Optimal. Leaf size=12 \[ -2 E\left (\left .\sin ^{-1}\left (\sqrt {-x}\right )\right |-1\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {111}
\begin {gather*} -2 E\left (\left .\text {ArcSin}\left (\sqrt {-x}\right )\right |-1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 111
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x}}{\sqrt {-x} \sqrt {1+x}} \, dx &=-2 E\left (\left .\sin ^{-1}\left (\sqrt {-x}\right )\right |-1\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.48, size = 66, normalized size = 5.50 \begin {gather*} -\frac {2 x \sqrt {1-x^2} \left (-3 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^2\right )+x \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};x^2\right )\right )}{3 \sqrt {1-x} \sqrt {-x (1+x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 17, normalized size = 1.42
method | result | size |
default | \(2 \sqrt {2}\, \EllipticE \left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )\) | \(17\) |
elliptic | \(\frac {\sqrt {x \left (x^{2}-1\right )}\, \left (\frac {\sqrt {x +1}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}-\frac {\sqrt {x +1}\, \sqrt {2-2 x}\, \sqrt {-x}\, \left (-2 \EllipticE \left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {x^{3}-x}}\right )}{\sqrt {-x}\, \sqrt {1-x}\, \sqrt {x +1}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.54, size = 16, normalized size = 1.33 \begin {gather*} 2 \, {\rm weierstrassPInverse}\left (4, 0, x\right ) + 2 \, {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, x\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 {\sqrt {1 - x}}{\sqrt {- x} \sqrt {x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.08 \begin {gather*} \int \frac {\sqrt {1-x}}{\sqrt {-x}\,\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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