3.238 \(\int \frac{1}{\sqrt{1-x^2} \left (4+x^2\right )} \, dx\)

Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{5} x}{2 \sqrt{1-x^2}}\right )}{2 \sqrt{5}} \]

[Out]

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

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Rubi [A]  time = 0.0258338, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{5} x}{2 \sqrt{1-x^2}}\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[1 - x^2]*(4 + x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.47719, size = 24, normalized size = 0.77 \[ \frac{\sqrt{5} \operatorname{atan}{\left (\frac{\sqrt{5} x}{2 \sqrt{- x^{2} + 1}} \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2+4)/(-x**2+1)**(1/2),x)

[Out]

sqrt(5)*atan(sqrt(5)*x/(2*sqrt(-x**2 + 1)))/10

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Mathematica [A]  time = 0.0426013, size = 33, normalized size = 1.06 \[ -\frac{\tan ^{-1}\left (\frac{x \sqrt{5-5 x^2}}{2 \left (x^2-1\right )}\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 29, normalized size = 0.9 \[ -{\frac{\sqrt{5}}{10}\arctan \left ({\frac{x\sqrt{5}}{2\,{x}^{2}-2}\sqrt{-{x}^{2}+1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+4)/(-x^2+1)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 4\right )} \sqrt{-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 4)*sqrt(-x^2 + 1)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.211857, size = 65, normalized size = 2.1 \[ \frac{1}{10} \, \sqrt{5} \arctan \left (\frac{2 \,{\left (\sqrt{5}{\left (x^{2} - 1\right )} + \sqrt{5} \sqrt{-x^{2} + 1}\right )}}{5 \,{\left (\sqrt{-x^{2} + 1} x - x\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 4)*sqrt(-x^2 + 1)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 4\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2+4)/(-x**2+1)**(1/2),x)

[Out]

Integral(1/(sqrt(-(x - 1)*(x + 1))*(x**2 + 4)), x)

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GIAC/XCAS [A]  time = 0.221259, size = 69, normalized size = 2.23 \[ \frac{1}{20} \, \sqrt{5}{\left (\pi{\rm sign}\left (x\right ) + 2 \, \arctan \left (-\frac{\sqrt{5} x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{5 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + 4)*sqrt(-x^2 + 1)),x, algorithm="giac")

[Out]

1/20*sqrt(5)*(pi*sign(x) + 2*arctan(-1/5*sqrt(5)*x*((sqrt(-x^2 + 1) - 1)^2/x^2 -
1)/(sqrt(-x^2 + 1) - 1)))