Integrand size = 22, antiderivative size = 18 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x+x^3}} \, dx=-\frac {2 \sqrt {x+x^3}}{1+x^2} \]
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Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2081, 460} \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x+x^3}} \, dx=-\frac {2 x}{\sqrt {x^3+x}} \]
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Rule 460
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {1+x^2}\right ) \int \frac {-1+x^2}{\sqrt {x} \left (1+x^2\right )^{3/2}} \, dx}{\sqrt {x+x^3}} \\ & = -\frac {2 x}{\sqrt {x+x^3}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x+x^3}} \, dx=-\frac {2 x}{\sqrt {x+x^3}} \]
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Time = 0.98 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(-\frac {2 x}{\sqrt {x^{3}+x}}\) | \(11\) |
default | \(-\frac {2 x}{\sqrt {\left (x^{2}+1\right ) x}}\) | \(13\) |
risch | \(-\frac {2 x}{\sqrt {\left (x^{2}+1\right ) x}}\) | \(13\) |
elliptic | \(-\frac {2 x}{\sqrt {\left (x^{2}+1\right ) x}}\) | \(13\) |
pseudoelliptic | \(-\frac {2 x}{\sqrt {\left (x^{2}+1\right ) x}}\) | \(13\) |
trager | \(-\frac {2 \sqrt {x^{3}+x}}{x^{2}+1}\) | \(17\) |
meijerg | \(\frac {2 x^{\frac {5}{2}} \operatorname {hypergeom}\left (\left [\frac {5}{4}, \frac {3}{2}\right ], \left [\frac {9}{4}\right ], -x^{2}\right )}{5}-2 \sqrt {x}\, \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {3}{2}\right ], \left [\frac {5}{4}\right ], -x^{2}\right )\) | \(34\) |
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none
Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x+x^3}} \, dx=-\frac {2 \, \sqrt {x^{3} + x}}{x^{2} + 1} \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x+x^3}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt {x \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x+x^3}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{3} + x} {\left (x^{2} + 1\right )}} \,d x } \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x+x^3}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{3} + x} {\left (x^{2} + 1\right )}} \,d x } \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.56 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {x+x^3}} \, dx=-\frac {2\,x}{\sqrt {x^3+x}} \]
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