3.396 \(\int \frac{x^8}{1-3 x^4+x^8} \, dx\)

Optimal. Leaf size=170 \[ x-\frac{\sqrt [4]{\frac{1}{2} \left (123+55 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{984-440 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (123+55 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{984-440 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt{5}} \]

[Out]

x - (((123 + 55*Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + ((984 - 440*Sqrt[5])^(1/4)*
ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(4*Sqrt[5]) - (((123 + 55*Sqrt[5])/2)^(1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)
*x])/(2*Sqrt[5]) + ((984 - 440*Sqrt[5])^(1/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(4*Sqrt[5])

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Rubi [A]  time = 0.114406, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1367, 1422, 212, 206, 203} \[ x-\frac{\sqrt [4]{\frac{1}{2} \left (123+55 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{984-440 \sqrt{5}} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (123+55 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{984-440 \sqrt{5}} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{4 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(1 - 3*x^4 + x^8),x]

[Out]

x - (((123 + 55*Sqrt[5])/2)^(1/4)*ArcTan[(2/(3 + Sqrt[5]))^(1/4)*x])/(2*Sqrt[5]) + ((984 - 440*Sqrt[5])^(1/4)*
ArcTan[((3 + Sqrt[5])/2)^(1/4)*x])/(4*Sqrt[5]) - (((123 + 55*Sqrt[5])/2)^(1/4)*ArcTanh[(2/(3 + Sqrt[5]))^(1/4)
*x])/(2*Sqrt[5]) + ((984 - 440*Sqrt[5])^(1/4)*ArcTanh[((3 + Sqrt[5])/2)^(1/4)*x])/(4*Sqrt[5])

Rule 1367

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

Rule 1422

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] ||  !IGtQ[n/2, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^8}{1-3 x^4+x^8} \, dx &=x-\int \frac{1-3 x^4}{1-3 x^4+x^8} \, dx\\ &=x-\frac{1}{10} \left (-15+7 \sqrt{5}\right ) \int \frac{1}{-\frac{3}{2}+\frac{\sqrt{5}}{2}+x^4} \, dx+\frac{1}{10} \left (15+7 \sqrt{5}\right ) \int \frac{1}{-\frac{3}{2}-\frac{\sqrt{5}}{2}+x^4} \, dx\\ &=x+\sqrt{\frac{1}{10} \left (9-4 \sqrt{5}\right )} \int \frac{1}{\sqrt{3-\sqrt{5}}-\sqrt{2} x^2} \, dx+\sqrt{\frac{1}{10} \left (9-4 \sqrt{5}\right )} \int \frac{1}{\sqrt{3-\sqrt{5}}+\sqrt{2} x^2} \, dx--\frac{\left (-15-7 \sqrt{5}\right ) \int \frac{1}{\sqrt{3+\sqrt{5}}-\sqrt{2} x^2} \, dx}{10 \sqrt{3+\sqrt{5}}}--\frac{\left (-15-7 \sqrt{5}\right ) \int \frac{1}{\sqrt{3+\sqrt{5}}+\sqrt{2} x^2} \, dx}{10 \sqrt{3+\sqrt{5}}}\\ &=x-\frac{\sqrt [4]{\frac{1}{2} \left (123+55 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (123-55 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}-\frac{\sqrt [4]{\frac{1}{2} \left (123+55 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{5}}+\frac{\sqrt [4]{\frac{1}{2} \left (123-55 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.274404, size = 160, normalized size = 0.94 \[ x+\frac{\left (\sqrt{5}-2\right ) \tan ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\left (2+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}}+\frac{\left (\sqrt{5}-2\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{\sqrt{5}-1}} x\right )}{\sqrt{10 \left (\sqrt{5}-1\right )}}-\frac{\left (2+\sqrt{5}\right ) \tanh ^{-1}\left (\sqrt{\frac{2}{1+\sqrt{5}}} x\right )}{\sqrt{10 \left (1+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(1 - 3*x^4 + x^8),x]

[Out]

x + ((-2 + Sqrt[5])*ArcTan[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[10*(-1 + Sqrt[5])] - ((2 + Sqrt[5])*ArcTan[Sqrt[2/(
1 + Sqrt[5])]*x])/Sqrt[10*(1 + Sqrt[5])] + ((-2 + Sqrt[5])*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x])/Sqrt[10*(-1 + Sq
rt[5])] - ((2 + Sqrt[5])*ArcTanh[Sqrt[2/(1 + Sqrt[5])]*x])/Sqrt[10*(1 + Sqrt[5])]

