Integrand size = 17, antiderivative size = 44 \[ \int \frac {x^2}{\sqrt {1-(1+x)^2}} \, dx=\frac {3}{2} \sqrt {1-(1+x)^2}-\frac {1}{2} x \sqrt {1-(1+x)^2}+\frac {3}{2} \arcsin (1+x) \]
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Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {378, 685, 655, 222} \[ \int \frac {x^2}{\sqrt {1-(1+x)^2}} \, dx=\frac {3}{2} \arcsin (x+1)-\frac {1}{2} \sqrt {1-(x+1)^2} x+\frac {3}{2} \sqrt {1-(x+1)^2} \]
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Rule 222
Rule 378
Rule 655
Rule 685
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {(-1+x)^2}{\sqrt {1-x^2}} \, dx,x,1+x\right ) \\ & = -\frac {1}{2} x \sqrt {1-(1+x)^2}-\frac {3}{2} \text {Subst}\left (\int \frac {-1+x}{\sqrt {1-x^2}} \, dx,x,1+x\right ) \\ & = \frac {3}{2} \sqrt {1-(1+x)^2}-\frac {1}{2} x \sqrt {1-(1+x)^2}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,1+x\right ) \\ & = \frac {3}{2} \sqrt {1-(1+x)^2}-\frac {1}{2} x \sqrt {1-(1+x)^2}+\frac {3}{2} \sin ^{-1}(1+x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.25 \[ \int \frac {x^2}{\sqrt {1-(1+x)^2}} \, dx=\frac {x \left (-6-x+x^2\right )-6 \sqrt {x} \sqrt {2+x} \log \left (-\sqrt {x}+\sqrt {2+x}\right )}{2 \sqrt {-x (2+x)}} \]
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Time = 0.99 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.57
method | result | size |
risch | \(\frac {\left (-3+x \right ) x \left (x +2\right )}{2 \sqrt {-x \left (x +2\right )}}+\frac {3 \arcsin \left (x +1\right )}{2}\) | \(25\) |
pseudoelliptic | \(-3 \arctan \left (\frac {\sqrt {-x \left (x +2\right )}}{x}\right )+\frac {\left (3-x \right ) \sqrt {-x \left (x +2\right )}}{2}\) | \(32\) |
default | \(-\frac {x \sqrt {-x^{2}-2 x}}{2}+\frac {3 \sqrt {-x^{2}-2 x}}{2}+\frac {3 \arcsin \left (x +1\right )}{2}\) | \(35\) |
meijerg | \(-\frac {4 i \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-5 x +15\right ) \sqrt {1+\frac {x}{2}}}{40}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {\sqrt {2}\, \sqrt {x}}{2}\right )}{4}\right )}{\sqrt {\pi }}\) | \(45\) |
trager | \(\left (\frac {3}{2}-\frac {x}{2}\right ) \sqrt {-x^{2}-2 x}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}-2 x}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{2}\) | \(54\) |
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Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{\sqrt {1-(1+x)^2}} \, dx=-\frac {1}{2} \, \sqrt {-x^{2} - 2 \, x} {\left (x - 3\right )} - 3 \, \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x}}{x}\right ) \]
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Time = 0.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.59 \[ \int \frac {x^2}{\sqrt {1-(1+x)^2}} \, dx=\left (\frac {3}{2} - \frac {x}{2}\right ) \sqrt {- x^{2} - 2 x} + \frac {3 \operatorname {asin}{\left (x + 1 \right )}}{2} \]
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Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int \frac {x^2}{\sqrt {1-(1+x)^2}} \, dx=-\frac {1}{2} \, \sqrt {-x^{2} - 2 \, x} x + \frac {3}{2} \, \sqrt {-x^{2} - 2 \, x} - \frac {3}{2} \, \arcsin \left (-x - 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.52 \[ \int \frac {x^2}{\sqrt {1-(1+x)^2}} \, dx=-\frac {1}{2} \, \sqrt {-x^{2} - 2 \, x} {\left (x - 3\right )} + \frac {3}{2} \, \arcsin \left (x + 1\right ) \]
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Timed out. \[ \int \frac {x^2}{\sqrt {1-(1+x)^2}} \, dx=\int \frac {x^2}{\sqrt {1-{\left (x+1\right )}^2}} \,d x \]
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