Integrand size = 21, antiderivative size = 51 \[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\frac {1}{2} \sqrt {\frac {1-x^2}{1+x^2}} \left (1+x^2\right )-\arctan \left (\sqrt {\frac {1-x^2}{1+x^2}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1981, 1979, 294, 210} \[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\frac {1}{2} \sqrt {\frac {1-x^2}{x^2+1}} \left (x^2+1\right )-\arctan \left (\sqrt {\frac {1-x^2}{x^2+1}}\right ) \]
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Rule 210
Rule 294
Rule 1979
Rule 1981
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \sqrt {\frac {1-x}{1+x}} \, dx,x,x^2\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {x^2}{\left (-1-x^2\right )^2} \, dx,x,\sqrt {\frac {1-x^2}{1+x^2}}\right )\right ) \\ & = \frac {1}{2} \sqrt {\frac {1-x^2}{1+x^2}} \left (1+x^2\right )+\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {\frac {1-x^2}{1+x^2}}\right ) \\ & = \frac {1}{2} \sqrt {\frac {1-x^2}{1+x^2}} \left (1+x^2\right )-\tan ^{-1}\left (\sqrt {\frac {1-x^2}{1+x^2}}\right ) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.86 \[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\frac {\sqrt {\frac {1-x^2}{1+x^2}} \left (\sqrt {1-x^2} \left (1+x^2\right )+4 \sqrt {1+x^2} \arctan \left (\frac {\sqrt {1-x^2}}{\sqrt {2}-\sqrt {1+x^2}}\right )\right )}{2 \sqrt {1-x^2}} \]
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Time = 0.41 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\sqrt {-\frac {x^{2}-1}{x^{2}+1}}\, \left (x^{2}+1\right ) \left (\arcsin \left (x^{2}\right )+\sqrt {-x^{4}+1}\right )}{2 \sqrt {-\left (x^{2}+1\right ) \left (x^{2}-1\right )}}\) | \(52\) |
risch | \(\frac {\left (x^{2}+1\right ) \sqrt {-\frac {x^{2}-1}{x^{2}+1}}}{2}-\frac {\arcsin \left (x^{2}\right ) \sqrt {-\frac {x^{2}-1}{x^{2}+1}}\, \sqrt {-\left (x^{2}+1\right ) \left (x^{2}-1\right )}}{2 \left (x^{2}-1\right )}\) | \(68\) |
trager | \(\left (\frac {x^{2}}{2}+\frac {1}{2}\right ) \sqrt {-\frac {x^{2}-1}{x^{2}+1}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {x^{2}-1}{x^{2}+1}}\, x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {x^{2}-1}{x^{2}+1}}+x^{2}\right )}{2}\) | \(88\) |
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Time = 0.31 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.08 \[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\frac {1}{2} \, {\left (x^{2} + 1\right )} \sqrt {-\frac {x^{2} - 1}{x^{2} + 1}} - \arctan \left (\frac {{\left (x^{2} + 1\right )} \sqrt {-\frac {x^{2} - 1}{x^{2} + 1}} - 1}{x^{2}}\right ) \]
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\[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\int x \sqrt {- \frac {\left (x - 1\right ) \left (x + 1\right )}{x^{2} + 1}}\, dx \]
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\[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\int { x \sqrt {-\frac {x^{2} - 1}{x^{2} + 1}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.35 \[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=\frac {1}{2} \, \sqrt {-x^{4} + 1} + \frac {1}{2} \, \arcsin \left (x^{2}\right ) \]
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Time = 17.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.08 \[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx=-\mathrm {atan}\left (\sqrt {-\frac {x^2-1}{x^2+1}}\right )-\frac {\sqrt {-\frac {x^2-1}{x^2+1}}}{\frac {x^2-1}{x^2+1}-1} \]
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