Optimal. Leaf size=17 \[ \sin ^{-1}(x)-\frac {x}{\sqrt {1-x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {288, 216} \[ \sin ^{-1}(x)-\frac {x}{\sqrt {1-x^2}} \]
Antiderivative was successfully verified.
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Rule 216
Rule 288
Rubi steps
\begin {align*} \int -\frac {x^2}{\left (1-x^2\right )^{3/2}} \, dx &=-\frac {x}{\sqrt {1-x^2}}+\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {x}{\sqrt {1-x^2}}+\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.88 \[ \frac {\sqrt {1-x^2} x+x^2 \sin ^{-1}(x)-\sin ^{-1}(x)}{x^2-1} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 45, normalized size = 2.65 \[ -\frac {2 \, {\left (x^{2} - 1\right )} \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) - \sqrt {-x^{2} + 1} x}{x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 21, normalized size = 1.24 \[ \frac {\sqrt {-x^{2} + 1} x}{x^{2} - 1} + \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.94 \[ -\frac {x}{\sqrt {-x^{2}+1}}+\arcsin \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.53, size = 15, normalized size = 0.88 \[ -\frac {x}{\sqrt {-x^{2} + 1}} + \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 37, normalized size = 2.18 \[ \mathrm {asin}\relax (x)+\frac {\sqrt {1-x^2}}{2\,\left (x-1\right )}+\frac {\sqrt {1-x^2}}{2\,\left (x+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.51, size = 34, normalized size = 2.00 \[ \frac {x^{2} \operatorname {asin}{\relax (x )}}{x^{2} - 1} + \frac {x \sqrt {1 - x^{2}}}{x^{2} - 1} - \frac {\operatorname {asin}{\relax (x )}}{x^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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