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

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

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\sin ^{-1}(5-2 x) \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.50 \begin {gather*} -2 \sin ^{-1}\left (\sqrt {3-x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*ArcSin[Sqrt[3 - x]]

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IntegrateAlgebraic [B]  time = 0.04, size = 20, normalized size = 2.50 \begin {gather*} -2 \tan ^{-1}\left (\frac {\sqrt {3-x}}{\sqrt {x-2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*ArcTan[Sqrt[3 - x]/Sqrt[-2 + x]]

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fricas [B]  time = 1.35, size = 32, normalized size = 4.00 \begin {gather*} -\arctan \left (\frac {{\left (2 \, x - 5\right )} \sqrt {x - 2} \sqrt {-x + 3}}{2 \, {\left (x^{2} - 5 \, x + 6\right )}}\right ) \end {gather*}

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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giac [A]  time = 1.02, size = 8, normalized size = 1.00 \begin {gather*} 2 \, \arcsin \left (\sqrt {x - 2}\right ) \end {gather*}

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))

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maple [B]  time = 0.00, size = 31, normalized size = 3.88 \begin {gather*} \frac {\sqrt {\left (x -2\right ) \left (-x +3\right )}\, \arcsin \left (2 x -5\right )}{\sqrt {x -2}\, \sqrt {-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.00, size = 6, normalized size = 0.75 \begin {gather*} \arcsin \left (2 \, x - 5\right ) \end {gather*}

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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mupad [B]  time = 0.18, size = 31, normalized size = 3.88 \begin {gather*} -4\,\mathrm {atan}\left (\frac {\sqrt {x-2}-\sqrt {2}\,1{}\mathrm {i}}{\sqrt {3}-\sqrt {3-x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*atan(((x - 2)^(1/2) - 2^(1/2)*1i)/(3^(1/2) - (3 - x)^(1/2)))

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sympy [A]  time = 1.61, size = 26, normalized size = 3.25 \begin {gather*} \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 {gather*}

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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