Optimal. Leaf size=45 \[ -\frac {\tanh ^{-1}\left (\frac {x \left (6-3 x^2\right )}{2 \sqrt {3} \sqrt {3 x^2-3 x^4+x^6}}\right )}{2 \sqrt {3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2021, 1918,
212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x \left (6-3 x^2\right )}{2 \sqrt {3} \sqrt {x^6-3 x^4+3 x^2}}\right )}{2 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 1918
Rule 2021
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-\left (1-x^2\right )^3}} \, dx &=\int \frac {1}{\sqrt {3 x^2-3 x^4+x^6}} \, dx\\ &=-\text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {x \left (6-3 x^2\right )}{\sqrt {3 x^2-3 x^4+x^6}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {x \left (6-3 x^2\right )}{2 \sqrt {3} \sqrt {3 x^2-3 x^4+x^6}}\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 66, normalized size = 1.47 \begin {gather*} \frac {x \sqrt {3-3 x^2+x^4} \tanh ^{-1}\left (\frac {x^2-\sqrt {3-3 x^2+x^4}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt {x^2 \left (3-3 x^2+x^4\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 58, normalized size = 1.29
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{3}+2 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +2 \sqrt {x^{6}-3 x^{4}+3 x^{2}}}{x^{3}}\right )}{6}\) | \(53\) |
default | \(\frac {x \sqrt {x^{4}-3 x^{2}+3}\, \sqrt {3}\, \arctanh \left (\frac {\left (x^{2}-2\right ) \sqrt {3}}{2 \sqrt {x^{4}-3 x^{2}+3}}\right )}{6 \sqrt {x^{6}-3 x^{4}+3 x^{2}}}\) | \(58\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 55, normalized size = 1.22 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (-\frac {3 \, x^{3} + 2 \, \sqrt {3} {\left (x^{3} - 2 \, x\right )} + 2 \, \sqrt {x^{6} - 3 \, x^{4} + 3 \, x^{2}} {\left (\sqrt {3} + 2\right )} - 6 \, x}{x^{3}}\right ) \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 \frac {1}{\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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Giac [A]
time = 5.52, size = 60, normalized size = 1.33 \begin {gather*} \frac {\sqrt {3} \log \left (x^{2} + \sqrt {3} - \sqrt {x^{4} - 3 \, x^{2} + 3}\right ) - \sqrt {3} \log \left (-x^{2} + \sqrt {3} + \sqrt {x^{4} - 3 \, x^{2} + 3}\right )}{6 \, \mathrm {sgn}\left (x\right )} \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 = -0.02 \begin {gather*} \int \frac {1}{\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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