Optimal. Leaf size=86 \[ -\frac{\sqrt{x^6-3 x^4+3 x^2} \left (3-2 x^2\right )}{8 x}-\frac{3 \sqrt{x^6-3 x^4+3 x^2} \sinh ^{-1}\left (\frac{3-2 x^2}{\sqrt{3}}\right )}{16 x \sqrt{x^4-3 x^2+3}} \]
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Rubi [A] time = 0.0411441, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {1996, 1903, 1107, 612, 619, 215} \[ -\frac{\sqrt{x^6-3 x^4+3 x^2} \left (3-2 x^2\right )}{8 x}-\frac{3 \sqrt{x^6-3 x^4+3 x^2} \sinh ^{-1}\left (\frac{3-2 x^2}{\sqrt{3}}\right )}{16 x \sqrt{x^4-3 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1996
Rule 1903
Rule 1107
Rule 612
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \sqrt{1-\left (1-x^2\right )^3} \, dx &=\int \sqrt{3 x^2-3 x^4+x^6} \, dx\\ &=\frac{\sqrt{3 x^2-3 x^4+x^6} \int x \sqrt{3-3 x^2+x^4} \, dx}{x \sqrt{3-3 x^2+x^4}}\\ &=\frac{\sqrt{3 x^2-3 x^4+x^6} \operatorname{Subst}\left (\int \sqrt{3-3 x+x^2} \, dx,x,x^2\right )}{2 x \sqrt{3-3 x^2+x^4}}\\ &=-\frac{\left (3-2 x^2\right ) \sqrt{3 x^2-3 x^4+x^6}}{8 x}+\frac{\left (3 \sqrt{3 x^2-3 x^4+x^6}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{3-3 x+x^2}} \, dx,x,x^2\right )}{16 x \sqrt{3-3 x^2+x^4}}\\ &=-\frac{\left (3-2 x^2\right ) \sqrt{3 x^2-3 x^4+x^6}}{8 x}+\frac{\left (\sqrt{3} \sqrt{3 x^2-3 x^4+x^6}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{3}}} \, dx,x,-3+2 x^2\right )}{16 x \sqrt{3-3 x^2+x^4}}\\ &=-\frac{\left (3-2 x^2\right ) \sqrt{3 x^2-3 x^4+x^6}}{8 x}-\frac{3 \sqrt{3 x^2-3 x^4+x^6} \sinh ^{-1}\left (\frac{3-2 x^2}{\sqrt{3}}\right )}{16 x \sqrt{3-3 x^2+x^4}}\\ \end{align*}
Mathematica [A] time = 0.0033421, size = 70, normalized size = 0.81 \[ \frac{x \left (4 x^6-18 x^4+30 x^2+3 \sqrt{x^4-3 x^2+3} \sinh ^{-1}\left (\frac{2 x^2-3}{\sqrt{3}}\right )-18\right )}{16 \sqrt{x^2 \left (x^4-3 x^2+3\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 81, normalized size = 0.9 \begin{align*}{\frac{1}{16\,x}\sqrt{{x}^{6}-3\,{x}^{4}+3\,{x}^{2}} \left ( 4\,\sqrt{{x}^{4}-3\,{x}^{2}+3}{x}^{2}-6\,\sqrt{{x}^{4}-3\,{x}^{2}+3}+3\,{\it Arcsinh} \left ( 1/3\,\sqrt{3} \left ( 2\,{x}^{2}-3 \right ) \right ) \right ){\frac{1}{\sqrt{{x}^{4}-3\,{x}^{2}+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{{\left (x^{2} - 1\right )}^{3} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29341, size = 157, normalized size = 1.83 \begin{align*} -\frac{12 \, x \log \left (-\frac{2 \, x^{3} - 3 \, x - 2 \, \sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}}}{x}\right ) - 8 \, \sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}}{\left (2 \, x^{2} - 3\right )} - 9 \, x}{64 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{1 - \left (1 - x^{2}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09643, size = 93, normalized size = 1.08 \begin{align*} \frac{1}{16} \,{\left (2 \, \sqrt{x^{4} - 3 \, x^{2} + 3}{\left (2 \, x^{2} - 3\right )} - 3 \, \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} - 3 \, x^{2} + 3} + 3\right )\right )} \mathrm{sgn}\left (x\right ) + \frac{3}{16} \,{\left (2 \, \sqrt{3} + \log \left (2 \, \sqrt{3} + 3\right )\right )} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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