Optimal. Leaf size=72 \[ \frac{\log \left (\sqrt [3]{a-b x^3}+\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}} \]
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Rubi [A] time = 0.0100193, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {239} \[ \frac{\log \left (\sqrt [3]{a-b x^3}+\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Rule 239
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{a-b x^3}} \, dx &=-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}}+\frac{\log \left (\sqrt [3]{b} x+\sqrt [3]{a-b x^3}\right )}{2 \sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.0695784, size = 116, normalized size = 1.61 \[ \frac{-\log \left (\frac{b^{2/3} x^2}{\left (a-b x^3\right )^{2/3}}-\frac{\sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}+1\right )+2 \log \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a-b x^3}}-1}{\sqrt{3}}\right )}{6 \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{-b{x}^{3}+a}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.56207, size = 772, normalized size = 10.72 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} b \sqrt{-\frac{1}{b^{\frac{2}{3}}}} \log \left (-3 \, b x^{3} - 3 \,{\left (-b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}} x^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (b^{\frac{4}{3}} x^{3} -{\left (-b x^{3} + a\right )}^{\frac{1}{3}} b x^{2} - 2 \,{\left (-b x^{3} + a\right )}^{\frac{2}{3}} b^{\frac{2}{3}} x\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} + 2 \, a\right ) + 2 \, b^{\frac{2}{3}} \log \left (\frac{b^{\frac{1}{3}} x +{\left (-b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) - b^{\frac{2}{3}} \log \left (\frac{b^{\frac{2}{3}} x^{2} -{\left (-b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{1}{3}} x +{\left (-b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right )}{6 \, b}, \frac{6 \, \sqrt{\frac{1}{3}} b^{\frac{2}{3}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (b^{\frac{1}{3}} x - 2 \,{\left (-b x^{3} + a\right )}^{\frac{1}{3}}\right )}}{b^{\frac{1}{3}} x}\right ) + 2 \, b^{\frac{2}{3}} \log \left (\frac{b^{\frac{1}{3}} x +{\left (-b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) - b^{\frac{2}{3}} \log \left (\frac{b^{\frac{2}{3}} x^{2} -{\left (-b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{1}{3}} x +{\left (-b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right )}{6 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.37568, size = 37, normalized size = 0.51 \begin{align*} \frac{x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{2 i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac{4}{3}\right )} \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 (-b x^{3} + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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