{2} \sqrt{-\sqrt{6} \sqrt{3} \sqrt{2} x \sqrt{-4 \, \sqrt{3} + 8} + 12 \, x^{2} + 12} \sqrt{-4 \, \sqrt{3} + 8} - \sqrt{3} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [A]  time = 1.2978, size = 26, normalized size = 0.07 \begin{align*} \operatorname{RootSum}{\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log{\left (- 442368 t^{7} - 192 t^{3} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(5308416*_t**8 - 2304*_t**4 + 1, Lambda(_t, _t*log(-442368*_t**7 - 192*_t**3 + x)))

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Giac [A]  time = 1.1451, 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^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) + 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)