3.371 \(\int \frac{x^5}{1+3 x^4+x^8} \, dx\)

Optimal. Leaf size=81 \[ \frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )-\frac{1}{2} \sqrt{\frac{1}{10} \left (3-\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

[Out]

(Sqrt[(3 + Sqrt[5])/10]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 - (Sqrt[(3 - Sqrt[5])/10]*ArcTan[Sqrt[(3 + Sqrt[5
])/2]*x^2])/2

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Rubi [A]  time = 0.084069, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1359, 1130, 203} \[ \frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )-\frac{1}{2} \sqrt{\frac{1}{10} \left (3-\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5/(1 + 3*x^4 + x^8),x]

[Out]

(Sqrt[(3 + Sqrt[5])/10]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 - (Sqrt[(3 - Sqrt[5])/10]*ArcTan[Sqrt[(3 + Sqrt[5
])/2]*x^2])/2

Rule 1359

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

Rule 1130

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

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^5}{1+3 x^4+x^8} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{1+3 x^2+x^4} \, dx,x,x^2\right )\\ &=\frac{1}{20} \left (5-3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx,x,x^2\right )+\frac{1}{20} \left (5+3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \sqrt{\frac{1}{10} \left (3+\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )-\frac{1}{2} \sqrt{\frac{1}{10} \left (3-\sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0468945, size = 75, normalized size = 0.93 \[ \frac{2 \sqrt{5} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\left (5-3 \sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )}{10 \sqrt{6-2 \sqrt{5}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(1 + 3*x^4 + x^8),x]

[Out]

(2*Sqrt[5]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2] + (5 - 3*Sqrt[5])*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/(10*Sqrt[6 -
2*Sqrt[5]])

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Maple [B]  time = 0.017, size = 110, normalized size = 1.4 \begin{align*}{\frac{1}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{-2+2\,\sqrt{5}}} \right ) }-{\frac{3\,\sqrt{5}}{-10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{-2+2\,\sqrt{5}}} \right ) }+{\frac{1}{2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{2+2\,\sqrt{5}}} \right ) }+{\frac{3\,\sqrt{5}}{10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{2+2\,\sqrt{5}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x^8+3*x^4+1),x)

[Out]

1/(-2+2*5^(1/2))*arctan(4*x^2/(-2+2*5^(1/2)))-3/5*5^(1/2)/(-2+2*5^(1/2))*arctan(4*x^2/(-2+2*5^(1/2)))+1/(2+2*5
^(1/2))*arctan(4*x^2/(2+2*5^(1/2)))+3/5*5^(1/2)/(2+2*5^(1/2))*arctan(4*x^2/(2+2*5^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^8+3*x^4+1),x, algorithm="maxima")

[Out]

integrate(x^5/(x^8 + 3*x^4 + 1), x)

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Fricas [B]  time = 1.59255, size = 517, normalized size = 6.38 \begin{align*} -\frac{1}{10} \, \sqrt{10} \sqrt{\sqrt{5} + 3} \arctan \left (\frac{1}{40} \, \sqrt{10} \sqrt{2 \, x^{4} + \sqrt{5} + 3}{\left (3 \, \sqrt{5} \sqrt{2} - 5 \, \sqrt{2}\right )} \sqrt{\sqrt{5} + 3} - \frac{1}{20} \, \sqrt{10}{\left (3 \, \sqrt{5} x^{2} - 5 \, x^{2}\right )} \sqrt{\sqrt{5} + 3}\right ) + \frac{1}{10} \, \sqrt{10} \sqrt{-\sqrt{5} + 3} \arctan \left (\frac{1}{40} \, \sqrt{10} \sqrt{2 \, x^{4} - \sqrt{5} + 3}{\left (3 \, \sqrt{5} \sqrt{2} + 5 \, \sqrt{2}\right )} \sqrt{-\sqrt{5} + 3} - \frac{1}{20} \, \sqrt{10}{\left (3 \, \sqrt{5} x^{2} + 5 \, x^{2}\right )} \sqrt{-\sqrt{5} + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^8+3*x^4+1),x, algorithm="fricas")

[Out]

-1/10*sqrt(10)*sqrt(sqrt(5) + 3)*arctan(1/40*sqrt(10)*sqrt(2*x^4 + sqrt(5) + 3)*(3*sqrt(5)*sqrt(2) - 5*sqrt(2)
)*sqrt(sqrt(5) + 3) - 1/20*sqrt(10)*(3*sqrt(5)*x^2 - 5*x^2)*sqrt(sqrt(5) + 3)) + 1/10*sqrt(10)*sqrt(-sqrt(5) +
3)*arctan(1/40*sqrt(10)*sqrt(2*x^4 - sqrt(5) + 3)*(3*sqrt(5)*sqrt(2) + 5*sqrt(2))*sqrt(-sqrt(5) + 3) - 1/20*s
qrt(10)*(3*sqrt(5)*x^2 + 5*x^2)*sqrt(-sqrt(5) + 3))

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Sympy [A]  time = 0.193788, size = 49, normalized size = 0.6 \begin{align*} - 2 \left (\frac{1}{8} - \frac{\sqrt{5}}{40}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{-1 + \sqrt{5}} \right )} + 2 \left (\frac{\sqrt{5}}{40} + \frac{1}{8}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{1 + \sqrt{5}} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(x**8+3*x**4+1),x)

[Out]

-2*(1/8 - sqrt(5)/40)*atan(2*x**2/(-1 + sqrt(5))) + 2*(sqrt(5)/40 + 1/8)*atan(2*x**2/(1 + sqrt(5)))

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Giac [A]  time = 1.2764, size = 63, normalized size = 0.78 \begin{align*} \frac{1}{20} \, x^{4}{\left (\sqrt{5} - 5\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} + 1}\right ) + \frac{1}{20} \, x^{4}{\left (\sqrt{5} + 5\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(x^8+3*x^4+1),x, algorithm="giac")

[Out]

1/20*x^4*(sqrt(5) - 5)*arctan(2*x^2/(sqrt(5) + 1)) + 1/20*x^4*(sqrt(5) + 5)*arctan(2*x^2/(sqrt(5) - 1))