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

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 = 0.0074626, 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, Rules used = {53, 619, 216} \[ -\sin ^{-1}\left (\frac{1}{4} (-x-1)\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[3 - x]*Sqrt[5 + x]),x]

[Out]

-ArcSin[(-1 - x)/4]

Rule 53

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{3-x} \sqrt{5+x}} \, dx &=\int \frac{1}{\sqrt{15-2 x-x^2}} \, dx\\ &=-\left (\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{64}}} \, dx,x,-2-2 x\right )\right )\\ &=-\sin ^{-1}\left (\frac{1}{4} (-1-x)\right )\\ \end{align*}

Mathematica [A]  time = 0.011606, size = 21, normalized size = 1.75 \[ -2 \sin ^{-1}\left (\frac{\sqrt{3-x}}{2 \sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*ArcSin[Sqrt[3 - x]/(2*Sqrt[2])]

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Maple [B]  time = 0.006, size = 31, normalized size = 2.6 \begin{align*}{\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}}}} \end{align*}

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 = 1.60581, size = 11, normalized size = 0.92 \begin{align*} -\arcsin \left (-\frac{1}{4} \, x - \frac{1}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-x)^(1/2)/(5+x)^(1/2),x, algorithm="maxima")

[Out]

-arcsin(-1/4*x - 1/4)

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Fricas [B]  time = 1.44129, size = 81, normalized size = 6.75 \begin{align*} -\arctan \left (\frac{\sqrt{x + 5}{\left (x + 1\right )} \sqrt{-x + 3}}{x^{2} + 2 \, x - 15}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-x)^(1/2)/(5+x)^(1/2),x, algorithm="fricas")

[Out]

-arctan(sqrt(x + 5)*(x + 1)*sqrt(-x + 3)/(x^2 + 2*x - 15))

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Sympy [B]  time = 1.01498, size = 41, normalized size = 3.42 \begin{align*} \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} \end{align*}

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 [B]  time = 1.14132, size = 18, normalized size = 1.5 \begin{align*} 2 \, \arcsin \left (\frac{1}{4} \, \sqrt{2} \sqrt{x + 5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-x)^(1/2)/(5+x)^(1/2),x, algorithm="giac")

[Out]

2*arcsin(1/4*sqrt(2)*sqrt(x + 5))