Optimal. Leaf size=42 \[ \frac{\sin ^{-1}(x)^{3/2}}{2 \left (1-x^2\right )}-\frac{3 x \sqrt{\sin ^{-1}(x)}}{4 \sqrt{1-x^2}} \]
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Rubi [A] time = 0.150535, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {4677, 4651} \[ \frac{\sin ^{-1}(x)^{3/2}}{2 \left (1-x^2\right )}-\frac{3 x \sqrt{\sin ^{-1}(x)}}{4 \sqrt{1-x^2}} \]
Antiderivative was successfully verified.
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Rule 4677
Rule 4651
Rubi steps
\begin{align*} \int \left (-\frac{3 x}{8 \left (1-x^2\right ) \sqrt{\sin ^{-1}(x)}}+\frac{x \sin ^{-1}(x)^{3/2}}{\left (1-x^2\right )^2}\right ) \, dx &=-\left (\frac{3}{8} \int \frac{x}{\left (1-x^2\right ) \sqrt{\sin ^{-1}(x)}} \, dx\right )+\int \frac{x \sin ^{-1}(x)^{3/2}}{\left (1-x^2\right )^2} \, dx\\ &=\frac{\sin ^{-1}(x)^{3/2}}{2 \left (1-x^2\right )}-\frac{3}{8} \int \frac{x}{\left (1-x^2\right ) \sqrt{\sin ^{-1}(x)}} \, dx-\frac{3}{4} \int \frac{\sqrt{\sin ^{-1}(x)}}{\left (1-x^2\right )^{3/2}} \, dx\\ &=-\frac{3 x \sqrt{\sin ^{-1}(x)}}{4 \sqrt{1-x^2}}+\frac{\sin ^{-1}(x)^{3/2}}{2 \left (1-x^2\right )}\\ \end{align*}
Mathematica [F] time = 3.52706, size = 0, normalized size = 0. \[ \int \left (-\frac{3 x}{8 \left (1-x^2\right ) \sqrt{\sin ^{-1}(x)}}+\frac{x \sin ^{-1}(x)^{3/2}}{\left (1-x^2\right )^2}\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.218, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{ \left ( -{x}^{2}+1 \right ) ^{2}} \left ( \arcsin \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{3\,x}{-8\,{x}^{2}+8}{\frac{1}{\sqrt{\arcsin \left ( x \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int - \frac{3 x}{x^{4} \sqrt{\operatorname{asin}{\left (x \right )}} - 2 x^{2} \sqrt{\operatorname{asin}{\left (x \right )}} + \sqrt{\operatorname{asin}{\left (x \right )}}}\, dx + \int \frac{3 x^{3}}{x^{4} \sqrt{\operatorname{asin}{\left (x \right )}} - 2 x^{2} \sqrt{\operatorname{asin}{\left (x \right )}} + \sqrt{\operatorname{asin}{\left (x \right )}}}\, dx + \int \frac{8 x \operatorname{asin}^{2}{\left (x \right )}}{x^{4} \sqrt{\operatorname{asin}{\left (x \right )}} - 2 x^{2} \sqrt{\operatorname{asin}{\left (x \right )}} + \sqrt{\operatorname{asin}{\left (x \right )}}}\, dx}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \arcsin \left (x\right )^{\frac{3}{2}}}{{\left (x^{2} - 1\right )}^{2}} + \frac{3 \, x}{8 \,{\left (x^{2} - 1\right )} \sqrt{\arcsin \left (x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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