Optimal. Leaf size=36 \[ \sqrt {1-x^2} \sin ^{-1}(x)+\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-x-\tanh ^{-1}(x) \]
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Rubi [A] time = 0.07, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {266, 43, 4689, 388, 206} \[ \sqrt {1-x^2} \sin ^{-1}(x)+\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-x-\tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 206
Rule 266
Rule 388
Rule 4689
Rubi steps
\begin {align*} \int \frac {x^3 \sin ^{-1}(x)}{\left (1-x^2\right )^{3/2}} \, dx &=\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}+\sqrt {1-x^2} \sin ^{-1}(x)-\int \frac {2-x^2}{1-x^2} \, dx\\ &=-x+\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}+\sqrt {1-x^2} \sin ^{-1}(x)-\int \frac {1}{1-x^2} \, dx\\ &=-x+\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}+\sqrt {1-x^2} \sin ^{-1}(x)-\tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.07, size = 40, normalized size = 1.11 \[ \frac {1}{2} \left (-\frac {2 \left (x^2-2\right ) \sin ^{-1}(x)}{\sqrt {1-x^2}}-2 x+\log (1-x)-\log (x+1)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^3 \sin ^{-1}(x)}{\left (1-x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.10, size = 57, normalized size = 1.58 \[ -\frac {2 \, x^{3} - 2 \, {\left (x^{2} - 2\right )} \sqrt {-x^{2} + 1} \arcsin \relax (x) + {\left (x^{2} - 1\right )} \log \left (x + 1\right ) - {\left (x^{2} - 1\right )} \log \left (x - 1\right ) - 2 \, x}{2 \, {\left (x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.35, size = 40, normalized size = 1.11 \[ {\left (\sqrt {-x^{2} + 1} + \frac {1}{\sqrt {-x^{2} + 1}}\right )} \arcsin \relax (x) - x - \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 61, normalized size = 1.69
method | result | size |
default | \(-x +\arcsin \relax (x ) \sqrt {-x^{2}+1}-\frac {\sqrt {-x^{2}+1}\, \arcsin \relax (x )}{x^{2}-1}-\ln \left (\frac {1}{\sqrt {-x^{2}+1}}+\frac {x}{\sqrt {-x^{2}+1}}\right )\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 45, normalized size = 1.25 \[ -{\left (\frac {x^{2}}{\sqrt {-x^{2} + 1}} - \frac {2}{\sqrt {-x^{2} + 1}}\right )} \arcsin \relax (x) - x - \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^3\,\mathrm {asin}\relax (x)}{{\left (1-x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.48, size = 37, normalized size = 1.03 \[ - x - \left (- \sqrt {1 - x^{2}} - \frac {1}{\sqrt {1 - x^{2}}}\right ) \operatorname {asin}{\relax (x )} + \frac {\log {\left (x - 1 \right )}}{2} - \frac {\log {\left (x + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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