Integrand size = 17, antiderivative size = 48 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{3}+\frac {1}{12} \log \left (9-x^4\right ) \]
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Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {1607, 1845, 272, 36, 31, 29, 304, 209, 212} \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{12} \log \left (9-x^4\right )-\frac {\log (x)}{3} \]
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Rule 29
Rule 31
Rule 36
Rule 209
Rule 212
Rule 272
Rule 304
Rule 1607
Rule 1845
Rubi steps \begin{align*} \text {integral}& = \int \frac {3+2 x^3}{x \left (-9+x^4\right )} \, dx \\ & = \int \left (\frac {3}{x \left (-9+x^4\right )}+\frac {2 x^2}{-9+x^4}\right ) \, dx \\ & = 2 \int \frac {x^2}{-9+x^4} \, dx+3 \int \frac {1}{x \left (-9+x^4\right )} \, dx \\ & = \frac {3}{4} \text {Subst}\left (\int \frac {1}{(-9+x) x} \, dx,x,x^4\right )-\int \frac {1}{3-x^2} \, dx+\int \frac {1}{3+x^2} \, dx \\ & = \frac {\arctan \left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{12} \text {Subst}\left (\int \frac {1}{-9+x} \, dx,x,x^4\right )-\frac {1}{12} \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right ) \\ & = \frac {\arctan \left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{3}+\frac {1}{12} \log \left (9-x^4\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.40 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\frac {1}{12} \left (4 \sqrt {3} \arctan \left (\frac {x}{\sqrt {3}}\right )-4 \log (x)+2 \sqrt {3} \log \left (3-\sqrt {3} x\right )-2 \sqrt {3} \log \left (3+\sqrt {3} x\right )+\log \left (9-x^4\right )\right ) \]
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Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {\ln \left (x^{2}+3\right )}{12}+\frac {\arctan \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{3}-\frac {\ln \left (x \right )}{3}+\frac {\ln \left (x^{2}-3\right )}{12}-\frac {\operatorname {arctanh}\left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{3}\) | \(46\) |
risch | \(\frac {\ln \left (x^{2}+3\right )}{12}+\frac {\arctan \left (\frac {x \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\ln \left (x -\sqrt {3}\right )}{12}+\frac {\sqrt {3}\, \ln \left (x -\sqrt {3}\right )}{6}+\frac {\ln \left (x +\sqrt {3}\right )}{12}-\frac {\sqrt {3}\, \ln \left (x +\sqrt {3}\right )}{6}-\frac {\ln \left (x \right )}{3}\) | \(68\) |
meijerg | \(-\frac {\ln \left (x \right )}{3}+\frac {\ln \left (3\right )}{6}-\frac {i \pi }{12}+\frac {\ln \left (1-\frac {x^{4}}{9}\right )}{12}+\frac {x^{3} \sqrt {3}\, \left (\ln \left (1-\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{3}\right )-\ln \left (1+\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{3}\right )+2 \arctan \left (\frac {\sqrt {3}\, \left (x^{4}\right )^{\frac {1}{4}}}{3}\right )\right )}{6 \left (x^{4}\right )^{\frac {3}{4}}}\) | \(79\) |
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Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.21 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + \frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{2} - 2 \, \sqrt {3} x + 3}{x^{2} - 3}\right ) + \frac {1}{12} \, \log \left (x^{2} + 3\right ) + \frac {1}{12} \, \log \left (x^{2} - 3\right ) - \frac {1}{3} \, \log \left (x\right ) \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 306, normalized size of antiderivative = 6.38 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=- \frac {\log {\left (x \right )}}{3} + \left (\frac {1}{12} + \frac {\sqrt {3} i}{6}\right ) \log {\left (x + \frac {17413}{11544} - \frac {943 \sqrt {3} i}{5772} + \frac {1368 \left (\frac {1}{12} + \frac {\sqrt {3} i}{6}\right )^{3}}{481} + \frac {4158 \left (\frac {1}{12} + \frac {\sqrt {3} i}{6}\right )^{2}}{481} - \frac {108000 \left (\frac {1}{12} + \frac {\sqrt {3} i}{6}\right )^{4}}{481} \right )} + \left (\frac {1}{12} - \frac {\sqrt {3} i}{6}\right ) \log {\left (x + \frac {17413}{11544} - \frac {108000 \left (\frac {1}{12} - \frac {\sqrt {3} i}{6}\right )^{4}}{481} + \frac {4158 \left (\frac {1}{12} - \frac {\sqrt {3} i}{6}\right )^{2}}{481} + \frac {1368 \left (\frac {1}{12} - \frac {\sqrt {3} i}{6}\right )^{3}}{481} + \frac {943 \sqrt {3} i}{5772} \right )} + \left (\frac {1}{12} - \frac {\sqrt {3}}{6}\right ) \log {\left (x - \frac {108000 \left (\frac {1}{12} - \frac {\sqrt {3}}{6}\right )^{4}}{481} + \frac {1368 \left (\frac {1}{12} - \frac {\sqrt {3}}{6}\right )^{3}}{481} + \frac {943 \sqrt {3}}{5772} + \frac {4158 \left (\frac {1}{12} - \frac {\sqrt {3}}{6}\right )^{2}}{481} + \frac {17413}{11544} \right )} + \left (\frac {1}{12} + \frac {\sqrt {3}}{6}\right ) \log {\left (x - \frac {108000 \left (\frac {1}{12} + \frac {\sqrt {3}}{6}\right )^{4}}{481} - \frac {943 \sqrt {3}}{5772} + \frac {1368 \left (\frac {1}{12} + \frac {\sqrt {3}}{6}\right )^{3}}{481} + \frac {4158 \left (\frac {1}{12} + \frac {\sqrt {3}}{6}\right )^{2}}{481} + \frac {17413}{11544} \right )} \]
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Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.12 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + \frac {1}{6} \, \sqrt {3} \log \left (\frac {x - \sqrt {3}}{x + \sqrt {3}}\right ) + \frac {1}{12} \, \log \left (x^{2} + 3\right ) + \frac {1}{12} \, \log \left (x^{2} - 3\right ) - \frac {1}{3} \, \log \left (x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.33 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) + \frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {3} \right |}}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}}\right ) + \frac {1}{12} \, \log \left (x^{2} + 3\right ) + \frac {1}{12} \, \log \left ({\left | x^{2} - 3 \right |}\right ) - \frac {1}{3} \, \log \left ({\left | x \right |}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.52 \[ \int \frac {3+2 x^3}{-9 x+x^5} \, dx=\ln \left (x-\sqrt {3}\right )\,\left (\frac {\sqrt {3}}{6}+\frac {1}{12}\right )-\ln \left (x+\sqrt {3}\right )\,\left (\frac {\sqrt {3}}{6}-\frac {1}{12}\right )-\frac {\ln \left (x\right )}{3}-\ln \left (x-\sqrt {3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\ln \left (x+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]
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