3.1162 \(\int \frac{1}{\sqrt{3-x} \sqrt{-2+x}} \, dx\)

Optimal. Leaf size=8 \[ -\sin ^{-1}(5-2 x) \]

[Out]

-ArcSin[5 - 2*x]

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Rubi [A]  time = 0.0041647, antiderivative size = 8, 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} \[ -\sin ^{-1}(5-2 x) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[3 - x]*Sqrt[-2 + x]),x]

[Out]

-ArcSin[5 - 2*x]

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{-2+x}} \, dx &=\int \frac{1}{\sqrt{-6+5 x-x^2}} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,5-2 x\right )\\ &=-\sin ^{-1}(5-2 x)\\ \end{align*}

Mathematica [A]  time = 0.0100673, size = 12, normalized size = 1.5 \[ -2 \sin ^{-1}\left (\sqrt{3-x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[3 - x]*Sqrt[-2 + x]),x]

[Out]

-2*ArcSin[Sqrt[3 - x]]

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Maple [B]  time = 0.003, size = 31, normalized size = 3.9 \begin{align*}{\arcsin \left ( 2\,x-5 \right ) \sqrt{ \left ( -2+x \right ) \left ( 3-x \right ) }{\frac{1}{\sqrt{3-x}}}{\frac{1}{\sqrt{-2+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3-x)^(1/2)/(-2+x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.48708, size = 8, normalized size = 1. \begin{align*} \arcsin \left (2 \, x - 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(2*x - 5)

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Fricas [B]  time = 1.58787, size = 88, normalized size = 11. \begin{align*} -\arctan \left (\frac{{\left (2 \, x - 5\right )} \sqrt{x - 2} \sqrt{-x + 3}}{2 \,{\left (x^{2} - 5 \, x + 6\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(1/2*(2*x - 5)*sqrt(x - 2)*sqrt(-x + 3)/(x^2 - 5*x + 6))

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Sympy [A]  time = 1.61958, size = 26, normalized size = 3.25 \begin{align*} \begin{cases} - 2 i \operatorname{acosh}{\left (\sqrt{x - 2} \right )} & \text{for}\: \left |{x - 2}\right | > 1 \\2 \operatorname{asin}{\left (\sqrt{x - 2} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08056, size = 11, normalized size = 1.38 \begin{align*} 2 \, \arcsin \left (\sqrt{x - 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arcsin(sqrt(x - 2))