Integrand size = 15, antiderivative size = 82 \[ \int \frac {1}{x \sqrt [3]{-a+b x}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{-a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}+\frac {\log (x)}{2 \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{-a+b x}\right )}{2 \sqrt [3]{a}} \]
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Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {58, 631, 210, 31} \[ \int \frac {1}{x \sqrt [3]{-a+b x}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b x-a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{2 \sqrt [3]{a}}+\frac {\log (x)}{2 \sqrt [3]{a}} \]
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Rule 31
Rule 58
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = \frac {\log (x)}{2 \sqrt [3]{a}}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{-a+b x}\right )-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+x} \, dx,x,\sqrt [3]{-a+b x}\right )}{2 \sqrt [3]{a}} \\ & = \frac {\log (x)}{2 \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{-a+b x}\right )}{2 \sqrt [3]{a}}+\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{-a+b x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a}} \\ & = -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{-a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {\log (x)}{2 \sqrt [3]{a}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{-a+b x}\right )}{2 \sqrt [3]{a}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x \sqrt [3]{-a+b x}} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{-a+b x}\right )+\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{-a+b x}+(-a+b x)^{2/3}\right )}{2 \sqrt [3]{a}} \]
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Time = 0.44 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {3}\, \arctan \left (\frac {\left (a^{\frac {1}{3}}-2 \left (b x -a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right )+\ln \left (a^{\frac {1}{3}}+\left (b x -a \right )^{\frac {1}{3}}\right )-\frac {\ln \left (\left (b x -a \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (b x -a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2}}{a^{\frac {1}{3}}}\) | \(79\) |
derivativedivides | \(-\frac {\ln \left (a^{\frac {1}{3}}+\left (b x -a \right )^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x -a \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (b x -a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x -a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{a^{\frac {1}{3}}}\) | \(83\) |
default | \(-\frac {\ln \left (a^{\frac {1}{3}}+\left (b x -a \right )^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x -a \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (b x -a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x -a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{a^{\frac {1}{3}}}\) | \(83\) |
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none
Time = 0.24 (sec) , antiderivative size = 285, normalized size of antiderivative = 3.48 \[ \int \frac {1}{x \sqrt [3]{-a+b x}} \, dx=\left [\frac {\sqrt {3} a \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x + \sqrt {3} {\left (2 \, {\left (b x - a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} + {\left (b x - a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x - a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} - 3 \, a}{x}\right ) + \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x - a\right )}^{\frac {2}{3}} + {\left (b x - a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x - a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )}{2 \, a}, \frac {2 \, \sqrt {3} a \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (b x - a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) + \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x - a\right )}^{\frac {2}{3}} + {\left (b x - a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x - a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )}{2 \, a}\right ] \]
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Result contains complex when optimal does not.
Time = 1.63 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.95 \[ \int \frac {1}{x \sqrt [3]{-a+b x}} \, dx=- \frac {2 e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{- \frac {a}{b} + x} e^{\frac {i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {2}{3}\right )}{3 \sqrt [3]{a} \Gamma \left (\frac {5}{3}\right )} - \frac {2 \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{- \frac {a}{b} + x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {2}{3}\right )}{3 \sqrt [3]{a} \Gamma \left (\frac {5}{3}\right )} - \frac {2 e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{- \frac {a}{b} + x} e^{\frac {5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {2}{3}\right )}{3 \sqrt [3]{a} \Gamma \left (\frac {5}{3}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x \sqrt [3]{-a+b x}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x - a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} + \frac {\log \left ({\left (b x - a\right )}^{\frac {2}{3}} - {\left (b x - a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{2 \, a^{\frac {1}{3}}} - \frac {\log \left ({\left (b x - a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} \]
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Time = 0.58 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x \sqrt [3]{-a+b x}} \, dx=-\frac {\sqrt {3} \left (-a\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x - a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right )}}{3 \, \left (-a\right )^{\frac {1}{3}}}\right )}{a} + \frac {\left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x - a\right )}^{\frac {2}{3}} + {\left (b x - a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right )}{2 \, a} - \frac {\left (-a\right )^{\frac {2}{3}} \log \left ({\left | {\left (b x - a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}} \right |}\right )}{a} \]
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Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x \sqrt [3]{-a+b x}} \, dx=\frac {\ln \left (9\,{\left (b\,x-a\right )}^{1/3}-9\,{\left (-a\right )}^{1/3}\right )}{{\left (-a\right )}^{1/3}}+\frac {\ln \left (9\,{\left (b\,x-a\right )}^{1/3}-\frac {9\,{\left (-a\right )}^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,{\left (-a\right )}^{1/3}}-\frac {\ln \left (9\,{\left (b\,x-a\right )}^{1/3}-\frac {9\,{\left (-a\right )}^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,{\left (-a\right )}^{1/3}} \]
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