3.369 \(\int \frac{x^9}{1+3 x^4+x^8} \, dx\)

Optimal. Leaf size=90 \[ \frac{x^2}{2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9+4 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

[Out]

x^2/2 - (Sqrt[(9 + 4*Sqrt[5])/5]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + (Sqrt[(9 - 4*Sqrt[5])/5]*ArcTan[Sqrt[(
3 + Sqrt[5])/2]*x^2])/2

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Rubi [A]  time = 0.142151, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1359, 1122, 1166, 203} \[ \frac{x^2}{2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9+4 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{2} \sqrt{\frac{1}{5} \left (9-4 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^9/(1 + 3*x^4 + x^8),x]

[Out]

x^2/2 - (Sqrt[(9 + 4*Sqrt[5])/5]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + (Sqrt[(9 - 4*Sqrt[5])/5]*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 1122

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

Rule 1166

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

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^9}{1+3 x^4+x^8} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{1+3 x^2+x^4} \, dx,x,x^2\right )\\ &=\frac{x^2}{2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1+3 x^2}{1+3 x^2+x^4} \, dx,x,x^2\right )\\ &=\frac{x^2}{2}-\frac{1}{20} \left (15-7 \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 (15+7 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx,x,x^2\right )\\ &=\frac{x^2}{2}-\frac{1}{2} \sqrt{\frac{1}{5} \left (9+4 \sqrt{5}\right )} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{20} \sqrt{180-80 \sqrt{5}} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.146126, size = 97, normalized size = 1.08 \[ \frac{1}{40} \left (20 x^2-\sqrt{6-2 \sqrt{5}} \left (15+7 \sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\sqrt{2 \left (3+\sqrt{5}\right )} \left (7 \sqrt{5}-15\right ) \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(1 + 3*x^4 + x^8),x]

[Out]

(20*x^2 - Sqrt[6 - 2*Sqrt[5]]*(15 + 7*Sqrt[5])*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^2] + Sqrt[2*(3 + Sqrt[5])]*(-15
+ 7*Sqrt[5])*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^2])/40

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Maple [B]  time = 0.041, size = 117, normalized size = 1.3 \begin{align*}{\frac{{x}^{2}}{2}}+{\frac{7\,\sqrt{5}}{-10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{-2+2\,\sqrt{5}}} \right ) }-3\,{\frac{1}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{-2+2\,\sqrt{5}}} \right ) }-{\frac{7\,\sqrt{5}}{10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{{x}^{2}}{2+2\,\sqrt{5}}} \right ) }-3\,{\frac{1}{2+2\,\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^9/(x^8+3*x^4+1),x)

[Out]

1/2*x^2+7/5*5^(1/2)/(-2+2*5^(1/2))*arctan(4*x^2/(-2+2*5^(1/2)))-3/(-2+2*5^(1/2))*arctan(4*x^2/(-2+2*5^(1/2)))-
7/5*5^(1/2)/(2+2*5^(1/2))*arctan(4*x^2/(2+2*5^(1/2)))-3/(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*} \frac{1}{2} \, x^{2} - \int \frac{{\left (3 \, x^{4} + 1\right )} x}{x^{8} + 3 \, x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(x^8+3*x^4+1),x, algorithm="maxima")

[Out]

1/2*x^2 - integrate((3*x^4 + 1)*x/(x^8 + 3*x^4 + 1), x)

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Fricas [B]  time = 1.58515, size = 456, normalized size = 5.07 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{5} \, \sqrt{5} \sqrt{-4 \, \sqrt{5} + 9} \arctan \left (\frac{1}{4} \, \sqrt{2 \, x^{4} - \sqrt{5} + 3}{\left (3 \, \sqrt{5} \sqrt{2} + 7 \, \sqrt{2}\right )} \sqrt{-4 \, \sqrt{5} + 9} - \frac{1}{2} \,{\left (3 \, \sqrt{5} x^{2} + 7 \, x^{2}\right )} \sqrt{-4 \, \sqrt{5} + 9}\right ) - \frac{1}{5} \, \sqrt{5} \sqrt{4 \, \sqrt{5} + 9} \arctan \left (-\frac{1}{4} \,{\left (6 \, \sqrt{5} x^{2} - 14 \, x^{2} - \sqrt{2 \, x^{4} + \sqrt{5} + 3}{\left (3 \, \sqrt{5} \sqrt{2} - 7 \, \sqrt{2}\right )}\right )} \sqrt{4 \, \sqrt{5} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(x^8+3*x^4+1),x, algorithm="fricas")

[Out]

1/2*x^2 - 1/5*sqrt(5)*sqrt(-4*sqrt(5) + 9)*arctan(1/4*sqrt(2*x^4 - sqrt(5) + 3)*(3*sqrt(5)*sqrt(2) + 7*sqrt(2)
)*sqrt(-4*sqrt(5) + 9) - 1/2*(3*sqrt(5)*x^2 + 7*x^2)*sqrt(-4*sqrt(5) + 9)) - 1/5*sqrt(5)*sqrt(4*sqrt(5) + 9)*a
rctan(-1/4*(6*sqrt(5)*x^2 - 14*x^2 - sqrt(2*x^4 + sqrt(5) + 3)*(3*sqrt(5)*sqrt(2) - 7*sqrt(2)))*sqrt(4*sqrt(5)
+ 9))

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Sympy [A]  time = 0.197984, size = 54, normalized size = 0.6 \begin{align*} \frac{x^{2}}{2} + 2 \left (\frac{1}{4} - \frac{\sqrt{5}}{10}\right ) \operatorname{atan}{\left (\frac{2 x^{2}}{-1 + \sqrt{5}} \right )} - 2 \left (\frac{\sqrt{5}}{10} + \frac{1}{4}\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**9/(x**8+3*x**4+1),x)

[Out]

x**2/2 + 2*(1/4 - sqrt(5)/10)*atan(2*x**2/(-1 + sqrt(5))) - 2*(sqrt(5)/10 + 1/4)*atan(2*x**2/(1 + sqrt(5)))

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Giac [A]  time = 1.23001, size = 89, normalized size = 0.99 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{20} \,{\left (3 \, x^{4}{\left (\sqrt{5} - 5\right )} + \sqrt{5} - 5\right )} \arctan \left (\frac{2 \, x^{2}}{\sqrt{5} + 1}\right ) - \frac{1}{20} \,{\left (3 \, x^{4}{\left (\sqrt{5} + 5\right )} + \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^9/(x^8+3*x^4+1),x, algorithm="giac")

[Out]

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