Integrand size = 19, antiderivative size = 23 \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x^2}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {641, 65, 212} \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x^2}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
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Rule 65
Rule 212
Rule 641
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {1-x} (1+x)} \, dx \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1-x}\right )\right ) \\ & = -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x^2}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {1+x}}{\sqrt {1-x^2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(39\) vs. \(2(18)=36\).
Time = 2.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74
method | result | size |
default | \(-\frac {\sqrt {-x^{2}+1}\, \operatorname {arctanh}\left (\frac {\sqrt {1-x}\, \sqrt {2}}{2}\right ) \sqrt {2}}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(40\) |
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Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (18) = 36\).
Time = 0.32 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.96 \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x^2}} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {x^{2} + 2 \, \sqrt {2} \sqrt {-x^{2} + 1} \sqrt {x + 1} - 2 \, x - 3}{x^{2} + 2 \, x + 1}\right ) \]
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\[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x^2}} \, dx=\int \frac {1}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {x + 1}}\, dx \]
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\[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + 1} \sqrt {x + 1}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x^2}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (\sqrt {2} + \sqrt {-x + 1}\right ) + \frac {1}{2} \, \sqrt {2} \log \left (\sqrt {2} - \sqrt {-x + 1}\right ) \]
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Timed out. \[ \int \frac {1}{\sqrt {1+x} \sqrt {1-x^2}} \, dx=\int \frac {1}{\sqrt {1-x^2}\,\sqrt {x+1}} \,d x \]
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