Integrand size = 15, antiderivative size = 32 \[ \int \frac {1}{\sqrt {(b-x) (-a+x)}} \, dx=-\arctan \left (\frac {a+b-2 x}{2 \sqrt {-a b+(a+b) x-x^2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1976, 635, 210} \[ \int \frac {1}{\sqrt {(b-x) (-a+x)}} \, dx=-\arctan \left (\frac {a+b-2 x}{2 \sqrt {x (a+b)-a b-x^2}}\right ) \]
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Rule 210
Rule 635
Rule 1976
Rubi steps \begin{align*} \text {integral}& = \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 ) \\ & = -\arctan \left (\frac {a+b-2 x}{2 \sqrt {-a b+(a+b) x-x^2}}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.72 \[ \int \frac {1}{\sqrt {(b-x) (-a+x)}} \, dx=\frac {2 \sqrt {b-x} \sqrt {-a+x} \arctan \left (\frac {\sqrt {-a+x}}{\sqrt {b-x}}\right )}{\sqrt {(a-x) (-b+x)}} \]
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Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88
method | result | size |
default | \(\arctan \left (\frac {x -\frac {b}{2}-\frac {a}{2}}{\sqrt {-a b +\left (a +b \right ) x -x^{2}}}\right )\) | \(28\) |
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none
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\sqrt {(b-x) (-a+x)}} \, dx=-\arctan \left (-\frac {\sqrt {-a b + {\left (a + b\right )} x - x^{2}} {\left (a + b - 2 \, x\right )}}{2 \, {\left (a b - {\left (a + b\right )} x + x^{2}\right )}}\right ) \]
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Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.22 \[ \int \frac {1}{\sqrt {(b-x) (-a+x)}} \, dx=\begin {cases} - i \log {\left (a + b - 2 x + 2 i \sqrt {- a b - x^{2} + x \left (a + b\right )} \right )} & \text {for}\: a b - \frac {\left (a + b\right )^{2}}{4} \neq 0 \\\frac {\left (- \frac {a}{2} - \frac {b}{2} + x\right ) \log {\left (- \frac {a}{2} - \frac {b}{2} + x \right )}}{\sqrt {- \left (- \frac {a}{2} - \frac {b}{2} + x\right )^{2}}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {1}{\sqrt {(b-x) (-a+x)}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (28) = 56\).
Time = 0.31 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {1}{\sqrt {(b-x) (-a+x)}} \, dx=\frac {1}{8} \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \arcsin \left (\frac {a + b - 2 \, x}{a - b}\right ) \mathrm {sgn}\left (-a + b\right ) - \frac {1}{4} \, \sqrt {-a b + a x + b x - x^{2}} {\left (a + b - 2 \, x\right )} \]
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Timed out. \[ \int \frac {1}{\sqrt {(b-x) (-a+x)}} \, dx=\int \frac {1}{\sqrt {-\left (a-x\right )\,\left (b-x\right )}} \,d x \]
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