Optimal. Leaf size=21 \[ \sin ^{-1}(x)-\sqrt {1-x} \sqrt {x+1} \]
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Rubi [A] time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {50, 41, 216} \begin {gather*} \sin ^{-1}(x)-\sqrt {1-x} \sqrt {x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 50
Rule 216
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx &=-\sqrt {1-x} \sqrt {1+x}+\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\sqrt {1-x} \sqrt {1+x}+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\sqrt {1-x} \sqrt {1+x}+\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.52 \begin {gather*} -\sqrt {1-x^2}-2 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.09, size = 45, normalized size = 2.14 \begin {gather*} -\sqrt {1-x} \sqrt {x+1}+2 i \log \left (\sqrt {1-x}-i \sqrt {x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.11, size = 37, normalized size = 1.76 \begin {gather*} -\sqrt {x + 1} \sqrt {-x + 1} - 2 \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 28, normalized size = 1.33 \begin {gather*} -\sqrt {x + 1} \sqrt {-x + 1} + 2 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 42, normalized size = 2.00 \begin {gather*} \frac {\sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {x +1}\, \sqrt {-x +1}}-\sqrt {-x +1}\, \sqrt {x +1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.90, size = 14, normalized size = 0.67 \begin {gather*} -\sqrt {-x^{2} + 1} + \arcsin \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 14, normalized size = 0.67 \begin {gather*} \mathrm {asin}\relax (x)-\sqrt {1-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.84, size = 100, normalized size = 4.76 \begin {gather*} \begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{\sqrt {x - 1}} + \frac {2 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{\sqrt {1 - x}} - \frac {2 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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