Integrand size = 20, antiderivative size = 32 \[ \int \frac {(1+x)^2}{x \sqrt {1-x^2}} \, dx=-\sqrt {1-x^2}+2 \arcsin (x)-\text {arctanh}\left (\sqrt {1-x^2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1823, 858, 222, 272, 65, 212} \[ \int \frac {(1+x)^2}{x \sqrt {1-x^2}} \, dx=2 \arcsin (x)-\text {arctanh}\left (\sqrt {1-x^2}\right )-\sqrt {1-x^2} \]
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Rule 65
Rule 212
Rule 222
Rule 272
Rule 858
Rule 1823
Rubi steps \begin{align*} \text {integral}& = -\sqrt {1-x^2}-\int \frac {-1-2 x}{x \sqrt {1-x^2}} \, dx \\ & = -\sqrt {1-x^2}+2 \int \frac {1}{\sqrt {1-x^2}} \, dx+\int \frac {1}{x \sqrt {1-x^2}} \, dx \\ & = -\sqrt {1-x^2}+2 \sin ^{-1}(x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right ) \\ & = -\sqrt {1-x^2}+2 \sin ^{-1}(x)-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right ) \\ & = -\sqrt {1-x^2}+2 \sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt {1-x^2}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.62 \[ \int \frac {(1+x)^2}{x \sqrt {1-x^2}} \, dx=-\sqrt {1-x^2}+4 \arctan \left (\frac {x}{-1+\sqrt {1-x^2}}\right )-\log (x)+\log \left (-1+\sqrt {1-x^2}\right ) \]
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Time = 0.38 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91
method | result | size |
default | \(2 \arcsin \left (x \right )-\sqrt {-x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right )\) | \(29\) |
trager | \(-\sqrt {-x^{2}+1}+\ln \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}+1}\right )\) | \(56\) |
meijerg | \(\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{2}+1}}{2}\right )+\left (-2 \ln \left (2\right )+2 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{2 \sqrt {\pi }}+2 \arcsin \left (x \right )-\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{2}+1}}{2 \sqrt {\pi }}\) | \(73\) |
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \[ \int \frac {(1+x)^2}{x \sqrt {1-x^2}} \, dx=-\sqrt {-x^{2} + 1} - 4 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
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Time = 2.52 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {(1+x)^2}{x \sqrt {1-x^2}} \, dx=- \sqrt {1 - x^{2}} + \begin {cases} - \operatorname {acosh}{\left (\frac {1}{x} \right )} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{x} \right )} & \text {otherwise} \end {cases} + 2 \operatorname {asin}{\left (x \right )} \]
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {(1+x)^2}{x \sqrt {1-x^2}} \, dx=-\sqrt {-x^{2} + 1} + 2 \, \arcsin \left (x\right ) - \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(1+x)^2}{x \sqrt {1-x^2}} \, dx=-\sqrt {-x^{2} + 1} + 2 \, \arcsin \left (x\right ) + \log \left (-\frac {\sqrt {-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \]
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Time = 11.44 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {(1+x)^2}{x \sqrt {1-x^2}} \, dx=2\,\mathrm {asin}\left (x\right )+\ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )-\sqrt {1-x^2} \]
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