Optimal. Leaf size=204 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c-3 d x^2}\right )}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{c-3 d x^2}+\sqrt [3]{c}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c}}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )}{2\ 2^{2/3} \sqrt{3} c^{5/6} \sqrt{d}} \]
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Rubi [A] time = 0.0287804, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {393} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c-3 d x^2}\right )}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{c-3 d x^2}+\sqrt [3]{c}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c}}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )}{2\ 2^{2/3} \sqrt{3} c^{5/6} \sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 393
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{c-3 d x^2} \left (c+d x^2\right )} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt{c}}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c-3 d x^2}\right )}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )}{2\ 2^{2/3} \sqrt{3} c^{5/6} \sqrt{d}}+\frac{\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{c-3 d x^2}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}\\ \end{align*}
Mathematica [C] time = 0.15142, size = 156, normalized size = 0.76 \[ \frac{3 c x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right )}{\sqrt [3]{c-3 d x^2} \left (c+d x^2\right ) \left (2 d x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right )\right )+3 c F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{3 d x^2}{c},-\frac{d x^2}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{2}+c}{\frac{1}{\sqrt [3]{-3\,d{x}^{2}+c}}}}\, 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} + c\right )}{\left (-3 \, d x^{2} + c\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*} \int \frac{1}{\sqrt [3]{c - 3 d x^{2}} \left (c + d x^{2}\right )}\, dx \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} + c\right )}{\left (-3 \, d x^{2} + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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