Integrand size = 28, antiderivative size = 14 \[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=2 \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{\sqrt {3-x}}\right ),2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {119} \[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=2 \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{\sqrt {3-x}}\right ),2\right ) \]
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Rule 119
Rubi steps \begin{align*} \text {integral}& = 2 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {3-x}}\right )\right |2\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.19 (sec) , antiderivative size = 67, normalized size of antiderivative = 4.79 \[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=\frac {2 i \sqrt {\frac {-3+x}{-1+x}} \sqrt {\frac {-2+x}{-1+x}} (-1+x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {1}{\sqrt {1-x}}\right ),2\right )}{\sqrt {2-x} \sqrt {3-x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(13)=26\).
Time = 1.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.93
method | result | size |
default | \(-\frac {F\left (\sqrt {3-x}, \frac {\sqrt {2}}{2}\right ) \sqrt {-2+x}\, \sqrt {-2+2 x}\, \sqrt {2-x}\, \sqrt {2}\, \sqrt {2-2 x}}{2 \left (x^{2}-3 x +2\right )}\) | \(55\) |
elliptic | \(-\frac {\sqrt {-\left (-1+x \right ) \left (-2+x \right ) \left (-3+x \right )}\, \sqrt {-2+2 x}\, \sqrt {-2+x}\, F\left (\sqrt {3-x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {1-x}\, \sqrt {2-x}\, \sqrt {-x^{3}+6 x^{2}-11 x +6}}\) | \(72\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=-2 i \, {\rm weierstrassPInverse}\left (4, 0, x - 2\right ) \]
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Result contains complex when optimal does not.
Time = 11.82 (sec) , antiderivative size = 66, normalized size of antiderivative = 4.71 \[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=\frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {{G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {1}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \]
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\[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=\int { \frac {1}{\sqrt {-x + 3} \sqrt {-x + 2} \sqrt {-x + 1}} \,d x } \]
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\[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=\int { \frac {1}{\sqrt {-x + 3} \sqrt {-x + 2} \sqrt {-x + 1}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=\int \frac {1}{\sqrt {1-x}\,\sqrt {2-x}\,\sqrt {3-x}} \,d x \]
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