Optimal. Leaf size=119 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac {\tanh ^{-1}\left (\frac {\left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d} \]
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Rubi [A] time = 0.02, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {395} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )}{x}\right )}{4\ 2^{5/6} d}-\frac {\tanh ^{-1}\left (\frac {\left (\sqrt [3]{3 x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d} \]
Antiderivative was successfully verified.
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Rule 395
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{-2+3 x^2} \left (-6 d+d x^2\right )} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2}+\sqrt [3]{-2+3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {6}}\right )}{4\ 2^{5/6} \sqrt {3} d}-\frac {\tanh ^{-1}\left (\frac {\left (\sqrt [3]{2}+\sqrt [3]{-2+3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt {3} x}\right )}{4\ 2^{5/6} \sqrt {3} d}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 136, normalized size = 1.14 \[ \frac {9 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {3 x^2}{2},\frac {x^2}{6}\right )}{d \left (x^2-6\right ) \sqrt [3]{3 x^2-2} \left (x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};\frac {3 x^2}{2},\frac {x^2}{6}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};\frac {3 x^2}{2},\frac {x^2}{6}\right )\right )+9 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {3 x^2}{2},\frac {x^2}{6}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{{\left (d x^{2} - 6 \, d\right )} {\left (3 \, x^{2} - 2\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 58.82, size = 1063, normalized size = 8.93 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{{\left (d x^{2} - 6 \, d\right )} {\left (3 \, x^{2} - 2\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ -\int \frac {1}{{\left (3\,x^2-2\right )}^{1/3}\,\left (6\,d-d\,x^2\right )} \,d x \]
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 \[ \frac {\int \frac {1}{x^{2} \sqrt [3]{3 x^{2} - 2} - 6 \sqrt [3]{3 x^{2} - 2}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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