Integrand size = 14, antiderivative size = 123 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=-\frac {1}{a x}-\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}} \]
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Time = 0.04 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {331, 298, 31, 648, 631, 210, 642} \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=-\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}-\frac {1}{a x} \]
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Rule 31
Rule 210
Rule 298
Rule 331
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{a x}+\frac {b \int \frac {x}{a-b x^3} \, dx}{a} \\ & = -\frac {1}{a x}+\frac {b^{2/3} \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{4/3}}-\frac {b^{2/3} \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{4/3}} \\ & = -\frac {1}{a x}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \int \frac {\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{4/3}}-\frac {b^{2/3} \int \frac {1}{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a} \\ & = -\frac {1}{a x}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}}+\frac {\sqrt [3]{b} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{4/3}} \\ & = -\frac {1}{a x}-\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=-\frac {6 \sqrt [3]{a}+2 \sqrt {3} \sqrt [3]{b} x \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{b} x \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )-\sqrt [3]{b} x \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} x} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.65 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.41
method | result | size |
risch | \(-\frac {1}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{4} \textit {\_Z}^{3}+b \right )}{\sum }\textit {\_R} \ln \left (\left (4 \textit {\_R}^{3} a^{4}+3 b \right ) x +a^{3} \textit {\_R}^{2}\right )\right )}{3}\) | \(50\) |
default | \(-\frac {\left (\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b}{a}-\frac {1}{a x}\) | \(107\) |
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Time = 0.26 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=-\frac {2 \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} + a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 6}{6 \, a x} \]
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Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.25 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=- \operatorname {RootSum} {\left (27 t^{3} a^{4} - b, \left ( t \mapsto t \log {\left (- \frac {9 t^{2} a^{3}}{b} + x \right )} \right )\right )} - \frac {1}{a x} \]
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Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {1}{a x} \]
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Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=-\frac {b \left (\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2}} - \frac {\sqrt {3} \left (a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} b} + \frac {\left (a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b} - \frac {1}{a x} \]
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Time = 5.48 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^2 \left (a-b x^3\right )} \, dx=\frac {{\left (-b\right )}^{1/3}\,\ln \left (b^3\,x-a^{1/3}\,{\left (-b\right )}^{8/3}\right )}{3\,a^{4/3}}-\frac {1}{a\,x}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (a\,b^3\,x-a^{4/3}\,{\left (-b\right )}^{8/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (a\,b^3\,x-9\,a^{4/3}\,{\left (-b\right )}^{8/3}\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{4/3}} \]
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