Integrand size = 15, antiderivative size = 115 \[ \int \frac {1}{x \sqrt [3]{b+a x^3}} \, dx=\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b+a x^3}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {\log \left (-\sqrt [3]{b}+\sqrt [3]{b+a x^3}\right )}{3 \sqrt [3]{b}}-\frac {\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b+a x^3}+\left (b+a x^3\right )^{2/3}\right )}{6 \sqrt [3]{b}} \]
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Time = 0.04 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.72, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 57, 631, 210, 31} \[ \int \frac {1}{x \sqrt [3]{b+a x^3}} \, dx=\frac {\arctan \left (\frac {2 \sqrt [3]{a x^3+b}+\sqrt [3]{b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a x^3+b}\right )}{2 \sqrt [3]{b}}-\frac {\log (x)}{2 \sqrt [3]{b}} \]
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Rule 31
Rule 57
Rule 210
Rule 272
Rule 631
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt [3]{b+a x}} \, dx,x,x^3\right ) \\ & = -\frac {\log (x)}{2 \sqrt [3]{b}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b+a x^3}\right )-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{b}-x} \, dx,x,\sqrt [3]{b+a x^3}\right )}{2 \sqrt [3]{b}} \\ & = -\frac {\log (x)}{2 \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{b+a x^3}\right )}{2 \sqrt [3]{b}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b+a x^3}}{\sqrt [3]{b}}\right )}{\sqrt [3]{b}} \\ & = \frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{b+a x^3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log (x)}{2 \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{b+a x^3}\right )}{2 \sqrt [3]{b}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x \sqrt [3]{b+a x^3}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b+a x^3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )+2 \log \left (-\sqrt [3]{b}+\sqrt [3]{b+a x^3}\right )-\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b+a x^3}+\left (b+a x^3\right )^{2/3}\right )}{6 \sqrt [3]{b}} \]
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Time = 0.57 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.72
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 \left (a \,x^{3}+b \right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}}}\right )+2 \ln \left (-b^{\frac {1}{3}}+\left (a \,x^{3}+b \right )^{\frac {1}{3}}\right )-\ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (a \,x^{3}+b \right )^{\frac {1}{3}}+\left (a \,x^{3}+b \right )^{\frac {2}{3}}\right )}{6 b^{\frac {1}{3}}}\) | \(83\) |
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none
Time = 0.29 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.05 \[ \int \frac {1}{x \sqrt [3]{b+a x^3}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} b \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (\frac {2 \, a x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (a x^{3} + b\right )}^{\frac {2}{3}} b^{\frac {2}{3}} - {\left (a x^{3} + b\right )}^{\frac {1}{3}} b - b^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} - 3 \, {\left (a x^{3} + b\right )}^{\frac {1}{3}} b^{\frac {2}{3}} + 3 \, b}{x^{3}}\right ) - b^{\frac {2}{3}} \log \left ({\left (a x^{3} + b\right )}^{\frac {2}{3}} + {\left (a x^{3} + b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}\right ) + 2 \, b^{\frac {2}{3}} \log \left ({\left (a x^{3} + b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}{6 \, b}, \frac {6 \, \sqrt {\frac {1}{3}} b^{\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (a x^{3} + b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}\right )}}{b^{\frac {1}{3}}}\right ) - b^{\frac {2}{3}} \log \left ({\left (a x^{3} + b\right )}^{\frac {2}{3}} + {\left (a x^{3} + b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}\right ) + 2 \, b^{\frac {2}{3}} \log \left ({\left (a x^{3} + b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}{6 \, b}\right ] \]
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Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.32 \[ \int \frac {1}{x \sqrt [3]{b+a x^3}} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt [3]{a} x \Gamma \left (\frac {4}{3}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x \sqrt [3]{b+a x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a x^{3} + b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{3 \, b^{\frac {1}{3}}} - \frac {\log \left ({\left (a x^{3} + b\right )}^{\frac {2}{3}} + {\left (a x^{3} + b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}\right )}{6 \, b^{\frac {1}{3}}} + \frac {\log \left ({\left (a x^{3} + b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}{3 \, b^{\frac {1}{3}}} \]
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Time = 0.53 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \sqrt [3]{b+a x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a x^{3} + b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{3 \, b^{\frac {1}{3}}} - \frac {\log \left ({\left (a x^{3} + b\right )}^{\frac {2}{3}} + {\left (a x^{3} + b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}\right )}{6 \, b^{\frac {1}{3}}} + \frac {\log \left ({\left | {\left (a x^{3} + b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}} \right |}\right )}{3 \, b^{\frac {1}{3}}} \]
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Time = 5.80 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \sqrt [3]{b+a x^3}} \, dx=\frac {\ln \left ({\left (a\,x^3+b\right )}^{1/3}-b^{1/3}\right )}{3\,b^{1/3}}+\frac {\ln \left ({\left (a\,x^3+b\right )}^{1/3}-\frac {b^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}}-\frac {\ln \left ({\left (a\,x^3+b\right )}^{1/3}-\frac {b^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}} \]
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