Integrand size = 18, antiderivative size = 52 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2 x^{3/2}}{3}+\sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {x}\right )-\sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {x}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1816, 841, 1176, 631, 210} \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {x}\right )-\sqrt {2} \arctan \left (\sqrt {2} \sqrt {x}+1\right )+\frac {2 x^{3/2}}{3} \]
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Rule 210
Rule 631
Rule 841
Rule 1176
Rule 1816
Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt {x}-\frac {1+x}{\sqrt {x} \left (1+x^2\right )}\right ) \, dx \\ & = \frac {2 x^{3/2}}{3}-\int \frac {1+x}{\sqrt {x} \left (1+x^2\right )} \, dx \\ & = \frac {2 x^{3/2}}{3}-2 \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 x^{3/2}}{3}-\text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right )-\text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 x^{3/2}}{3}-\sqrt {2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {x}\right )+\sqrt {2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {x}\right ) \\ & = \frac {2 x^{3/2}}{3}+\sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {x}\right )-\sqrt {2} \tan ^{-1}\left (1+\sqrt {2} \sqrt {x}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.62 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2 x^{3/2}}{3}-\sqrt {2} \arctan \left (\frac {-1+x}{\sqrt {2} \sqrt {x}}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.17
method | result | size |
trager | \(\frac {2 x^{\frac {3}{2}}}{3}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x -4 x^{\frac {3}{2}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )+4 \sqrt {x}}{x^{2}+1}\right )}{2}\) | \(61\) |
risch | \(\frac {2 x^{\frac {3}{2}}}{3}-\arctan \left (1+\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}-\arctan \left (-1+\sqrt {2}\, \sqrt {x}\right ) \sqrt {2}-\frac {\sqrt {2}\, \ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )}{4}-\frac {\sqrt {2}\, \ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )}{4}\) | \(97\) |
derivativedivides | \(\frac {2 x^{\frac {3}{2}}}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{4}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{4}\) | \(117\) |
default | \(\frac {2 x^{\frac {3}{2}}}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x +\sqrt {2}\, \sqrt {x}+1}{x -\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{4}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x -\sqrt {2}\, \sqrt {x}+1}{x +\sqrt {2}\, \sqrt {x}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {x}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {x}\right )\right )}{4}\) | \(117\) |
meijerg | \(\frac {2 x^{\frac {3}{2}}}{3}-\frac {x^{\frac {3}{2}} \left (\frac {\sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {3}{4}}}-\frac {\sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{2 \left (x^{2}\right )^{\frac {3}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{\left (x^{2}\right )^{\frac {3}{4}}}\right )}{2}+\frac {\sqrt {x}\, \sqrt {2}\, \ln \left (1-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{4 \left (x^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {x}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2-\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {x}\, \sqrt {2}\, \ln \left (1+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}+\sqrt {x^{2}}\right )}{4 \left (x^{2}\right )^{\frac {1}{4}}}-\frac {\sqrt {x}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}{2+\sqrt {2}\, \left (x^{2}\right )^{\frac {1}{4}}}\right )}{2 \left (x^{2}\right )^{\frac {1}{4}}}\) | \(273\) |
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.44 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2}{3} \, x^{\frac {3}{2}} - \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 1\right )}}{2 \, \sqrt {x}}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2 x^{\frac {3}{2}}}{3} - \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} - 1 \right )} - \sqrt {2} \operatorname {atan}{\left (\sqrt {2} \sqrt {x} + 1 \right )} \]
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Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2}{3} \, x^{\frac {3}{2}} - \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2}{3} \, x^{\frac {3}{2}} - \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {x}\right )}\right ) - \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {x}\right )}\right ) \]
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Time = 18.59 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.83 \[ \int \frac {-1+x^3}{\sqrt {x} \left (1+x^2\right )} \, dx=\frac {2\,x^{3/2}}{3}-\frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x}}{2}+\frac {\sqrt {2}\,x^{3/2}}{2}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x}}{2}\right )\right )}{2} \]
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