Optimal. Leaf size=17 \[ \sin ^{-1}(x)-\frac{x}{\sqrt{1-x^2}} \]
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Rubi [A] time = 0.0050743, antiderivative size = 17, normalized size of antiderivative = 1., 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 288
Rule 216
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*}
Mathematica [A] time = 0.0068797, 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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Maple [A] time = 0.004, size = 16, normalized size = 0.9 \begin{align*} \arcsin \left ( x \right ) -{x{\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41049, size = 20, normalized size = 1.18 \begin{align*} -\frac{x}{\sqrt{-x^{2} + 1}} + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.16875, size = 103, normalized size = 6.06 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.5255, size = 34, normalized size = 2. \begin{align*} \frac{x^{2} \operatorname{asin}{\left (x \right )}}{x^{2} - 1} + \frac{x \sqrt{1 - x^{2}}}{x^{2} - 1} - \frac{\operatorname{asin}{\left (x \right )}}{x^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07129, size = 28, normalized size = 1.65 \begin{align*} \frac{\sqrt{-x^{2} + 1} x}{x^{2} - 1} + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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