Optimal. Leaf size=21 \[ -\sqrt {1-x} \sqrt {1+x}+\sin ^{-1}(x) \]
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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 = {52, 41, 222}
\begin {gather*} \text {ArcSin}(x)-\sqrt {1-x} \sqrt {x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 52
Rule 222
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 = 0.00, size = 34, normalized size = 1.62 \begin {gather*} -\sqrt {1-x^2}+2 \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs.
\(2(17)=34\).
time = 0.17, size = 42, normalized size = 2.00
method | result | size |
default | \(-\sqrt {1-x}\, \sqrt {1+x}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(42\) |
risch | \(\frac {\sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{\sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 14, normalized size = 0.67 \begin {gather*} -\sqrt {-x^{2} + 1} + \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (17) = 34\).
time = 0.56, 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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Sympy [C] Result contains complex when optimal does not.
time = 0.93, size = 99, normalized size = 4.71 \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}\: \left |{x + 1}\right | > 2 \\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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Giac [A]
time = 1.29, 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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Mupad [B]
time = 0.14, size = 14, normalized size = 0.67 \begin {gather*} \mathrm {asin}\left (x\right )-\sqrt {1-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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