Optimal. Leaf size=129 \[ -\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {3 x^2+2}+2\ 2^{3/4}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {3 x^2+2}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 129, 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 = {397} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {3 x^2+2}+2\ 2^{3/4}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {3 x^2+2}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{2\ 2^{3/4} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 397
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{2+3 x^2} \left (4+3 x^2\right )} \, dx &=-\frac {\tan ^{-1}\left (\frac {2\ 2^{3/4}+2 \sqrt [4]{2} \sqrt {2+3 x^2}}{2 \sqrt {3} x \sqrt [4]{2+3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {2+3 x^2}}{2 \sqrt {3} x \sqrt [4]{2+3 x^2}}\right )}{2\ 2^{3/4} \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 135, normalized size = 1.05 \begin {gather*} -\frac {4 x F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {3 x^2}{2},-\frac {3 x^2}{4}\right )}{\sqrt [4]{3 x^2+2} \left (3 x^2+4\right ) \left (x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};-\frac {3 x^2}{2},-\frac {3 x^2}{4}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};-\frac {3 x^2}{2},-\frac {3 x^2}{4}\right )\right )-4 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};-\frac {3 x^2}{2},-\frac {3 x^2}{4}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.33, size = 137, normalized size = 1.06 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\frac {\sqrt {3} x^2}{2 \sqrt [4]{2}}-\frac {\sqrt [4]{2} \sqrt {3 x^2+2}}{\sqrt {3}}}{x \sqrt [4]{3 x^2+2}}\right )}{4\ 2^{3/4} \sqrt {3}}+\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4} \sqrt {3} x \sqrt [4]{3 x^2+2}}{3 \sqrt {2} x^2+4 \sqrt {3 x^2+2}}\right )}{4\ 2^{3/4} \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 9.14, size = 553, normalized size = 4.29 \begin {gather*} \frac {1}{72} \cdot 18^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {6 \cdot 18^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{3} + 54 \, x^{4} + 24 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}} x + 12 \, \sqrt {2} {\left (3 \, x^{2} + 4\right )} \sqrt {3 \, x^{2} + 2} + 72 \, x^{2} - {\left (18^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{3} - 4 \, x\right )} \sqrt {3 \, x^{2} + 2} + 72 \, {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{2} + 6 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{3} + 4 \, x\right )} + 48 \, \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {3 \, \sqrt {2} x^{2} + 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {3 \, x^{2} + 2}}{3 \, x^{2} + 4}}}{6 \, {\left (9 \, x^{4} - 24 \, x^{2} - 16\right )}}\right ) - \frac {1}{72} \cdot 18^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {6 \cdot 18^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{3} - 54 \, x^{4} + 24 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}} x - 12 \, \sqrt {2} {\left (3 \, x^{2} + 4\right )} \sqrt {3 \, x^{2} + 2} - 72 \, x^{2} - {\left (18^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{3} - 4 \, x\right )} \sqrt {3 \, x^{2} + 2} - 72 \, {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{2} + 6 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{3} + 4 \, x\right )} - 48 \, \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {3 \, \sqrt {2} x^{2} - 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {3 \, x^{2} + 2}}{3 \, x^{2} + 4}}}{6 \, {\left (9 \, x^{4} - 24 \, x^{2} - 16\right )}}\right ) + \frac {1}{288} \cdot 18^{\frac {3}{4}} \sqrt {2} \log \left (\frac {36 \, {\left (3 \, \sqrt {2} x^{2} + 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {3 \, x^{2} + 2}\right )}}{3 \, x^{2} + 4}\right ) - \frac {1}{288} \cdot 18^{\frac {3}{4}} \sqrt {2} \log \left (\frac {36 \, {\left (3 \, \sqrt {2} x^{2} - 2 \cdot 18^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 4 \, \sqrt {3 \, x^{2} + 2}\right )}}{3 \, 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} + 4\right )} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.67, size = 186, normalized size = 1.44 \begin {gather*} \frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+72\right )^{2}\right ) \ln \left (-\frac {3 x \RootOf \left (\textit {\_Z}^{4}+72\right )^{2}+\left (3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+72\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+72\right )^{2}-18 \sqrt {3 x^{2}+2}\, x -6 \left (3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+72\right )^{2}\right )}{3 x^{2}+4}\right )}{24}+\frac {\RootOf \left (\textit {\_Z}^{4}+72\right ) \ln \left (\frac {3 x \RootOf \left (\textit {\_Z}^{4}+72\right )^{2}+\left (3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+72\right )^{3}+18 \sqrt {3 x^{2}+2}\, x +6 \left (3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+72\right )}{3 x^{2}+4}\right )}{24} \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} + 4\right )} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}\,{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 (3\,x^2+2\right )}^{1/4}\,\left (3\,x^2+4\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}{\sqrt [4]{3 x^{2} + 2} \left (3 x^{2} + 4\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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