3.690 \(\int \frac{1}{\sqrt{-3-x} \sqrt{5+x}} \, dx\)

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

[Out]

ArcSin[4 + x]

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Rubi [A]  time = 0.0197263, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \sin ^{-1}(x+4) \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[-3 - x]*Sqrt[5 + x]),x]

[Out]

ArcSin[4 + x]

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Rubi in Sympy [A]  time = 1.52303, 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/(-3-x)**(1/2)/(5+x)**(1/2),x)

[Out]

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

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Mathematica [B]  time = 0.0163908, size = 42, normalized size = 10.5 \[ \frac{2 \sqrt{x+3} \sqrt{x+5} \sinh ^{-1}\left (\frac{\sqrt{x+3}}{\sqrt{2}}\right )}{\sqrt{-(x+3) (x+5)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[-3 - x]*Sqrt[5 + x]),x]

[Out]

(2*Sqrt[3 + x]*Sqrt[5 + x]*ArcSinh[Sqrt[3 + x]/Sqrt[2]])/Sqrt[-((3 + x)*(5 + x))
]

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Maple [B]  time = 0.008, size = 29, normalized size = 7.3 \[{\arcsin \left ( 4+x \right ) \sqrt{ \left ( -3-x \right ) \left ( 5+x \right ) }{\frac{1}{\sqrt{-3-x}}}{\frac{1}{\sqrt{5+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3-x)^(1/2)/(5+x)^(1/2),x)

[Out]

((-3-x)*(5+x))^(1/2)/(-3-x)^(1/2)/(5+x)^(1/2)*arcsin(4+x)

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Maxima [A]  time = 0.754736, 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 + 5)*sqrt(-x - 3)),x, algorithm="maxima")

[Out]

-arcsin(-x - 4)

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Fricas [A]  time = 0.27047, size = 23, normalized size = 5.75 \[ \arctan \left (\frac{x + 4}{\sqrt{x + 5} \sqrt{-x - 3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 5)*sqrt(-x - 3)),x, algorithm="fricas")

[Out]

arctan((x + 4)/(sqrt(x + 5)*sqrt(-x - 3)))

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Sympy [A]  time = 3.75162, size = 41, normalized size = 10.25 \[ \begin{cases} - 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 5}}{2} \right )} & \text{for}\: \frac{\left |{x + 5}\right |}{2} > 1 \\2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 5}}{2} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-3-x)**(1/2)/(5+x)**(1/2),x)

[Out]

Piecewise((-2*I*acosh(sqrt(2)*sqrt(x + 5)/2), Abs(x + 5)/2 > 1), (2*asin(sqrt(2)
*sqrt(x + 5)/2), True))

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GIAC/XCAS [A]  time = 0.268886, size = 18, normalized size = 4.5 \[ 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 5)*sqrt(-x - 3)),x, algorithm="giac")

[Out]

2*arcsin(1/2*sqrt(2)*sqrt(x + 5))