3.689 \(\int \frac{1}{\sqrt{-15-8 x-x^2}} \, dx\)

Optimal. Leaf size=4 \[ \sin ^{-1}(x+4) \]

[Out]

ArcSin[4 + x]

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Rubi [A]  time = 0.0113972, antiderivative size = 4, 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 \[ \sin ^{-1}(x+4) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-15 - 8*x - x^2],x]

[Out]

ArcSin[4 + x]

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Rubi in Sympy [A]  time = 0.705083, size = 24, normalized size = 6. \[ \operatorname{atan}{\left (- \frac{- 2 x - 8}{2 \sqrt{- x^{2} - 8 x - 15}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**2-8*x-15)**(1/2),x)

[Out]

atan(-(-2*x - 8)/(2*sqrt(-x**2 - 8*x - 15)))

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Mathematica [A]  time = 0.00934286, size = 4, normalized size = 1. \[ \sin ^{-1}(x+4) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[-15 - 8*x - x^2],x]

[Out]

ArcSin[4 + x]

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Maple [A]  time = 0.004, size = 5, normalized size = 1.3 \[ \arcsin \left ( 4+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^2-8*x-15)^(1/2),x)

[Out]

arcsin(4+x)

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Maxima [A]  time = 0.763192, size = 11, normalized size = 2.75 \[ -\arcsin \left (-x - 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-x^2 - 8*x - 15),x, algorithm="maxima")

[Out]

-arcsin(-x - 4)

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Fricas [A]  time = 0.272359, size = 23, normalized size = 5.75 \[ \arctan \left (\frac{x + 4}{\sqrt{-x^{2} - 8 \, x - 15}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-x^2 - 8*x - 15),x, algorithm="fricas")

[Out]

arctan((x + 4)/sqrt(-x^2 - 8*x - 15))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- x^{2} - 8 x - 15}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**2-8*x-15)**(1/2),x)

[Out]

Integral(1/sqrt(-x**2 - 8*x - 15), x)

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GIAC/XCAS [A]  time = 0.267783, size = 5, normalized size = 1.25 \[ \arcsin \left (x + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-x^2 - 8*x - 15),x, algorithm="giac")

[Out]

arcsin(x + 4)