Optimal. Leaf size=21 \[ -x^2+\frac {2}{3} \left (1-x^2\right )^{3/2} \]
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Rubi [A]
time = 0.08, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6820, 6874,
267} \begin {gather*} \frac {2}{3} \left (1-x^2\right )^{3/2}-x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 6820
Rule 6874
Rubi steps
\begin {align*} \int x \left (-\sqrt {1-x}-\sqrt {1+x}\right ) \left (\sqrt {1-x}+\sqrt {1+x}\right ) \, dx &=-\int x \left (\sqrt {1-x}+\sqrt {1+x}\right )^2 \, dx\\ &=-\int \left (2 x+2 x \sqrt {1-x^2}\right ) \, dx\\ &=-x^2-2 \int x \sqrt {1-x^2} \, dx\\ &=-x^2+\frac {2}{3} \left (1-x^2\right )^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.19 \begin {gather*} -\frac {1}{3} (-1+x) (1+x) \left (3+2 \sqrt {1-x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 26, normalized size = 1.24
method | result | size |
default | \(-x^{2}-\frac {2 \sqrt {1-x}\, \sqrt {1+x}\, \left (x^{2}-1\right )}{3}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 17, normalized size = 0.81 \begin {gather*} -x^{2} + \frac {2}{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 = 0.35, size = 25, normalized size = 1.19 \begin {gather*} -x^{2} - \frac {2}{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 = 54.64, size = 144, normalized size = 6.86 \begin {gather*} \frac {x^{3}}{3} + x - \frac {\left (x + 1\right )^{3}}{3} + 4 \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 ) - 4 \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 = 3.04, size = 54, normalized size = 2.57 \begin {gather*} -{\left (x + 1\right )}^{2} - \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + 2 \, x + 2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.06, size = 25, normalized size = 1.19 \begin {gather*} -x^2-\frac {2\,\left (x^2-1\right )\,\sqrt {1-x}\,\sqrt {x+1}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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