Integrand size = 15, antiderivative size = 16 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {1-x^2}}{x} \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {1-x^2}}{x} \]
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Rule 270
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-x^2}}{x} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {1-x^2}}{x} \]
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Time = 0.14 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {\sqrt {-x^{2}+1}}{x}\) | \(15\) |
trager | \(-\frac {\sqrt {-x^{2}+1}}{x}\) | \(15\) |
meijerg | \(-\frac {\sqrt {-x^{2}+1}}{x}\) | \(15\) |
pseudoelliptic | \(-\frac {\sqrt {-x^{2}+1}}{x}\) | \(15\) |
risch | \(\frac {x^{2}-1}{x \sqrt {-x^{2}+1}}\) | \(19\) |
gosper | \(\frac {\left (-1+x \right ) \left (1+x \right )}{x \sqrt {-x^{2}+1}}\) | \(20\) |
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {-x^{2} + 1}}{x} \]
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Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=\begin {cases} - \frac {i \sqrt {x^{2} - 1}}{x} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {\sqrt {1 - x^{2}}}{x} & \text {otherwise} \end {cases} \]
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none
Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {-x^{2} + 1}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=\frac {x}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} - \frac {\sqrt {-x^{2} + 1} - 1}{2 \, x} \]
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Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \sqrt {1-x^2}} \, dx=-\frac {\sqrt {1-x^2}}{x} \]
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