Integrand size = 17, antiderivative size = 40 \[ \int \frac {(1+x)^2}{\sqrt {1-x^2}} \, dx=-\frac {3}{2} \sqrt {1-x^2}-\frac {1}{2} (1+x) \sqrt {1-x^2}+\frac {3 \arcsin (x)}{2} \]
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Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {685, 655, 222} \[ \int \frac {(1+x)^2}{\sqrt {1-x^2}} \, dx=\frac {3 \arcsin (x)}{2}-\frac {1}{2} \sqrt {1-x^2} (x+1)-\frac {3 \sqrt {1-x^2}}{2} \]
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Rule 222
Rule 655
Rule 685
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} (1+x) \sqrt {1-x^2}+\frac {3}{2} \int \frac {1+x}{\sqrt {1-x^2}} \, dx \\ & = -\frac {3}{2} \sqrt {1-x^2}-\frac {1}{2} (1+x) \sqrt {1-x^2}+\frac {3}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx \\ & = -\frac {3}{2} \sqrt {1-x^2}-\frac {1}{2} (1+x) \sqrt {1-x^2}+\frac {3}{2} \sin ^{-1}(x) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {(1+x)^2}{\sqrt {1-x^2}} \, dx=\frac {1}{2} (-4-x) \sqrt {1-x^2}-3 \arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.62
method | result | size |
risch | \(\frac {\left (4+x \right ) \left (x^{2}-1\right )}{2 \sqrt {-x^{2}+1}}+\frac {3 \arcsin \left (x \right )}{2}\) | \(25\) |
default | \(\frac {3 \arcsin \left (x \right )}{2}-\frac {x \sqrt {-x^{2}+1}}{2}-2 \sqrt {-x^{2}+1}\) | \(29\) |
trager | \(\left (-2-\frac {x}{2}\right ) \sqrt {-x^{2}+1}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )}{2}\) | \(44\) |
meijerg | \(\arcsin \left (x \right )-\frac {-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-x^{2}+1}}{\sqrt {\pi }}+\frac {i \left (i \sqrt {\pi }\, x \sqrt {-x^{2}+1}-i \sqrt {\pi }\, \arcsin \left (x \right )\right )}{2 \sqrt {\pi }}\) | \(60\) |
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Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {(1+x)^2}{\sqrt {1-x^2}} \, dx=-\frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x + 4\right )} - 3 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.68 \[ \int \frac {(1+x)^2}{\sqrt {1-x^2}} \, dx=- \frac {x \sqrt {1 - x^{2}}}{2} - 2 \sqrt {1 - x^{2}} + \frac {3 \operatorname {asin}{\left (x \right )}}{2} \]
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int \frac {(1+x)^2}{\sqrt {1-x^2}} \, dx=-\frac {1}{2} \, \sqrt {-x^{2} + 1} x - 2 \, \sqrt {-x^{2} + 1} + \frac {3}{2} \, \arcsin \left (x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.48 \[ \int \frac {(1+x)^2}{\sqrt {1-x^2}} \, dx=-\frac {1}{2} \, \sqrt {-x^{2} + 1} {\left (x + 4\right )} + \frac {3}{2} \, \arcsin \left (x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.52 \[ \int \frac {(1+x)^2}{\sqrt {1-x^2}} \, dx=\frac {3\,\mathrm {asin}\left (x\right )}{2}-\left (\frac {x}{2}+2\right )\,\sqrt {1-x^2} \]
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