Optimal. Leaf size=38 \[ -\frac{x^4}{2}-\frac{2}{5} \left (1-x^2\right )^{5/2}+\frac{2}{3} \left (1-x^2\right )^{3/2} \]
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Rubi [A] time = 0.324781, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {6688, 6742, 266, 43} \[ -\frac{x^4}{2}-\frac{2}{5} \left (1-x^2\right )^{5/2}+\frac{2}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 6688
Rule 6742
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^3 \left (-\sqrt{1-x}-\sqrt{1+x}\right ) \left (\sqrt{1-x}+\sqrt{1+x}\right ) \, dx &=-\int x^3 \left (\sqrt{1-x}+\sqrt{1+x}\right )^2 \, dx\\ &=-\int \left (2 x^3+2 x^3 \sqrt{1-x^2}\right ) \, dx\\ &=-\frac{x^4}{2}-2 \int x^3 \sqrt{1-x^2} \, dx\\ &=-\frac{x^4}{2}-\operatorname{Subst}\left (\int \sqrt{1-x} x \, dx,x,x^2\right )\\ &=-\frac{x^4}{2}-\operatorname{Subst}\left (\int \left (\sqrt{1-x}-(1-x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac{x^4}{2}+\frac{2}{3} \left (1-x^2\right )^{3/2}-\frac{2}{5} \left (1-x^2\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0417771, size = 38, normalized size = 1. \[ -\frac{x^4}{2}-\frac{2}{5} \left (1-x^2\right )^{5/2}+\frac{2}{3} \left (1-x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 33, normalized size = 0.9 \begin{align*} -{\frac{{x}^{4}}{2}}-{\frac{ \left ( 2\,{x}^{2}-2 \right ) \left ( 3\,{x}^{2}+2 \right ) }{15}\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 = 1.6875, size = 42, normalized size = 1.11 \begin{align*} -\frac{1}{2} \, x^{4} + \frac{2}{5} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{2} + \frac{4}{15} \,{\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 = 1.22415, size = 81, normalized size = 2.13 \begin{align*} -\frac{1}{2} \, x^{4} - \frac{2}{15} \,{\left (3 \, x^{4} - x^{2} - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19235, size = 76, normalized size = 2. \begin{align*} -\frac{1}{2} \,{\left (x + 1\right )}^{4} + 2 \,{\left (x + 1\right )}^{3} - \frac{2}{15} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - 3 \,{\left (x + 1\right )}^{2} + 2 \, x + 2 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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