Optimal. Leaf size=29 \[ -\sqrt {1-x} \left (2-\sqrt {x}\right )-\sin ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {1398, 785, 780, 216} \[ -\sqrt {1-x} \left (2-\sqrt {x}\right )-\sin ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 785
Rule 1398
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x}}{1+\sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \sqrt {1-x^2}}{1+x} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {(1-x) x}{\sqrt {1-x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\left (2-\sqrt {x}\right ) \sqrt {1-x}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\sqrt {x}\right )\\ &=-\left (2-\sqrt {x}\right ) \sqrt {1-x}-\sin ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 26, normalized size = 0.90 \[ \left (\sqrt {x}-2\right ) \sqrt {1-x}-\sin ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 33, normalized size = 1.14 \[ \sqrt {x} \sqrt {-x + 1} - 2 \, \sqrt {-x + 1} + \arctan \left (\frac {\sqrt {-x + 1}}{\sqrt {x}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 29, normalized size = 1.00 \[ \sqrt {x} \sqrt {-x + 1} - 2 \, \sqrt {-x + 1} + \arcsin \left (\sqrt {-x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 48, normalized size = 1.66 \[ -\frac {\sqrt {-x +1}\, \left (\arcsin \left (2 x -1\right )-2 \sqrt {-\left (x -1\right ) x}\right ) \sqrt {x}}{2 \sqrt {-\left (x -1\right ) x}}-2 \sqrt {-x +1} \]
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 \[ \int \frac {\sqrt {-x + 1}}{\sqrt {x} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.90, size = 39, normalized size = 1.34 \[ \sqrt {x}\,\sqrt {1-x}-2\,\sqrt {1-x}-2\,\mathrm {atan}\left (\frac {\sqrt {x}}{\sqrt {1-x}-1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.79, size = 32, normalized size = 1.10 \[ i \sqrt {x} \sqrt {x - 1} - 2 i \sqrt {x - 1} + i \operatorname {asinh}{\left (\sqrt {x - 1} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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