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 = 0.04, antiderivative size = 86, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rule 1107
Rule 1903
Rule 1996
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*}
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Mathematica [A] time = 0.00, size = 70, normalized size = 0.81 \begin {gather*} \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 )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 73, normalized size = 0.85 \begin {gather*} \frac {\sqrt {x^6-3 x^4+3 x^2} \left (2 x^2-3\right )}{8 x}-\frac {3}{16} \log \left (-2 x^3+2 \sqrt {x^6-3 x^4+3 x^2}+3 x\right )+\frac {3 \log (x)}{16} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 70, normalized size = 0.81 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 69, normalized size = 0.80 \begin {gather*} \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}\relax (x) + \frac {3}{16} \, {\left (2 \, \sqrt {3} + \log \left (2 \, \sqrt {3} + 3\right )\right )} \mathrm {sgn}\relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 81, normalized size = 0.94 \begin {gather*} \frac {\sqrt {x^{6}-3 x^{4}+3 x^{2}}\, \left (4 \sqrt {x^{4}-3 x^{2}+3}\, x^{2}+3 \arcsinh \left (\frac {\sqrt {3}\, \left (2 x^{2}-3\right )}{3}\right )-6 \sqrt {x^{4}-3 x^{2}+3}\right )}{16 \sqrt {x^{4}-3 x^{2}+3}\, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {{\left (x^{2} - 1\right )}^{3} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {{\left (x^2-1\right )}^3+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {1 - \left (1 - x^{2}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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