Optimal. Leaf size=81 \[ \frac {1}{4} \tan ^{-1}\left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )-\frac {\tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {395} \[ \frac {1}{4} \tan ^{-1}\left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )-\frac {\tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 395
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{1-3 x^2} \left (3-x^2\right )} \, dx &=\frac {1}{4} \tan ^{-1}\left (\frac {1-\sqrt [3]{1-3 x^2}}{x}\right )+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {\left (1-\sqrt [3]{1-3 x^2}\right )^2}{3 \sqrt {3} x}\right )}{4 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 126, normalized size = 1.56 \[ -\frac {9 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};3 x^2,\frac {x^2}{3}\right )}{\sqrt [3]{1-3 x^2} \left (x^2-3\right ) \left (2 x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};3 x^2,\frac {x^2}{3}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};3 x^2,\frac {x^2}{3}\right )\right )+9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};3 x^2,\frac {x^2}{3}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 3.82, size = 1792, normalized size = 22.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 10.06, size = 538, normalized size = 6.64 \[ -\RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right ) \ln \left (\frac {-3 x^{2}+48 \left (-3 x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right )+96 x \RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right )+2 \left (-3 x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{2}-3\right )+4 x \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}+6 \left (-3 x^{2}+1\right )^{\frac {1}{3}}-3}{x^{2}-3}\right )-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {-3 x^{2}+48 \left (-3 x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right )+96 x \RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right )+2 \left (-3 x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{2}-3\right )+4 x \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}+6 \left (-3 x^{2}+1\right )^{\frac {1}{3}}-3}{x^{2}-3}\right )}{12}-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {12 x^{2} \RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right )+192 \left (-3 x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}-3\right )-384 x \RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}-3\right )+8 \left (-3 x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right )^{2}-16 x \RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right )^{2}+3 x^{2}-96 x \RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right )-4 x \RootOf \left (\textit {\_Z}^{2}-3\right )+24 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right )+12 \RootOf \left (48 \textit {\_Z}^{2}+4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}+3}{x^{2}-3}\right )}{12} \]
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 \[ -\int \frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ -\int \frac {1}{\left (x^2-3\right )\,{\left (1-3\,x^2\right )}^{1/3}} \,d x \]
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 \[ - \int \frac {1}{x^{2} \sqrt [3]{1 - 3 x^{2}} - 3 \sqrt [3]{1 - 3 x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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