3.41 \(\int \frac{1+2 x^2}{1+6 x^2+4 x^4} \, dx\)

Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{\sqrt{10}}+\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{\sqrt{10}} \]

[Out]

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

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Rubi [A]  time = 0.0593532, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1163, 203} \[ \frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{\sqrt{10}}+\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x^2)/(1 + 6*x^2 + 4*x^4),x]

[Out]

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

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+2 x^2}{1+6 x^2+4 x^4} \, dx &=\frac{1}{5} \left (5-\sqrt{5}\right ) \int \frac{1}{3-\sqrt{5}+4 x^2} \, dx+\frac{1}{5} \left (5+\sqrt{5}\right ) \int \frac{1}{3+\sqrt{5}+4 x^2} \, dx\\ &=\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{\sqrt{10}}+\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{\sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0743086, size = 83, normalized size = 1.84 \[ \frac{\left (\sqrt{5}-1\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{2 \sqrt{5 \left (3-\sqrt{5}\right )}}+\frac{\left (1+\sqrt{5}\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{2 \sqrt{5 \left (3+\sqrt{5}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x^2)/(1 + 6*x^2 + 4*x^4),x]

[Out]

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

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Maple [B]  time = 0.075, size = 136, normalized size = 3. \begin{align*} -{\frac{2\,\sqrt{5}}{10\,\sqrt{10}-10\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) }+2\,{\frac{1}{2\,\sqrt{10}-2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) }+{\frac{2\,\sqrt{5}}{10\,\sqrt{10}+10\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}+2\,\sqrt{2}}} \right ) }+2\,{\frac{1}{2\,\sqrt{10}+2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}+2\,\sqrt{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2+1)/(4*x^4+6*x^2+1),x)

[Out]

-2/5*5^(1/2)/(2*10^(1/2)-2*2^(1/2))*arctan(8*x/(2*10^(1/2)-2*2^(1/2)))+2/(2*10^(1/2)-2*2^(1/2))*arctan(8*x/(2*
10^(1/2)-2*2^(1/2)))+2/5*5^(1/2)/(2*10^(1/2)+2*2^(1/2))*arctan(8*x/(2*10^(1/2)+2*2^(1/2)))+2/(2*10^(1/2)+2*2^(
1/2))*arctan(8*x/(2*10^(1/2)+2*2^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x^{2} + 1}{4 \, x^{4} + 6 \, x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+1)/(4*x^4+6*x^2+1),x, algorithm="maxima")

[Out]

integrate((2*x^2 + 1)/(4*x^4 + 6*x^2 + 1), x)

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Fricas [A]  time = 1.31568, size = 117, normalized size = 2.6 \begin{align*} \frac{1}{10} \, \sqrt{10} \arctan \left (\frac{2}{5} \, \sqrt{10}{\left (x^{3} + 2 \, x\right )}\right ) + \frac{1}{10} \, \sqrt{10} \arctan \left (\frac{1}{5} \, \sqrt{10} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+1)/(4*x^4+6*x^2+1),x, algorithm="fricas")

[Out]

1/10*sqrt(10)*arctan(2/5*sqrt(10)*(x^3 + 2*x)) + 1/10*sqrt(10)*arctan(1/5*sqrt(10)*x)

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Sympy [A]  time = 0.115144, size = 42, normalized size = 0.93 \begin{align*} \frac{\sqrt{10} \left (2 \operatorname{atan}{\left (\frac{\sqrt{10} x}{5} \right )} + 2 \operatorname{atan}{\left (\frac{2 \sqrt{10} x^{3}}{5} + \frac{4 \sqrt{10} x}{5} \right )}\right )}{20} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2+1)/(4*x**4+6*x**2+1),x)

[Out]

sqrt(10)*(2*atan(sqrt(10)*x/5) + 2*atan(2*sqrt(10)*x**3/5 + 4*sqrt(10)*x/5))/20

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Giac [A]  time = 1.17602, size = 53, normalized size = 1.18 \begin{align*} \frac{1}{10} \, \sqrt{10} \arctan \left (\frac{4 \, x}{\sqrt{10} + \sqrt{2}}\right ) + \frac{1}{10} \, \sqrt{10} \arctan \left (\frac{4 \, x}{\sqrt{10} - \sqrt{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+1)/(4*x^4+6*x^2+1),x, algorithm="giac")

[Out]

1/10*sqrt(10)*arctan(4*x/(sqrt(10) + sqrt(2))) + 1/10*sqrt(10)*arctan(4*x/(sqrt(10) - sqrt(2)))