Optimal. Leaf size=32 \[ -\tan ^{-1}\left (\frac {a+b-2 x}{2 \sqrt {-a b+(a+b) x-x^2}}\right ) \]
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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 = {1976, 635, 210}
\begin {gather*} -\text {ArcTan}\left (\frac {a+b-2 x}{2 \sqrt {x (a+b)-a b-x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 635
Rule 1976
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 \text {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 [A]
time = 0.05, size = 55, normalized size = 1.72 \begin {gather*} -\frac {2 \sqrt {b-x} \sqrt {-a+x} \tan ^{-1}\left (\frac {\sqrt {b-x}}{\sqrt {-a+x}}\right )}{\sqrt {(b-x) (-a+x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 28, normalized size = 0.88
method | result | size |
default | \(\arctan \left (\frac {x -\frac {a}{2}-\frac {b}{2}}{\sqrt {-a b +\left (a +b \right ) x -x^{2}}}\right )\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 43, normalized size = 1.34 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (- a + x\right ) \left (b - x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (28) = 56\).
time = 4.79, size = 61, normalized size = 1.91 \begin {gather*} \frac {1}{8} \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \arcsin \left (\frac {a + b - 2 \, x}{a - b}\right ) \mathrm {sgn}\left (-a + b\right ) - \frac {1}{4} \, \sqrt {-a b + a x + b x - x^{2}} {\left (a + b - 2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {-\left (a-x\right )\,\left (b-x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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