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

Optimal. Leaf size=10 \[ \frac{1}{2} \sin ^{-1}\left (x+\frac{1}{2}\right ) \]

[Out]

ArcSin[1/2 + x]/2

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Rubi [A]  time = 0.0146968, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{2} \sin ^{-1}\left (x+\frac{1}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[3 - 4*x - 4*x^2],x]

[Out]

ArcSin[1/2 + x]/2

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Rubi in Sympy [A]  time = 1.5281, size = 26, normalized size = 2.6 \[ \frac{\operatorname{atan}{\left (- \frac{- 8 x - 4}{4 \sqrt{- 4 x^{2} - 4 x + 3}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-4*x**2-4*x+3)**(1/2),x)

[Out]

atan(-(-8*x - 4)/(4*sqrt(-4*x**2 - 4*x + 3)))/2

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Mathematica [A]  time = 0.0100993, size = 14, normalized size = 1.4 \[ -\frac{1}{2} \sin ^{-1}\left (\frac{1}{2} (-2 x-1)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[3 - 4*x - 4*x^2],x]

[Out]

-ArcSin[(-1 - 2*x)/2]/2

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Maple [A]  time = 0.003, size = 7, normalized size = 0.7 \[{\frac{1}{2}\arcsin \left ({\frac{1}{2}}+x \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-4*x^2-4*x+3)^(1/2),x)

[Out]

1/2*arcsin(1/2+x)

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Maxima [A]  time = 0.804628, size = 11, normalized size = 1.1 \[ -\frac{1}{2} \, \arcsin \left (-x - \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-4*x^2 - 4*x + 3),x, algorithm="maxima")

[Out]

-1/2*arcsin(-x - 1/2)

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Fricas [A]  time = 0.228652, size = 28, normalized size = 2.8 \[ \frac{1}{2} \, \arctan \left (\frac{2 \, x + 1}{\sqrt{-4 \, x^{2} - 4 \, x + 3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-4*x^2 - 4*x + 3),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 4 x^{2} - 4 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-4*x**2-4*x+3)**(1/2),x)

[Out]

Integral(1/sqrt(-4*x**2 - 4*x + 3), x)

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GIAC/XCAS [A]  time = 0.211983, size = 8, normalized size = 0.8 \[ \frac{1}{2} \, \arcsin \left (x + \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-4*x^2 - 4*x + 3),x, algorithm="giac")

[Out]

1/2*arcsin(x + 1/2)