Optimal. Leaf size=28 \[ \frac {1}{2} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{2} \sin ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 28, 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 = {38, 41, 222}
\begin {gather*} \frac {1}{2} \sqrt {1-x} \sqrt {x+1} x+\frac {1}{2} \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 222
Rubi steps
\begin {align*} \int \sqrt {1-x} \sqrt {1+x} \, dx &=\frac {1}{2} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {1}{2} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {1}{2} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 37, normalized size = 1.32 \begin {gather*} \frac {1}{2} x \sqrt {1-x^2}-\tan ^{-1}\left (\frac {\sqrt {1-x^2}}{1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 3.42, size = 106, normalized size = 3.79 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-x \sqrt {1+x}+x^2 \sqrt {1+x}-2 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ] \sqrt {-1+x}\right )}{2 \sqrt {-1+x}},\text {Abs}\left [1+x\right ]>2\right \}\right \},\text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]-\frac {\sqrt {1+x}}{\sqrt {1-x}}-\frac {\left (1+x\right )^{\frac {5}{2}}}{2 \sqrt {1-x}}+\frac {3 \left (1+x\right )^{\frac {3}{2}}}{2 \sqrt {1-x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs.
\(2(20)=40\).
time = 0.16, size = 57, normalized size = 2.04
method | result | size |
default | \(\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{2}-\frac {\sqrt {1-x}\, \sqrt {1+x}}{2}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(57\) |
risch | \(-\frac {x \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{2 \sqrt {1+x}\, \sqrt {1-x}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 17, normalized size = 0.61 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{2} + 1} x + \frac {1}{2} \, \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 38, normalized size = 1.36 \begin {gather*} \frac {1}{2} \, \sqrt {x + 1} x \sqrt {-x + 1} - \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 = 1.61, size = 131, normalized size = 4.68 \begin {gather*} \begin {cases} - i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {i \left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {x - 1}} - \frac {3 i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} + \frac {i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} + \frac {3 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} - \frac {\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs.
\(2 (20) = 40\).
time = 0.01, size = 103, normalized size = 3.68 \begin {gather*} 2 \left (2 \left (\frac {3}{8}-\frac {1}{8} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )}{2}\right )-2 \left (\frac {1}{2} \sqrt {-x+1} \sqrt {x+1}+\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 37, normalized size = 1.32 \begin {gather*} \frac {x\,\sqrt {1-x}\,\sqrt {x+1}}{2}-\frac {\ln \left (x-\sqrt {1-x}\,\sqrt {x+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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