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

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 = 0.008832, 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, Rules used = {1981, 619, 216} \[ -\sin ^{-1}\left (\frac{1}{4} (-x-1)\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(3 - x)*(5 + x)],x]

[Out]

-ArcSin[(-1 - x)/4]

Rule 1981

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && QuadraticQ[u, x] &&  !QuadraticMatch
Q[u, x]

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) (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 = 0.002709, 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)*(5 + x)],x]

[Out]

-2*ArcSin[Sqrt[3 - x]/(2*Sqrt[2])]

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

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 = 2.10107, 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)*(5+x))^(1/2),x, algorithm="maxima")

[Out]

-arcsin(-1/4*x - 1/4)

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Fricas [B]  time = 1.45716, size = 77, normalized size = 6.42 \begin{align*} -\arctan \left (\frac{\sqrt{-x^{2} - 2 \, x + 15}{\left (x + 1\right )}}{x^{2} + 2 \, x - 15}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt((3 - x)*(x + 5)), x)

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Giac [A]  time = 1.32539, size = 8, normalized size = 0.67 \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)*(5+x))^(1/2),x, algorithm="giac")

[Out]

arcsin(1/4*x + 1/4)