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

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 = 0.0234365, antiderivative size = 12, 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 \[ -\sin ^{-1}\left (\frac{1}{4} (-x-1)\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcSin[(-1 - x)/4]

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Rubi in Sympy [A]  time = 1.51967, 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)**(1/2)/(5+x)**(1/2),x)

[Out]

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

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Mathematica [B]  time = 0.0190806, 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]*Sqrt[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 [B]  time = 0.009, size = 31, normalized size = 2.6 \[{1\sqrt{ \left ( 3-x \right ) \left ( 5+x \right ) }\arcsin \left ({\frac{1}{4}}+{\frac{x}{4}} \right ){\frac{1}{\sqrt{3-x}}}{\frac{1}{\sqrt{5+x}}}} \]

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(1/4+1/4*x)

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Maxima [A]  time = 0.819908, 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)*sqrt(-x + 3)),x, algorithm="maxima")

[Out]

-arcsin(-1/4*x - 1/4)

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Fricas [A]  time = 0.268892, size = 23, normalized size = 1.92 \[ \arctan \left (\frac{x + 1}{\sqrt{x + 5} \sqrt{-x + 3}}\right ) \]

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 + 1)/(sqrt(x + 5)*sqrt(-x + 3)))

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Sympy [A]  time = 3.79521, size = 41, normalized size = 3.42 \[ \begin{cases} - 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 5}}{4} \right )} & \text{for}\: \frac{\left |{x + 5}\right |}{8} > 1 \\2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 5}}{4} \right )} & \text{otherwise} \end{cases} \]

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)/4), Abs(x + 5)/8 > 1), (2*asin(sqrt(2)
*sqrt(x + 5)/4), True))

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GIAC/XCAS [A]  time = 0.269612, size = 18, normalized size = 1.5 \[ 2 \, \arcsin \left (\frac{1}{4} \, \sqrt{2} \sqrt{x + 5}\right ) \]

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(1/4*sqrt(2)*sqrt(x + 5))