Optimal. Leaf size=32 \[ -\tan ^{-1}\left (\frac{a+b-2 x}{2 \sqrt{x (a+b)-a b-x^2}}\right ) \]
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Rubi [A] time = 0.0130767, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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.
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Rule 1981
Rule 621
Rule 204
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*}
Mathematica [B] time = 0.0260302, 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.
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Maple [A] time = 0.003, size = 28, normalized size = 0.9 \begin{align*} \arctan \left ({ \left ( x-{\frac{b}{2}}-{\frac{a}{2}} \right ){\frac{1}{\sqrt{-ab+ \left ( a+b \right ) x-{x}^{2}}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34545, size = 111, normalized size = 3.47 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (- a + x\right ) \left (b - x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24874, size = 30, normalized size = 0.94 \begin{align*} \arcsin \left (\frac{a + b - 2 \, x}{a - b}\right ) \mathrm{sgn}\left (-a + b\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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