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

Optimal. Leaf size=24 \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3-\sqrt{5}\right )} x\right ) \]

[Out]

ArcTan[Sqrt[(3 - Sqrt[5])/2]*x]/2

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Rubi [A]  time = 0.0524535, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{1}{2} \left (3-\sqrt{5}\right )} x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + Sqrt[5] - x^2 + Sqrt[5]*x^2)^(-1),x]

[Out]

ArcTan[Sqrt[(3 - Sqrt[5])/2]*x]/2

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Rubi in Sympy [A]  time = 2.00668, size = 42, normalized size = 1.75 \[ \frac{\operatorname{atan}{\left (\frac{x \sqrt{-1 + \sqrt{5}}}{\sqrt{1 + \sqrt{5}}} \right )}}{\sqrt{-1 + \sqrt{5}} \sqrt{1 + \sqrt{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x**2+5**(1/2)+x**2*5**(1/2)),x)

[Out]

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

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Mathematica [C]  time = 0.0357357, size = 39, normalized size = 1.62 \[ \frac{1}{4} i \log \left (-2 i x+\sqrt{5}+1\right )-\frac{1}{4} i \log \left (2 i x+\sqrt{5}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + Sqrt[5] - x^2 + Sqrt[5]*x^2)^(-1),x]

[Out]

(I/4)*Log[1 + Sqrt[5] - (2*I)*x] - (I/4)*Log[1 + Sqrt[5] + (2*I)*x]

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Maple [B]  time = 0.018, size = 32, normalized size = 1.3 \[ 4\,{\frac{1}{ \left ( \sqrt{5}-1 \right ) \left ( 2\,\sqrt{5}+2 \right ) }\arctan \left ( 4\,{\frac{x}{2\,\sqrt{5}+2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x^2+5^(1/2)+5^(1/2)*x^2),x)

[Out]

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

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Maxima [A]  time = 0.761452, size = 15, normalized size = 0.62 \[ \frac{1}{2} \, \arctan \left (\frac{1}{2} \, x{\left (\sqrt{5} - 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(5)*x^2 - x^2 + sqrt(5) + 1),x, algorithm="maxima")

[Out]

1/2*arctan(1/2*x*(sqrt(5) - 1))

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(5)*x^2 - x^2 + sqrt(5) + 1),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.611955, size = 14, normalized size = 0.58 \[ \frac{\operatorname{atan}{\left (x \left (- \frac{1}{2} + \frac{\sqrt{5}}{2}\right ) \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-x**2+5**(1/2)+x**2*5**(1/2)),x)

[Out]

atan(x*(-1/2 + sqrt(5)/2))/2

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GIAC/XCAS [A]  time = 0.262188, size = 18, normalized size = 0.75 \[ \frac{1}{2} \, \arctan \left (\frac{2 \, x}{\sqrt{5} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(5)*x^2 - x^2 + sqrt(5) + 1),x, algorithm="giac")

[Out]

1/2*arctan(2*x/(sqrt(5) + 1))