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Maple [A]  time = 0.053, size = 205, normalized size = 1.2 \begin{align*} x-{\frac{2\,\sqrt{5}}{5\,\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{\sqrt{2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{2\,\sqrt{5}}{5\,\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{2\,\sqrt{5}}{5\,\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }+{\frac{1}{\sqrt{-2+2\,\sqrt{5}}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{-2+2\,\sqrt{5}}}} \right ) }-{\frac{2\,\sqrt{5}}{5\,\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) }-{\frac{1}{\sqrt{2+2\,\sqrt{5}}}\arctan \left ( 2\,{\frac{x}{\sqrt{2+2\,\sqrt{5}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(x^8-3*x^4+1),x)

[Out]

x-2/5*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctanh(2*x/(2+2*5^(1/2))^(1/2))-1/(2+2*5^(1/2))^(1/2)*arctanh(2*x/(2+2*5^(1
/2))^(1/2))-2/5*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctan(2*x/(-2+2*5^(1/2))^(1/2))+1/(-2+2*5^(1/2))^(1/2)*arctan(2*
x/(-2+2*5^(1/2))^(1/2))-2/5*5^(1/2)/(-2+2*5^(1/2))^(1/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))+1/(-2+2*5^(1/2))^(1
/2)*arctanh(2*x/(-2+2*5^(1/2))^(1/2))-2/5*5^(1/2)/(2+2*5^(1/2))^(1/2)*arctan(2*x/(2+2*5^(1/2))^(1/2))-1/(2+2*5
^(1/2))^(1/2)*arctan(2*x/(2+2*5^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^8-3*x^4+1),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.00951, size = 988, normalized size = 5.81 \begin{align*} -\frac{1}{10} \, \sqrt{10} \sqrt{5 \, \sqrt{5} + 11} \arctan \left (\frac{1}{20} \,{\left (\sqrt{10} \sqrt{2 \, x^{2} + \sqrt{5} + 1}{\left (2 \, \sqrt{5} \sqrt{2} - 5 \, \sqrt{2}\right )} - 2 \, \sqrt{10}{\left (2 \, \sqrt{5} x - 5 \, x\right )}\right )} \sqrt{5 \, \sqrt{5} + 11}\right ) - \frac{1}{10} \, \sqrt{10} \sqrt{5 \, \sqrt{5} - 11} \arctan \left (\frac{1}{20} \,{\left (\sqrt{10} \sqrt{2 \, x^{2} + \sqrt{5} - 1}{\left (2 \, \sqrt{5} \sqrt{2} + 5 \, \sqrt{2}\right )} - 2 \, \sqrt{10}{\left (2 \, \sqrt{5} x + 5 \, x\right )}\right )} \sqrt{5 \, \sqrt{5} - 11}\right ) + \frac{1}{40} \, \sqrt{10} \sqrt{5 \, \sqrt{5} - 11} \log \left (\sqrt{10} \sqrt{5 \, \sqrt{5} - 11}{\left (3 \, \sqrt{5} + 5\right )} + 20 \, x\right ) - \frac{1}{40} \, \sqrt{10} \sqrt{5 \, \sqrt{5} - 11} \log \left (-\sqrt{10} \sqrt{5 \, \sqrt{5} - 11}{\left (3 \, \sqrt{5} + 5\right )} + 20 \, x\right ) - \frac{1}{40} \, \sqrt{10} \sqrt{5 \, \sqrt{5} + 11} \log \left (\sqrt{10} \sqrt{5 \, \sqrt{5} + 11}{\left (3 \, \sqrt{5} - 5\right )} + 20 \, x\right ) + \frac{1}{40} \, \sqrt{10} \sqrt{5 \, \sqrt{5} + 11} \log \left (-\sqrt{10} \sqrt{5 \, \sqrt{5} + 11}{\left (3 \, \sqrt{5} - 5\right )} + 20 \, x\right ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^8-3*x^4+1),x, algorithm="fricas")

[Out]

