3.915 \(\int \frac{x}{1+x^2+x^4} \, dx\)

Optimal. Leaf size=20 \[ \frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

ArcTan[(1 + 2*x^2)/Sqrt[3]]/Sqrt[3]

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Rubi [A]  time = 0.0198906, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1107, 618, 204} \[ \frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[x/(1 + x^2 + x^4),x]

[Out]

ArcTan[(1 + 2*x^2)/Sqrt[3]]/Sqrt[3]

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

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

Mathematica [A]  time = 0.005332, size = 20, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 + x^2 + x^4),x]

[Out]

ArcTan[(1 + 2*x^2)/Sqrt[3]]/Sqrt[3]

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Maple [A]  time = 0.042, size = 19, normalized size = 1. \begin{align*}{\frac{\sqrt{3}}{3}\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/(x^4+x^2+1),x)

[Out]

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

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Maxima [A]  time = 1.46805, size = 24, normalized size = 1.2 \begin{align*} \frac{1}{3} \, \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/(x^4+x^2+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.46866, size = 61, normalized size = 3.05 \begin{align*} \frac{1}{3} \, \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/(x^4+x^2+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.102124, size = 26, normalized size = 1.3 \begin{align*} \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(3)*atan(2*sqrt(3)*x**2/3 + sqrt(3)/3)/3

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Giac [A]  time = 1.36519, size = 24, normalized size = 1.2 \begin{align*} \frac{1}{3} \, \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/(x^4+x^2+1),x, algorithm="giac")

[Out]

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