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

Optimal. Leaf size=39 \[ \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )-\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0327162, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1163, 203} \[ \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x\right )-\tan ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)/(1 + 3*x^2 + x^4),x]

[Out]

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

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

Mathematica [A]  time = 0.006976, size = 10, normalized size = 0.26 \[ \tan ^{-1}\left (\frac{x}{x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[x/(1 + x^2)]

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Maple [B]  time = 0.054, size = 104, normalized size = 2.7 \begin{align*} -2\,{\frac{1}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) }+2\,{\frac{\sqrt{5}}{-2+2\,\sqrt{5}}\arctan \left ( 4\,{\frac{x}{-2+2\,\sqrt{5}}} \right ) }-2\,{\frac{\sqrt{5}}{2+2\,\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/(-2+2*5^(1/2))*arctan(4*x/(-2+2*5^(1/2)))+2*5^(1/2)/(-2+2*5^(1/2))*arctan(4*x/(-2+2*5^(1/2)))-2*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 = 1.2397, size = 42, normalized size = 1.08 \begin{align*} \arctan \left (x^{3} + 2 \, x\right ) - \arctan \left (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]

arctan(x^3 + 2*x) - arctan(x)

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Sympy [A]  time = 0.098433, size = 10, normalized size = 0.26 \begin{align*} - \operatorname{atan}{\left (x \right )} + \operatorname{atan}{\left (x^{3} + 2 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)

[Out]

-atan(x) + atan(x**3 + 2*x)

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Giac [A]  time = 1.15351, size = 35, normalized size = 0.9 \begin{align*} \frac{1}{4} \, \pi \mathrm{sgn}\left (x\right ) - \frac{1}{2} \, \arctan \left (\frac{x^{4} + x^{2} + 1}{2 \,{\left (x^{3} + 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]

1/4*pi*sgn(x) - 1/2*arctan(1/2*(x^4 + x^2 + 1)/(x^3 + x))