Integrand size = 14, antiderivative size = 124 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=-\frac {1}{2 a x^2}+\frac {b^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}-\frac {b^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac {b^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}} \]
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Time = 0.04 (sec) , antiderivative size = 124, 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, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=\frac {b^{2/3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}+\frac {b^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}}-\frac {b^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{5/3}}-\frac {1}{2 a x^2} \]
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Rule 31
Rule 206
Rule 210
Rule 331
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 a x^2}+\frac {b \int \frac {1}{a-b x^3} \, dx}{a} \\ & = -\frac {1}{2 a x^2}+\frac {b \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{3 a^{5/3}}+\frac {b \int \frac {2 \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^{5/3}} \\ & = -\frac {1}{2 a x^2}-\frac {b^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac {b^{2/3} \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^{5/3}}+\frac {b \int \frac {1}{a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{4/3}} \\ & = -\frac {1}{2 a x^2}-\frac {b^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac {b^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}}-\frac {b^{2/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{5/3}} \\ & = -\frac {1}{2 a x^2}+\frac {b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}-\frac {b^{2/3} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac {b^{2/3} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=\frac {-3 a^{2/3}+2 \sqrt {3} b^{2/3} x^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b^{2/3} x^2 \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+b^{2/3} x^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} x^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.65 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.44
method | result | size |
risch | \(-\frac {1}{2 a \,x^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{5} \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (4 \textit {\_R}^{3} a^{5}+3 b^{2}\right ) x -a^{2} b \textit {\_R} \right )\right )}{3}\) | \(54\) |
default | \(-\frac {\left (\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b}{a}-\frac {1}{2 a \,x^{2}}\) | \(107\) |
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Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + \sqrt {3} b}{3 \, b}\right ) + x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} - a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - 2 \, x^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 3}{6 \, a x^{2}} \]
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Time = 0.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.27 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=- \operatorname {RootSum} {\left (27 t^{3} a^{5} - b^{2}, \left ( t \mapsto t \log {\left (- \frac {3 t a^{2}}{b} + x \right )} \right )\right )} - \frac {1}{2 a x^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {1}{2 \, a x^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=-\frac {b \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2}} + \frac {\sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2}} + \frac {\left (a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, a^{2}} - \frac {1}{2 \, a x^{2}} \]
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Time = 5.42 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^3 \left (a-b x^3\right )} \, dx=\frac {b^{2/3}\,\ln \left ({\left (-a\right )}^{7/3}+a^2\,b^{1/3}\,x\right )}{3\,{\left (-a\right )}^{5/3}}-\frac {1}{2\,a\,x^2}-\frac {b^{2/3}\,\ln \left (3\,a^2\,b^3\,x-3\,{\left (-a\right )}^{7/3}\,b^{8/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,{\left (-a\right )}^{5/3}}+\frac {b^{2/3}\,\ln \left (3\,a^2\,b^3\,x+9\,{\left (-a\right )}^{7/3}\,b^{8/3}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{{\left (-a\right )}^{5/3}} \]
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