3.105 \(\int \frac {1}{\sqrt {(b-x) (-a+x)}} \, dx\)

Optimal. Leaf size=32 \[ -\tan ^{-1}\left (\frac {a+b-2 x}{2 \sqrt {x (a+b)-a b-x^2}}\right ) \]

[Out]

-arctan(1/2*(a+b-2*x)/(-a*b+(a+b)*x-x^2)^(1/2))

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1981, 621, 204} \[ -\tan ^{-1}\left (\frac {a+b-2 x}{2 \sqrt {x (a+b)-a b-x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(b - x)*(-a + x)],x]

[Out]

-ArcTan[(a + b - 2*x)/(2*Sqrt[-(a*b) + (a + b)*x - x^2])]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 621

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

Rule 1981

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {(b-x) (-a+x)}} \, dx &=\int \frac {1}{\sqrt {-a b+(a+b) x-x^2}} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {a+b-2 x}{\sqrt {-a b+(a+b) x-x^2}}\right )\\ &=-\tan ^{-1}\left (\frac {a+b-2 x}{2 \sqrt {-a b+(a+b) x-x^2}}\right )\\ \end {align*}

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Mathematica [B]  time = 0.03, size = 72, normalized size = 2.25 \[ -\frac {2 \sqrt {a-b} \sqrt {b-x} \sqrt {\frac {a-x}{a-b}} \sinh ^{-1}\left (\frac {\sqrt {b-x}}{\sqrt {a-b}}\right )}{\sqrt {(a-x) (x-b)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(b - x)*(-a + x)],x]

[Out]

(-2*Sqrt[a - b]*Sqrt[(a - x)/(a - b)]*Sqrt[b - x]*ArcSinh[Sqrt[b - x]/Sqrt[a - b]])/Sqrt[(a - x)*(-b + x)]

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fricas [A]  time = 0.39, size = 43, normalized size = 1.34 \[ -\arctan \left (-\frac {\sqrt {-a b + {\left (a + b\right )} x - x^{2}} {\left (a + b - 2 \, x\right )}}{2 \, {\left (a b - {\left (a + b\right )} x + x^{2}\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.12, size = 22, normalized size = 0.69 \[ \arcsin \left (\frac {a + b - 2 \, x}{a - b}\right ) \mathrm {sgn}\left (-a + b\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((b-x)*(-a+x))^(1/2),x, algorithm="giac")

[Out]

arcsin((a + b - 2*x)/(a - b))*sgn(-a + b)

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maple [A]  time = 0.00, size = 28, normalized size = 0.88 \[ \arctan \left (\frac {-\frac {a}{2}-\frac {b}{2}+x}{\sqrt {-a b -x^{2}+\left (a +b \right ) x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b-x)*(-a+x))^(1/2),x)

[Out]

arctan((-1/2*a-1/2*b+x)/(-a*b-x^2+(a+b)*x)^(1/2))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((b-x)*(-a+x))^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is b-a zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {-\left (a-x\right )\,\left (b-x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-(a - x)*(b - x))^(1/2),x)

[Out]

int(1/(-(a - x)*(b - x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\left (- a + x\right ) \left (b - x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((b-x)*(-a+x))**(1/2),x)

[Out]

Integral(1/sqrt((-a + x)*(b - x)), x)

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