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

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

[Out]

-ArcSin[(-1 - x)/4]

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Rubi [A]  time = 0.019837, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\sin ^{-1}\left (\frac{1}{4} (-x-1)\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[(3 - x)*(5 + x)],x]

[Out]

-ArcSin[(-1 - x)/4]

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Rubi in Sympy [A]  time = 0.772345, 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/((3-x)*(5+x))**(1/2),x)

[Out]

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

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Mathematica [B]  time = 0.0102718, size = 45, normalized size = 3.75 \[ \frac{2 \sqrt{x-3} \sqrt{x+5} \sinh ^{-1}\left (\frac{\sqrt{x-3}}{2 \sqrt{2}}\right )}{\sqrt{-(x-3) (x+5)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, 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/((3-x)*(5+x))^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.765043, 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 + 5)*(x - 3)),x, algorithm="maxima")

[Out]

-arcsin(-1/4*x - 1/4)

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Fricas [A]  time = 0.27316, 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 + 5)*(x - 3)),x, algorithm="fricas")

[Out]

arctan((x + 1)/sqrt(-x^2 - 2*x + 15))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (- x + 3\right ) \left (x + 5\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt((-x + 3)*(x + 5)), x)

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GIAC/XCAS [A]  time = 0.267677, 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 + 5)*(x - 3)),x, algorithm="giac")

[Out]

arcsin(1/4*x + 1/4)