-1/10*sqrt(10)*sqrt(5*sqrt(5) + 11)*arctan(1/20*(sqrt(10)*sqrt(2*x^2 + sqrt(5) + 1)*(2*sqrt(5)*sqrt(2) - 5*sqr
t(2)) - 2*sqrt(10)*(2*sqrt(5)*x - 5*x))*sqrt(5*sqrt(5) + 11)) - 1/10*sqrt(10)*sqrt(5*sqrt(5) - 11)*arctan(1/20
*(sqrt(10)*sqrt(2*x^2 + sqrt(5) - 1)*(2*sqrt(5)*sqrt(2) + 5*sqrt(2)) - 2*sqrt(10)*(2*sqrt(5)*x + 5*x))*sqrt(5*
sqrt(5) - 11)) + 1/40*sqrt(10)*sqrt(5*sqrt(5) - 11)*log(sqrt(10)*sqrt(5*sqrt(5) - 11)*(3*sqrt(5) + 5) + 20*x)
- 1/40*sqrt(10)*sqrt(5*sqrt(5) - 11)*log(-sqrt(10)*sqrt(5*sqrt(5) - 11)*(3*sqrt(5) + 5) + 20*x) - 1/40*sqrt(10
)*sqrt(5*sqrt(5) + 11)*log(sqrt(10)*sqrt(5*sqrt(5) + 11)*(3*sqrt(5) - 5) + 20*x) + 1/40*sqrt(10)*sqrt(5*sqrt(5
) + 11)*log(-sqrt(10)*sqrt(5*sqrt(5) + 11)*(3*sqrt(5) - 5) + 20*x) + x

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Sympy [A]  time = 0.921476, size = 58, normalized size = 0.34 \begin{align*} x + \operatorname{RootSum}{\left (6400 t^{4} - 880 t^{2} - 1, \left ( t \mapsto t \log{\left (- \frac{15360 t^{5}}{11} + \frac{1288 t}{55} + x \right )} \right )\right )} + \operatorname{RootSum}{\left (6400 t^{4} + 880 t^{2} - 1, \left ( t \mapsto t \log{\left (- \frac{15360 t^{5}}{11} + \frac{1288 t}{55} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(x**8-3*x**4+1),x)

[Out]

x + RootSum(6400*_t**4 - 880*_t**2 - 1, Lambda(_t, _t*log(-15360*_t**5/11 + 1288*_t/55 + x))) + RootSum(6400*_
t**4 + 880*_t**2 - 1, Lambda(_t, _t*log(-15360*_t**5/11 + 1288*_t/55 + x)))

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Giac [A]  time = 1.22195, size = 200, normalized size = 1.18 \begin{align*} -\frac{1}{20} \, \sqrt{50 \, \sqrt{5} + 110} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}}}\right ) + \frac{1}{20} \, \sqrt{50 \, \sqrt{5} - 110} \arctan \left (\frac{x}{\sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}}}\right ) - \frac{1}{40} \, \sqrt{50 \, \sqrt{5} + 110} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{50 \, \sqrt{5} + 110} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} + \frac{1}{2}} \right |}\right ) + \frac{1}{40} \, \sqrt{50 \, \sqrt{5} - 110} \log \left ({\left | x + \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) - \frac{1}{40} \, \sqrt{50 \, \sqrt{5} - 110} \log \left ({\left | x - \sqrt{\frac{1}{2} \, \sqrt{5} - \frac{1}{2}} \right |}\right ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^8-3*x^4+1),x, algorithm="giac")

[Out]

-1/20*sqrt(50*sqrt(5) + 110)*arctan(x/sqrt(1/2*sqrt(5) + 1/2)) + 1/20*sqrt(50*sqrt(5) - 110)*arctan(x/sqrt(1/2
*sqrt(5) - 1/2)) - 1/40*sqrt(50*sqrt(5) + 110)*log(abs(x + sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(50*sqrt(5) +
110)*log(abs(x - sqrt(1/2*sqrt(5) + 1/2))) + 1/40*sqrt(50*sqrt(5) - 110)*log(abs(x + sqrt(1/2*sqrt(5) - 1/2)))
- 1/40*sqrt(50*sqrt(5) - 110)*log(abs(x - sqrt(1/2*sqrt(5) - 1/2))) + x