Optimal. Leaf size=61 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt {6}} \]
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Rubi [A] time = 0.01, antiderivative size = 61, 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 = {398} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt {6}} \end {gather*}
Antiderivative was successfully verified.
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Rule 398
Rubi steps
\begin {align*} \int \frac {1}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )}{2 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )}{2 \sqrt {6}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 127, normalized size = 2.08 \begin {gather*} \frac {2 x F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};3 x^2,\frac {3 x^2}{2}\right )}{\left (3 x^2-2\right ) \sqrt [4]{3 x^2-1} \left (x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};3 x^2,\frac {3 x^2}{2}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};3 x^2,\frac {3 x^2}{2}\right )\right )+2 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};3 x^2,\frac {3 x^2}{2}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.12, size = 63, normalized size = 1.03 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {\frac {2}{3}} \sqrt [4]{3 x^2-1}}{x}\right )}{2 \sqrt {6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )}{2 \sqrt {6}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 9.42, size = 104, normalized size = 1.70 \begin {gather*} \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {\sqrt {6} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}}}{3 \, x}\right ) + \frac {1}{24} \, \sqrt {6} \log \left (-\frac {9 \, x^{4} - 6 \, \sqrt {6} {\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 12 \, \sqrt {3 \, x^{2} - 1} x^{2} - 4 \, \sqrt {6} {\left (3 \, x^{2} - 1\right )}^{\frac {3}{4}} x + 12 \, x^{2} - 4}{9 \, x^{4} - 12 \, x^{2} + 4}\right ) \end {gather*}
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 (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.14, size = 138, normalized size = 2.26 \begin {gather*} \frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {-3 \sqrt {3 x^{2}-1}\, x -3 x +\left (3 x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}-6\right )+\left (3 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}-6\right )}{3 x^{2}-2}\right )}{12}-\frac {\RootOf \left (\textit {\_Z}^{2}+6\right ) \ln \left (\frac {3 \sqrt {3 x^{2}-1}\, x -3 x +\left (3 x^{2}-1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+6\right )-\left (3 x^{2}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+6\right )}{3 x^{2}-2}\right )}{12} \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 \frac {1}{{\left (3 \, x^{2} - 1\right )}^{\frac {1}{4}} {\left (3 \, x^{2} - 2\right )}}\,{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 = -0.02 \begin {gather*} \int \frac {1}{{\left (3\,x^2-1\right )}^{1/4}\,\left (3\,x^2-2\right )} \,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}{\left (3 x^{2} - 2\right ) \sqrt [4]{3 x^{2} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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