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

Optimal. Leaf size=12 \[ -\sin ^{-1}\left (\frac{1}{4} (-x-1)\right ) \]

[Out]

-ArcSin[(-1 - x)/4]

_______________________________________________________________________________________

Rubi [A]  time = 0.016483, antiderivative size = 12, 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 \[ -\sin ^{-1}\left (\frac{1}{4} (-x-1)\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[15 - 2*x - x^2],x]

[Out]

-ArcSin[(-1 - x)/4]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 0.701885, size = 22, normalized size = 1.83 \[ \operatorname{atan}{\left (- \frac{- 2 x - 2}{2 \sqrt{- x^{2} - 2 x + 15}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00954637, size = 12, normalized size = 1. \[ -\sin ^{-1}\left (\frac{1}{4} (-x-1)\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcSin[(-1 - x)/4]

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 7, normalized size = 0.6 \[ \arcsin \left ({\frac{1}{4}}+{\frac{x}{4}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^2-2*x+15)^(1/2),x)

[Out]

arcsin(1/4+1/4*x)

_______________________________________________________________________________________

Maxima [A]  time = 0.79663, size = 11, normalized size = 0.92 \[ -\arcsin \left (-\frac{1}{4} \, x - \frac{1}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arcsin(-1/4*x - 1/4)

_______________________________________________________________________________________

Fricas [A]  time = 0.266934, size = 23, normalized size = 1.92 \[ \arctan \left (\frac{x + 1}{\sqrt{-x^{2} - 2 \, x + 15}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan((x + 1)/sqrt(-x^2 - 2*x + 15))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(-x**2 - 2*x + 15), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.268775, size = 8, normalized size = 0.67 \[ \arcsin \left (\frac{1}{4} \, x + \frac{1}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsin(1/4*x + 1/4)