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

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

[Out]

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

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Rubi [A]  time = 0.0150571, antiderivative size = 24, normalized size of antiderivative = 1., 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} \[ \frac{1}{2} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

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 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]

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{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*}

Mathematica [A]  time = 0.0110083, size = 24, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}(x)-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

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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Maple [A]  time = 0.049, size = 18, normalized size = 0.8 \begin{align*}{\frac{\arctan \left ( x \right ) }{2}}-{\frac{\sqrt{3}}{6}\arctan \left ({\frac{x\sqrt{3}}{3}} \right ) } \end{align*}

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 = 1.44783, size = 23, normalized size = 0.96 \begin{align*} -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) + \frac{1}{2} \, \arctan \left (x\right ) \end{align*}

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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Fricas [A]  time = 1.34928, size = 70, normalized size = 2.92 \begin{align*} -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) + \frac{1}{2} \, \arctan \left (x\right ) \end{align*}

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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Sympy [A]  time = 0.145124, size = 20, normalized size = 0.83 \begin{align*} \frac{\operatorname{atan}{\left (x \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )}}{6} \end{align*}

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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Giac [A]  time = 1.16407, size = 23, normalized size = 0.96 \begin{align*} -\frac{1}{6} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) + \frac{1}{2} \, \arctan \left (x\right ) \end{align*}

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)