Optimal. Leaf size=21 \[ \frac{2}{3} \left (1-x^2\right )^{3/2}-x^2 \]
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Rubi [A] time = 0.11288, antiderivative size = 21, normalized size of antiderivative = 1., 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 = {6688, 6742, 261} \[ \frac{2}{3} \left (1-x^2\right )^{3/2}-x^2 \]
Antiderivative was successfully verified.
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Rule 6688
Rule 6742
Rule 261
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*}
Mathematica [A] time = 0.0175248, size = 21, normalized size = 1. \[ \frac{2}{3} \left (1-x^2\right )^{3/2}-x^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 26, normalized size = 1.2 \begin{align*} -{x}^{2}-{\frac{2\,{x}^{2}-2}{3}\sqrt{1-x}\sqrt{1+x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53081, size = 23, normalized size = 1.1 \begin{align*} -x^{2} + \frac{2}{3} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.945034, size = 63, normalized size = 3. \begin{align*} -x^{2} - \frac{2}{3} \,{\left (x^{2} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 76.971, size = 110, normalized size = 5.24 \begin{align*} \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}\: x \geq -1 \wedge x < 1 \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}\: x \geq -1 \wedge x < 1 \end{cases}\right ) + 1 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11091, size = 39, normalized size = 1.86 \begin{align*} -\frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 1\right )} \sqrt{-x + 1} -{\left (x + 1\right )}^{2} + 2 \, x + 2 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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