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

Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{\sqrt{2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0312964, antiderivative size = 46, 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} \[ \frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{\sqrt{2}} \]

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

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

Mathematica [A]  time = 0.0705627, size = 84, normalized size = 1.83 \[ \frac{-\left (\sqrt{5}-5\right ) \sqrt{3+\sqrt{5}} \tan ^{-1}\left (\frac{2 x}{\sqrt{3-\sqrt{5}}}\right )-\sqrt{3-\sqrt{5}} \left (5+\sqrt{5}\right ) \tan ^{-1}\left (\frac{2 x}{\sqrt{3+\sqrt{5}}}\right )}{4 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((-5 + Sqrt[5])*Sqrt[3 + Sqrt[5]]*ArcTan[(2*x)/Sqrt[3 - Sqrt[5]]]) - Sqrt[3 - Sqrt[5]]*(5 + Sqrt[5])*ArcTan[
(2*x)/Sqrt[3 + Sqrt[5]]])/(4*Sqrt[5])

________________________________________________________________________________________

Maple [B]  time = 0.054, size = 136, normalized size = 3. \begin{align*} -2\,{\frac{1}{2\,\sqrt{10}-2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) }+2\,{\frac{\sqrt{5}}{2\,\sqrt{10}-2\,\sqrt{2}}\arctan \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) }-2\,{\frac{\sqrt{5}}{2\,\sqrt{10}+2\,\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/(2*10^(1/2)-2*2^(1/2))*arctan(8*x/(2*10^(1/2)-2*2^(1/2)))+2*5^(1/2)/(2*10^(1/2)-2*2^(1/2))*arctan(8*x/(2*10
^(1/2)-2*2^(1/2)))-2*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)))

________________________________________________________________________________________

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)

________________________________________________________________________________________

Fricas [A]  time = 1.38737, size = 99, normalized size = 2.15 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (2 \, \sqrt{2}{\left (x^{3} + x\right )}\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (\sqrt{2} 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]

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

________________________________________________________________________________________

Sympy [A]  time = 0.113693, size = 39, normalized size = 0.85 \begin{align*} - \frac{\sqrt{2} \left (2 \operatorname{atan}{\left (\sqrt{2} x \right )} - 2 \operatorname{atan}{\left (2 \sqrt{2} x^{3} + 2 \sqrt{2} x \right )}\right )}{4} \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(2)*(2*atan(sqrt(2)*x) - 2*atan(2*sqrt(2)*x**3 + 2*sqrt(2)*x))/4

________________________________________________________________________________________

Giac [A]  time = 1.16846, size = 53, normalized size = 1.15 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{4 \, x}{\sqrt{10} + \sqrt{2}}\right ) + \frac{1}{2} \, \sqrt{2} \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]

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