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

Optimal. Leaf size=74 \[ \frac{\left (3+\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{10}}-\frac{1}{10} \sqrt{180-80 \sqrt{5}} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right ) \]

[Out]

-(Sqrt[180 - 80*Sqrt[5]]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x])/10 + ((3 + Sqrt[5])^(3/2)*ArcTan[Sqrt[(3 + Sqrt[5])/
2]*x])/(2*Sqrt[10])

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Rubi [A]  time = 0.0450255, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1166, 203} \[ \frac{\left (3+\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{10}}-\frac{1}{10} \sqrt{180-80 \sqrt{5}} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(3 + x^2)/(1 + 3*x^2 + x^4),x]

[Out]

-(Sqrt[180 - 80*Sqrt[5]]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x])/10 + ((3 + Sqrt[5])^(3/2)*ArcTan[Sqrt[(3 + Sqrt[5])/
2]*x])/(2*Sqrt[10])

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 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{3+x^2}{1+3 x^2+x^4} \, dx &=\frac{1}{10} \left (5-3 \sqrt{5}\right ) \int \frac{1}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx+\frac{1}{10} \left (5+3 \sqrt{5}\right ) \int \frac{1}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx\\ &=-\frac{1}{5} \sqrt{45-20 \sqrt{5}} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )+\frac{\left (3+\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{2 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.099855, size = 73, normalized size = 0.99 \[ \frac{\left (3+\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )-\left (3-\sqrt{5}\right )^{3/2} \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + x^2)/(1 + 3*x^2 + x^4),x]

[Out]

(-((3 - Sqrt[5])^(3/2)*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x]) + (3 + Sqrt[5])^(3/2)*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x])
/(2*Sqrt[10])

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Maple [B]  time = 0.064, size = 104, normalized size = 1.4 \begin{align*} 2\,{\frac{1}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) }+{\frac{6\,\sqrt{5}}{-10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) }+2\,{\frac{1}{2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{2+2\,\sqrt{5}}} \right ) }-{\frac{6\,\sqrt{5}}{10+10\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{2+2\,\sqrt{5}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(-2+2*5^(1/2))*arctan(4*x/(-2+2*5^(1/2)))+6/5*5^(1/2)/(-2+2*5^(1/2))*arctan(4*x/(-2+2*5^(1/2)))+2/(2+2*5^(1/
2))*arctan(4*x/(2+2*5^(1/2)))-6/5*5^(1/2)/(2+2*5^(1/2))*arctan(4*x/(2+2*5^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} + 3}{x^{4} + 3 \, x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.54818, size = 421, normalized size = 5.69 \begin{align*} \frac{2}{5} \, \sqrt{5} \sqrt{-4 \, \sqrt{5} + 9} \arctan \left (\frac{1}{4} \, \sqrt{2 \, x^{2} + \sqrt{5} + 3}{\left (\sqrt{5} \sqrt{2} + 3 \, \sqrt{2}\right )} \sqrt{-4 \, \sqrt{5} + 9} - \frac{1}{2} \,{\left (\sqrt{5} x + 3 \, x\right )} \sqrt{-4 \, \sqrt{5} + 9}\right ) + \frac{2}{5} \, \sqrt{5} \sqrt{4 \, \sqrt{5} + 9} \arctan \left (\frac{1}{4} \,{\left (\sqrt{2 \, x^{2} - \sqrt{5} + 3}{\left (\sqrt{5} \sqrt{2} - 3 \, \sqrt{2}\right )} - 2 \, \sqrt{5} x + 6 \, x\right )} \sqrt{4 \, \sqrt{5} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*sqrt(5)*sqrt(-4*sqrt(5) + 9)*arctan(1/4*sqrt(2*x^2 + sqrt(5) + 3)*(sqrt(5)*sqrt(2) + 3*sqrt(2))*sqrt(-4*sq
rt(5) + 9) - 1/2*(sqrt(5)*x + 3*x)*sqrt(-4*sqrt(5) + 9)) + 2/5*sqrt(5)*sqrt(4*sqrt(5) + 9)*arctan(1/4*(sqrt(2*
x^2 - sqrt(5) + 3)*(sqrt(5)*sqrt(2) - 3*sqrt(2)) - 2*sqrt(5)*x + 6*x)*sqrt(4*sqrt(5) + 9))

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Sympy [A]  time = 0.165928, size = 46, normalized size = 0.62 \begin{align*} 2 \left (\frac{\sqrt{5}}{5} + \frac{1}{2}\right ) \operatorname{atan}{\left (\frac{2 x}{-1 + \sqrt{5}} \right )} - 2 \left (\frac{1}{2} - \frac{\sqrt{5}}{5}\right ) \operatorname{atan}{\left (\frac{2 x}{1 + \sqrt{5}} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(sqrt(5)/5 + 1/2)*atan(2*x/(-1 + sqrt(5))) - 2*(1/2 - sqrt(5)/5)*atan(2*x/(1 + sqrt(5)))

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Giac [A]  time = 1.12001, size = 55, normalized size = 0.74 \begin{align*} \frac{1}{5} \,{\left (2 \, \sqrt{5} - 5\right )} \arctan \left (\frac{2 \, x}{\sqrt{5} + 1}\right ) + \frac{1}{5} \,{\left (2 \, \sqrt{5} + 5\right )} \arctan \left (\frac{2 \, x}{\sqrt{5} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(2*sqrt(5) - 5)*arctan(2*x/(sqrt(5) + 1)) + 1/5*(2*sqrt(5) + 5)*arctan(2*x/(sqrt(5) - 1))