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

Optimal. Leaf size=43 \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{\sqrt{6}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{\sqrt{6}} \]

[Out]

ArcTan[x/Sqrt[2 - Sqrt[3]]]/Sqrt[6] + ArcTan[x/Sqrt[2 + Sqrt[3]]]/Sqrt[6]

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Rubi [A]  time = 0.049938, antiderivative size = 43, 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} \[ \frac{\tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{\sqrt{6}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{\sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^2)/(1 + 4*x^2 + x^4),x]

[Out]

ArcTan[x/Sqrt[2 - Sqrt[3]]]/Sqrt[6] + ArcTan[x/Sqrt[2 + Sqrt[3]]]/Sqrt[6]

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+4 x^2+x^4} \, dx &=\frac{1}{6} \left (3-\sqrt{3}\right ) \int \frac{1}{2-\sqrt{3}+x^2} \, dx+\frac{1}{6} \left (3+\sqrt{3}\right ) \int \frac{1}{2+\sqrt{3}+x^2} \, dx\\ &=\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{\sqrt{6}}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{\sqrt{6}}\\ \end{align*}

Mathematica [A]  time = 0.0673384, size = 81, normalized size = 1.88 \[ \frac{\left (\sqrt{3}-1\right ) \tan ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}+\frac{\left (1+\sqrt{3}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + Sqrt[3])*ArcTan[x/Sqrt[2 - Sqrt[3]]])/(2*Sqrt[3*(2 - Sqrt[3])]) + ((1 + Sqrt[3])*ArcTan[x/Sqrt[2 + Sqrt
[3]]])/(2*Sqrt[3*(2 + Sqrt[3])])

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Maple [B]  time = 0.065, size = 110, normalized size = 2.6 \begin{align*} -{\frac{\sqrt{3}}{3\,\sqrt{6}-3\,\sqrt{2}}\arctan \left ( 2\,{\frac{x}{\sqrt{6}-\sqrt{2}}} \right ) }+{\frac{1}{\sqrt{6}-\sqrt{2}}\arctan \left ( 2\,{\frac{x}{\sqrt{6}-\sqrt{2}}} \right ) }+{\frac{\sqrt{3}}{3\,\sqrt{2}+3\,\sqrt{6}}\arctan \left ( 2\,{\frac{x}{\sqrt{2}+\sqrt{6}}} \right ) }+{\frac{1}{\sqrt{2}+\sqrt{6}}\arctan \left ( 2\,{\frac{x}{\sqrt{2}+\sqrt{6}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.3479, size = 109, normalized size = 2.53 \begin{align*} \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{6} \, \sqrt{6}{\left (x^{3} + 5 \, x\right )}\right ) + \frac{1}{6} \, \sqrt{6} \arctan \left (\frac{1}{6} \, \sqrt{6} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(6)*arctan(1/6*sqrt(6)*(x^3 + 5*x)) + 1/6*sqrt(6)*arctan(1/6*sqrt(6)*x)

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Sympy [A]  time = 0.108176, size = 41, normalized size = 0.95 \begin{align*} \frac{\sqrt{6} \left (2 \operatorname{atan}{\left (\frac{\sqrt{6} x}{6} \right )} + 2 \operatorname{atan}{\left (\frac{\sqrt{6} x^{3}}{6} + \frac{5 \sqrt{6} x}{6} \right )}\right )}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(6)*(2*atan(sqrt(6)*x/6) + 2*atan(sqrt(6)*x**3/6 + 5*sqrt(6)*x/6))/12

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Giac [A]  time = 1.11523, size = 35, normalized size = 0.81 \begin{align*} \frac{1}{12} \, \sqrt{6}{\left (\pi \mathrm{sgn}\left (x\right ) + 2 \, \arctan \left (\frac{\sqrt{6}{\left (x^{2} - 1\right )}}{6 \, x}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(6)*(pi*sgn(x) + 2*arctan(1/6*sqrt(6)*(x^2 - 1)/x))