Integrand size = 24, antiderivative size = 20 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {-x+x^3}} \, dx=-\frac {2 \sqrt {-x+x^3}}{-1+x^2} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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 = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {-x+x^3}} \, dx=-\frac {2 x}{\sqrt {x \left (-1+x^2\right )}} \]
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Time = 0.98 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.65
method | result | size |
gosper | \(-\frac {2 x}{\sqrt {x^{3}-x}}\) | \(13\) |
risch | \(-\frac {2 x}{\sqrt {x \left (x^{2}-1\right )}}\) | \(13\) |
elliptic | \(-\frac {2 x}{\sqrt {x \left (x^{2}-1\right )}}\) | \(13\) |
pseudoelliptic | \(-\frac {2 x}{\sqrt {x^{3}-x}}\) | \(13\) |
trager | \(-\frac {2 \sqrt {x^{3}-x}}{x^{2}-1}\) | \(19\) |
default | \(-\frac {x^{2}+x}{\sqrt {\left (x -1\right ) \left (x^{2}+x \right )}}+\frac {x^{2}-x}{\sqrt {\left (1+x \right ) \left (x^{2}-x \right )}}\) | \(41\) |
meijerg | \(-\frac {2 \sqrt {-\operatorname {signum}\left (x^{2}-1\right )}\, \sqrt {x}\, \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {3}{2}\right ], \left [\frac {5}{4}\right ], x^{2}\right )}{\sqrt {\operatorname {signum}\left (x^{2}-1\right )}}-\frac {2 \sqrt {-\operatorname {signum}\left (x^{2}-1\right )}\, x^{\frac {5}{2}} \operatorname {hypergeom}\left (\left [\frac {5}{4}, \frac {3}{2}\right ], \left [\frac {9}{4}\right ], x^{2}\right )}{5 \sqrt {\operatorname {signum}\left (x^{2}-1\right )}}\) | \(66\) |
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none
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {-x+x^3}} \, dx=-\frac {2 \, \sqrt {x^{3} - x}}{x^{2} - 1} \]
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\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {-x+x^3}} \, dx=\int \frac {x^{2} + 1}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}\, dx \]
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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 = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.60 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {-x+x^3}} \, dx=-\frac {2\,x}{\sqrt {x^3-x}} \]
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