3.1.12 \(\int \frac {1}{3+4 x^2+x^4} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{2} \tan ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{2 \sqrt {3}} \]

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1093, 203} \begin {gather*} \frac {1}{2} \tan ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{2 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 4*x^2 + x^4)^(-1),x]

[Out]

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

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])

Rule 1093

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} \frac {1}{2} \tan ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{2 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 4*x^2 + x^4)^(-1),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{3+4 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 4*x^2 + x^4)^(-1),x]

[Out]

IntegrateAlgebraic[(3 + 4*x^2 + x^4)^(-1), x]

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fricas [A]  time = 1.16, size = 17, normalized size = 0.71 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + \frac {1}{2} \, \arctan \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4+4*x^2+3),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 17, normalized size = 0.71 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + \frac {1}{2} \, \arctan \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4+4*x^2+3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 18, normalized size = 0.75 \begin {gather*} \frac {\arctan \relax (x )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{3}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4+4*x^2+3),x)

[Out]

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

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maxima [A]  time = 3.00, size = 17, normalized size = 0.71 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + \frac {1}{2} \, \arctan \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^4+4*x^2+3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.12, size = 17, normalized size = 0.71 \begin {gather*} \frac {\mathrm {atan}\relax (x)}{2}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,x}{3}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x^2 + x^4 + 3),x)

[Out]

atan(x)/2 - (3^(1/2)*atan((3^(1/2)*x)/3))/6

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sympy [A]  time = 0.16, size = 20, normalized size = 0.83 \begin {gather*} \frac {\operatorname {atan}{\relax (x )}}{2} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} \right )}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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