3.81 \(\int \frac{1-x^2}{1+5 x^2+x^4} \, dx\)

Optimal. Leaf size=50 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (5+\sqrt{21}\right )} x\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{3}} \]

[Out]

-(ArcTan[Sqrt[2/(5 + Sqrt[21])]*x]/Sqrt[3]) + ArcTan[Sqrt[(5 + Sqrt[21])/2]*x]/Sqrt[3]

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Rubi [A]  time = 0.0398747, antiderivative size = 50, 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} \[ \frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (5+\sqrt{21}\right )} x\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^2)/(1 + 5*x^2 + x^4),x]

[Out]

-(ArcTan[Sqrt[2/(5 + Sqrt[21])]*x]/Sqrt[3]) + ArcTan[Sqrt[(5 + Sqrt[21])/2]*x]/Sqrt[3]

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+5 x^2+x^4} \, dx &=\frac{1}{6} \left (-3+\sqrt{21}\right ) \int \frac{1}{\frac{5}{2}-\frac{\sqrt{21}}{2}+x^2} \, dx-\frac{1}{6} \left (3+\sqrt{21}\right ) \int \frac{1}{\frac{5}{2}+\frac{\sqrt{21}}{2}+x^2} \, dx\\ &=-\frac{\tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\sqrt{\frac{1}{2} \left (5+\sqrt{21}\right )} x\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.12987, size = 87, normalized size = 1.74 \[ \frac{\left (7-\sqrt{21}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{5-\sqrt{21}}} x\right )}{\sqrt{42 \left (5-\sqrt{21}\right )}}+\frac{\left (-7-\sqrt{21}\right ) \tan ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{21}}} x\right )}{\sqrt{42 \left (5+\sqrt{21}\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x^2)/(1 + 5*x^2 + x^4),x]

[Out]

((7 - Sqrt[21])*ArcTan[Sqrt[2/(5 - Sqrt[21])]*x])/Sqrt[42*(5 - Sqrt[21])] + ((-7 - Sqrt[21])*ArcTan[Sqrt[2/(5
+ Sqrt[21])]*x])/Sqrt[42*(5 + Sqrt[21])]

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Maple [B]  time = 0.058, size = 136, normalized size = 2.7 \begin{align*}{\frac{2\,\sqrt{21}}{6\,\sqrt{7}-6\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}-2\,\sqrt{3}}} \right ) }-2\,{\frac{1}{2\,\sqrt{7}-2\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}-2\,\sqrt{3}}} \right ) }-{\frac{2\,\sqrt{21}}{6\,\sqrt{7}+6\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}+2\,\sqrt{3}}} \right ) }-2\,{\frac{1}{2\,\sqrt{7}+2\,\sqrt{3}}\arctan \left ( 4\,{\frac{x}{2\,\sqrt{7}+2\,\sqrt{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+1)/(x^4+5*x^2+1),x)

[Out]

2/3*21^(1/2)/(2*7^(1/2)-2*3^(1/2))*arctan(4*x/(2*7^(1/2)-2*3^(1/2)))-2/(2*7^(1/2)-2*3^(1/2))*arctan(4*x/(2*7^(
1/2)-2*3^(1/2)))-2/3*21^(1/2)/(2*7^(1/2)+2*3^(1/2))*arctan(4*x/(2*7^(1/2)+2*3^(1/2)))-2/(2*7^(1/2)+2*3^(1/2))*
arctan(4*x/(2*7^(1/2)+2*3^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{2} - 1}{x^{4} + 5 \, x^{2} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)/(x^4+5*x^2+1),x, algorithm="maxima")

[Out]

-integrate((x^2 - 1)/(x^4 + 5*x^2 + 1), x)

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Fricas [A]  time = 1.4109, size = 109, normalized size = 2.18 \begin{align*} \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x^{3} + 4 \, x\right )}\right ) - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)/(x^4+5*x^2+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.112025, size = 42, normalized size = 0.84 \begin{align*} - \frac{\sqrt{3} \left (2 \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )} - 2 \operatorname{atan}{\left (\frac{\sqrt{3} x^{3}}{3} + \frac{4 \sqrt{3} x}{3} \right )}\right )}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+1)/(x**4+5*x**2+1),x)

[Out]

-sqrt(3)*(2*atan(sqrt(3)*x/3) - 2*atan(sqrt(3)*x**3/3 + 4*sqrt(3)*x/3))/6

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Giac [A]  time = 1.1387, size = 35, normalized size = 0.7 \begin{align*} \frac{1}{6} \, \sqrt{3}{\left (\pi \mathrm{sgn}\left (x\right ) - 2 \, \arctan \left (\frac{\sqrt{3}{\left (x^{2} + 1\right )}}{3 \, x}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)/(x^4+5*x^2+1),x, algorithm="giac")

[Out]

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