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

Optimal. Leaf size=54 \[ \frac{x^2}{2}+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

x^2/2 + ArcTan[(1 - 2*x^2)/Sqrt[3]]/(2*Sqrt[3]) - ArcTan[(1 + 2*x^2)/Sqrt[3]]/(2*Sqrt[3])

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Rubi [A]  time = 0.0577644, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {1359, 1122, 1161, 618, 204} \[ \frac{x^2}{2}+\frac{\tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[x^9/(1 + x^4 + x^8),x]

[Out]

x^2/2 + ArcTan[(1 - 2*x^2)/Sqrt[3]]/(2*Sqrt[3]) - ArcTan[(1 + 2*x^2)/Sqrt[3]]/(2*Sqrt[3])

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 1161

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

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])

Rubi steps

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

Mathematica [C]  time = 0.173811, size = 98, normalized size = 1.81 \[ \frac{x^2}{2}-\frac{\left (\sqrt{3}+i\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}-i\right ) x^2\right )}{2 \sqrt{6+6 i \sqrt{3}}}-\frac{\left (\sqrt{3}-i\right ) \tan ^{-1}\left (\frac{1}{2} \left (\sqrt{3}+i\right ) x^2\right )}{2 \sqrt{6-6 i \sqrt{3}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^9/(1 + x^4 + x^8),x]

[Out]

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

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Maple [A]  time = 0.005, size = 43, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(x^8+x^4+1),x)

[Out]

1/2*x^2-1/6*arctan(1/3*(2*x^2+1)*3^(1/2))*3^(1/2)-1/6*3^(1/2)*arctan(1/3*(2*x^2-1)*3^(1/2))

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Maxima [A]  time = 1.49526, size = 57, normalized size = 1.06 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1))

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Fricas [A]  time = 1.43287, size = 128, normalized size = 2.37 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x^{2}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x^{6} + 2 \, x^{2}\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*x^2) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(x^6 + 2*x^2))

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Sympy [A]  time = 0.137822, size = 51, normalized size = 0.94 \begin{align*} \frac{x^{2}}{2} + \frac{\sqrt{3} \left (- 2 \operatorname{atan}{\left (\frac{\sqrt{3} x^{2}}{3} \right )} - 2 \operatorname{atan}{\left (\frac{\sqrt{3} x^{6}}{3} + \frac{2 \sqrt{3} x^{2}}{3} \right )}\right )}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(x**8+x**4+1),x)

[Out]

x**2/2 + sqrt(3)*(-2*atan(sqrt(3)*x**2/3) - 2*atan(sqrt(3)*x**6/3 + 2*sqrt(3)*x**2/3))/12

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Giac [A]  time = 1.10704, size = 57, normalized size = 1.06 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) - \frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(x^8+x^4+1),x, algorithm="giac")

[Out]

1/2*x^2 - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1)) - 1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1))