Integrand size = 13, antiderivative size = 135 \[ \int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx=\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{a} x}{\sqrt [3]{a} x+2 \sqrt [3]{-b+a x^3}}\right )}{\sqrt {3} \sqrt [3]{a}}-\frac {\log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{3 \sqrt [3]{a}}+\frac {\log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 \sqrt [3]{a}} \]
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Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.55, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {245} \[ \int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx=\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a}}-\frac {\log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{2 \sqrt [3]{a}} \]
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Rule 245
Rubi steps \begin{align*} \text {integral}& = \frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a}}-\frac {\log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{a}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (1-\frac {\sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )+\log \left (1+\frac {a^{2/3} x^2}{\left (-b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}\right )}{6 \sqrt [3]{a}} \]
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Time = 0.68 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (a^{\frac {1}{3}} x +2 \left (a \,x^{3}-b \right )^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}} x}\right )+\ln \left (\frac {-a^{\frac {1}{3}} x +\left (a \,x^{3}-b \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {a^{\frac {2}{3}} x^{2}+a^{\frac {1}{3}} x \left (a \,x^{3}-b \right )^{\frac {1}{3}}+\left (a \,x^{3}-b \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}}{3 a^{\frac {1}{3}}}\) | \(107\) |
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none
Time = 0.25 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.59 \[ \int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (-3 \, a x^{3} + 3 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} x^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (\left (-a\right )^{\frac {1}{3}} a x^{3} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} a x^{2} + 2 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} x\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} + 2 \, b\right ) - 2 \, \left (-a\right )^{\frac {2}{3}} \log \left (\frac {\left (-a\right )^{\frac {1}{3}} x + {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right ) + \left (-a\right )^{\frac {2}{3}} \log \left (\frac {\left (-a\right )^{\frac {2}{3}} x^{2} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} x + {\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a}, -\frac {6 \, \sqrt {\frac {1}{3}} a \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-a\right )^{\frac {1}{3}} x - 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}}{x}\right ) + 2 \, \left (-a\right )^{\frac {2}{3}} \log \left (\frac {\left (-a\right )^{\frac {1}{3}} x + {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right ) - \left (-a\right )^{\frac {2}{3}} \log \left (\frac {\left (-a\right )^{\frac {2}{3}} x^{2} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} x + {\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a}\right ] \]
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Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.27 \[ \int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx=\frac {x e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {a x^{3}}{b}} \right )}}{3 \sqrt [3]{b} \Gamma \left (\frac {4}{3}\right )} \]
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none
Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a^{\frac {1}{3}} + \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {1}{3}}} + \frac {\log \left (a^{\frac {2}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} a^{\frac {1}{3}}}{x} + \frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a^{\frac {1}{3}}} - \frac {\log \left (-a^{\frac {1}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}{3 \, a^{\frac {1}{3}}} \]
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\[ \int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx=\int { \frac {1}{{\left (a x^{3} - b\right )}^{\frac {1}{3}}} \,d x } \]
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Time = 6.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx=\frac {x\,{\left (1-\frac {a\,x^3}{b}\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ \frac {a\,x^3}{b}\right )}{{\left (a\,x^3-b\right )}^{1/3}} \]
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