Optimal. Leaf size=38 \[ \sqrt {\frac {1-x}{x+1}} (x+1)-2 \tan ^{-1}\left (\sqrt {\frac {1-x}{x+1}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 38, 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 = {1959, 288, 204} \[ \sqrt {\frac {1-x}{x+1}} (x+1)-2 \tan ^{-1}\left (\sqrt {\frac {1-x}{x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 288
Rule 1959
Rubi steps
\begin {align*} \int \sqrt {\frac {1-x}{1+x}} \, dx &=-\left (4 \operatorname {Subst}\left (\int \frac {x^2}{\left (-1-x^2\right )^2} \, dx,x,\sqrt {\frac {1-x}{1+x}}\right )\right )\\ &=\sqrt {\frac {1-x}{1+x}} (1+x)+2 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {\frac {1-x}{1+x}}\right )\\ &=\sqrt {\frac {1-x}{1+x}} (1+x)-2 \tan ^{-1}\left (\sqrt {\frac {1-x}{1+x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 67, normalized size = 1.76 \[ \frac {\sqrt {\frac {1-x}{x+1}} \sqrt {x+1} \left (\sqrt {x+1} (x-1)+2 \sqrt {1-x} \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )\right )}{x-1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 32, normalized size = 0.84 \[ {\left (x + 1\right )} \sqrt {-\frac {x - 1}{x + 1}} - 2 \, \arctan \left (\sqrt {-\frac {x - 1}{x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 29, normalized size = 0.76 \[ \frac {1}{2} \, \pi \mathrm {sgn}\left (x + 1\right ) + \arcsin \relax (x) \mathrm {sgn}\left (x + 1\right ) + \sqrt {-x^{2} + 1} \mathrm {sgn}\left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 39, normalized size = 1.03 \[ \frac {\sqrt {-\frac {x -1}{x +1}}\, \left (x +1\right ) \left (\arcsin \relax (x )+\sqrt {-x^{2}+1}\right )}{\sqrt {-\left (x -1\right ) \left (x +1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.85, size = 43, normalized size = 1.13 \[ -\frac {2 \, \sqrt {-\frac {x - 1}{x + 1}}}{\frac {x - 1}{x + 1} - 1} - 2 \, \arctan \left (\sqrt {-\frac {x - 1}{x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 43, normalized size = 1.13 \[ -2\,\mathrm {atan}\left (\sqrt {-\frac {x-1}{x+1}}\right )-\frac {2\,\sqrt {-\frac {x-1}{x+1}}}{\frac {x-1}{x+1}-1} \]
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 \[ \int \sqrt {\frac {1 - x}{x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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