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

Optimal. Leaf size=49 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt{5}}+\frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{\sqrt{5}} \]

[Out]

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

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Rubi [A]  time = 0.0618606, antiderivative size = 49, 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 = {1163, 203} \[ \frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt{5}}+\frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(1 + 3*x^2 + x^4),x]

[Out]

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

Rule 1163

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] && EqQ[c*d^2 - a*e^2, 0] && GtQ[b^2
- 4*a*c, 0]

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+x^2}{1+3 x^2+x^4} \, dx &=\frac{1}{10} \left (5-\sqrt{5}\right ) \int \frac{1}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx+\frac{1}{10} \left (5+\sqrt{5}\right ) \int \frac{1}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx\\ &=\frac{\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt{5}}+\frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )}{\sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0963296, size = 83, normalized size = 1.69 \[ \frac{\left (\sqrt{5}-1\right ) \tan ^{-1}\left (\sqrt{\frac{2}{3-\sqrt{5}}} x\right )}{\sqrt{10 \left (3-\sqrt{5}\right )}}+\frac{\left (1+\sqrt{5}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right )}{\sqrt{10 \left (3+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + Sqrt[5])*ArcTan[Sqrt[2/(3 - Sqrt[5])]*x])/Sqrt[10*(3 - Sqrt[5])] + ((1 + Sqrt[5])*ArcTan[Sqrt[2/(3 + Sq
rt[5])]*x])/Sqrt[10*(3 + Sqrt[5])]

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Maple [B]  time = 0.069, size = 104, normalized size = 2.1 \begin{align*} -{\frac{2\,\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{2\,\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*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)))+2/5*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)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.32716, size = 109, normalized size = 2.22 \begin{align*} \frac{1}{5} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5}{\left (x^{3} + 4 \, x\right )}\right ) + \frac{1}{5} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(5)*arctan(1/5*sqrt(5)*(x^3 + 4*x)) + 1/5*sqrt(5)*arctan(1/5*sqrt(5)*x)

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Sympy [A]  time = 0.109102, size = 41, normalized size = 0.84 \begin{align*} \frac{\sqrt{5} \left (2 \operatorname{atan}{\left (\frac{\sqrt{5} x}{5} \right )} + 2 \operatorname{atan}{\left (\frac{\sqrt{5} x^{3}}{5} + \frac{4 \sqrt{5} x}{5} \right )}\right )}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(5)*(2*atan(sqrt(5)*x/5) + 2*atan(sqrt(5)*x**3/5 + 4*sqrt(5)*x/5))/10

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Giac [A]  time = 1.13601, size = 35, normalized size = 0.71 \begin{align*} \frac{1}{10} \, \sqrt{5}{\left (\pi \mathrm{sgn}\left (x\right ) + 2 \, \arctan \left (\frac{\sqrt{5}{\left (x^{2} - 1\right )}}{5 \, x}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*sqrt(5)*(pi*sgn(x) + 2*arctan(1/5*sqrt(5)*(x^2 - 1)/x))