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

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

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1107, 618, 204} \begin {gather*} \frac {\tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(1 + x^2 + x^4),x]

[Out]

ArcTan[(1 + 2*x^2)/Sqrt[3]]/Sqrt[3]

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])

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {2 x^2+1}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{1+x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x/(1 + x^2 + x^4),x]

[Out]

IntegrateAlgebraic[x/(1 + x^2 + x^4), x]

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fricas [A]  time = 0.93, size = 18, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) \end {gather*}

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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giac [A]  time = 0.15, size = 18, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) \end {gather*}

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))

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maple [A]  time = 0.00, size = 19, normalized size = 0.95 \begin {gather*} \frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}+1\right ) \sqrt {3}}{3}\right )}{3} \end {gather*}

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 = 2.84, size = 18, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + 1\right )}\right ) \end {gather*}

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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mupad [B]  time = 0.06, size = 20, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x^2}{3}+\frac {\sqrt {3}}{3}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^2 + x^4 + 1),x)

[Out]

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

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sympy [A]  time = 0.17, size = 26, normalized size = 1.30 \begin {gather*} \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{2}}{3} + \frac {\sqrt {3}}{3} \right )}}{3} \end {gather*}

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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