Optimal. Leaf size=194 \[ -\frac {1}{8} \tan ^{-1}\left (\frac {-\frac {\sqrt [3]{1-3 x^2}}{\sqrt {3}}+x+\frac {1}{\sqrt {3}}}{\sqrt [3]{1-3 x^2}}\right )-\frac {1}{8} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{1-3 x^2}}{\sqrt {3}}+x-\frac {1}{\sqrt {3}}}{\sqrt [3]{1-3 x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{1-3 x^2}+1}\right )}{4 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt {3} x \sqrt [3]{1-3 x^2}-2 \sqrt {3} x}{3 x^2+4 \left (1-3 x^2\right )^{2/3}-2 \sqrt [3]{1-3 x^2}+1}\right )}{8 \sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 81, normalized size of antiderivative = 0.42, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {395} \begin {gather*} -\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}} \end {gather*}
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 = 0.65 \begin {gather*} \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 )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 3.97, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{1-3 x^2} \left (-3+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 2.23, size = 1790, normalized size = 9.23
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 \begin {gather*} \int \frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 11.37, size = 538, normalized size = 2.77
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (\frac {-192 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-3\right ) \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )^{2} x -8 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-3\right )^{2} \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x +384 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}-3\right ) x +16 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right )^{2} x +12 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+24 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}+12 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) \RootOf \left (\textit {\_Z}^{2}-3\right )+96 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x +4 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +3 x^{2}+3}{x^{2}-3}\right )}{12}-\frac {\ln \left (\frac {-48 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -2 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}-96 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -4 \RootOf \left (\textit {\_Z}^{2}-3\right ) x -3 x^{2}+6 \left (-3 x^{2}+1\right )^{\frac {1}{3}}-3}{x^{2}-3}\right ) \RootOf \left (\textit {\_Z}^{2}-3\right )}{12}-\ln \left (\frac {-48 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -2 \left (-3 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \left (-3 x^{2}+1\right )^{\frac {2}{3}}-96 \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right ) x -4 \RootOf \left (\textit {\_Z}^{2}-3\right ) x -3 x^{2}+6 \left (-3 x^{2}+1\right )^{\frac {1}{3}}-3}{x^{2}-3}\right ) \RootOf \left (4 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}-3\right )+48 \textit {\_Z}^{2}+1\right )\) | \(538\) |
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 \frac {1}{{\left (x^{2} - 3\right )} {\left (-3 \, x^{2} + 1\right )}^{\frac {1}{3}}}\,{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 \frac {1}{\left (x^2-3\right )\,{\left (1-3\,x^2\right )}^{1/3}} \,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 \frac {1}{\sqrt [3]{1 - 3 x^{2}} \left (x^{2} - 3\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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