Optimal. Leaf size=23 \[ \frac {x^2}{2}-\frac {1}{3} \left (1-x^2\right )^{3/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 267}
\begin {gather*} \frac {x^2}{2}-\frac {1}{3} \left (1-x^2\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 267
Rubi steps
\begin {align*} \int x \left (1+\sqrt {1-x} \sqrt {1+x}\right ) \, dx &=\int \left (x+x \sqrt {1-x^2}\right ) \, dx\\ &=\frac {x^2}{2}+\int x \sqrt {1-x^2} \, dx\\ &=\frac {x^2}{2}-\frac {1}{3} \left (1-x^2\right )^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{6} \left (-1+x^2\right ) \left (3+2 \sqrt {1-x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 26, normalized size = 1.13
method | result | size |
default | \(\frac {\sqrt {1-x}\, \sqrt {1+x}\, \left (x^{2}-1\right )}{3}+\frac {x^{2}}{2}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{3} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 25, normalized size = 1.09 \begin {gather*} \frac {1}{2} \, x^{2} + \frac {1}{3} \, {\left (x^{2} - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 53.32, size = 139, normalized size = 6.04 \begin {gather*} - x + \frac {\left (x + 1\right )^{2}}{2} - 2 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) + 2 \left (\begin {cases} \frac {x \sqrt {1 - x} \sqrt {x + 1}}{4} - \frac {\left (1 - x\right )^{\frac {3}{2}} \left (x + 1\right )^{\frac {3}{2}}}{6} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{2} & \text {for}\: \sqrt {x + 1} > - \sqrt {2} \wedge \sqrt {x + 1} < \sqrt {2} \end {cases}\right ) - 1 \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. 54 vs.
\(2 (17) = 34\).
time = 2.56, size = 54, normalized size = 2.35 \begin {gather*} \frac {1}{2} \, {\left (x + 1\right )}^{2} + \frac {1}{6} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{2} \, \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} - x - 1 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.69, size = 35, normalized size = 1.52 \begin {gather*} \frac {x^2}{2}-\frac {\sqrt {1-x}\,\left (-\frac {x^3}{3}-\frac {x^2}{3}+\frac {x}{3}+\frac {1}{3}\right )}{\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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