Optimal. Leaf size=123 \[ \frac{\tan ^{-1}\left (\frac{\left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )^2}{3 \sqrt [6]{2} \sqrt{3} x}\right )}{4\ 2^{5/6} \sqrt{3} d}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )}{x}\right )}{4\ 2^{5/6} d}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{6}}\right )}{4\ 2^{5/6} \sqrt{3} d} \]
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Rubi [A] time = 0.0204762, antiderivative size = 123, normalized size of antiderivative = 1., 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 = {394} \[ \frac{\tan ^{-1}\left (\frac{\left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )^2}{3 \sqrt [6]{2} \sqrt{3} x}\right )}{4\ 2^{5/6} \sqrt{3} d}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )}{x}\right )}{4\ 2^{5/6} d}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{6}}\right )}{4\ 2^{5/6} \sqrt{3} d} \]
Antiderivative was successfully verified.
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Rule 394
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{x}{\sqrt{6}}\right )}{4\ 2^{5/6} \sqrt{3} d}+\frac{\tan ^{-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}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{2+3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d}\\ \end{align*}
Mathematica [C] time = 0.112846, size = 136, normalized size = 1.11 \[ -\frac{9 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right )}{d \left (x^2+6\right ) \sqrt [3]{3 x^2+2} \left (x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right )\right )-9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{2}+6\,d}{\frac{1}{\sqrt [3]{3\,{x}^{2}+2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x^{2} + 6 \, d\right )}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{x^{2} \sqrt [3]{3 x^{2} + 2} + 6 \sqrt [3]{3 x^{2} + 2}}\, dx}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x^{2} + 6 \, d\right )}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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