Integrand size = 9, antiderivative size = 119 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {x}{a}+\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{4/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{4/3}} \]
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Time = 0.05 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {199, 327, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{4/3}}+\frac {\sqrt [3]{b} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{4/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{4/3}}+\frac {x}{a} \]
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Rule 31
Rule 199
Rule 206
Rule 210
Rule 327
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{b+a x^3} \, dx \\ & = \frac {x}{a}-\frac {b \int \frac {1}{b+a x^3} \, dx}{a} \\ & = \frac {x}{a}-\frac {\sqrt [3]{b} \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 a}-\frac {\sqrt [3]{b} \int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 a} \\ & = \frac {x}{a}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{6 a^{4/3}}-\frac {b^{2/3} \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 a} \\ & = \frac {x}{a}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{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]{a} x}{\sqrt [3]{b}}\right )}{a^{4/3}} \\ & = \frac {x}{a}+\frac {\sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{4/3}}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{4/3}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.91 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {6 \sqrt [3]{a} x+2 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-2 \sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )+\sqrt [3]{b} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{4/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.29
method | result | size |
risch | \(\frac {x}{a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3 a^{2}}\) | \(34\) |
default | \(\frac {x}{a}-\frac {\left (\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\right ) b}{a}\) | \(103\) |
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Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {2 \, \sqrt {3} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 6 \, x}{6 \, a} \]
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Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.18 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{4} + b, \left ( t \mapsto t \log {\left (- 3 t a + x \right )} \right )\right )} + \frac {x}{a} \]
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Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {x}{a} - \frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {b \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {b \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}}} \]
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Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.93 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {\left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a} + \frac {x}{a} - \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2}} - \frac {\left (-a^{2} b\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{2}} \]
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Time = 5.82 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.96 \[ \int \frac {1}{a+\frac {b}{x^3}} \, dx=\frac {x}{a}+\frac {{\left (-b\right )}^{1/3}\,\ln \left ({\left (-b\right )}^{4/3}+a^{1/3}\,b\,x\right )}{3\,a^{4/3}}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (3\,a^{2/3}\,{\left (-b\right )}^{4/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-3\,a\,b\,x\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (9\,a^{2/3}\,{\left (-b\right )}^{4/3}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+3\,a\,b\,x\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{4/3}} \]
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