Optimal. Leaf size=36 \[ -x+\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}+\sqrt {1-x^2} \sin ^{-1}(x)-\tanh ^{-1}(x) \]
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Rubi [A]
time = 0.04, 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 = {272, 45, 4779,
396, 212} \begin {gather*} \sqrt {1-x^2} \sin ^{-1}(x)+\frac {\sin ^{-1}(x)}{\sqrt {1-x^2}}-x-\tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 212
Rule 272
Rule 396
Rule 4779
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 = 0.03, size = 40, normalized size = 1.11 \begin {gather*} \frac {1}{2} \left (-2 x-\frac {2 \left (-2+x^2\right ) \sin ^{-1}(x)}{\sqrt {1-x^2}}+\log (1-x)-\log (1+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 9.97, size = 45, normalized size = 1.25 \begin {gather*} \frac {-\text {ArcSin}\left [x\right ] \left (-2+x^2\right )+\frac {\left (-2 x+\text {Log}\left [-1+x\right ]-\text {Log}\left [1+x\right ]\right ) \sqrt {1-x^2}}{2}}{\sqrt {1-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 61, normalized size = 1.69
method | result | size |
default | \(-x +\arcsin \left (x \right ) \sqrt {-x^{2}+1}-\frac {\sqrt {-x^{2}+1}\, \arcsin \left (x \right )}{x^{2}-1}-\ln \left (\frac {1}{\sqrt {-x^{2}+1}}+\frac {x}{\sqrt {-x^{2}+1}}\right )\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 45, normalized size = 1.25 \begin {gather*} -{\left (\frac {x^{2}}{\sqrt {-x^{2} + 1}} - \frac {2}{\sqrt {-x^{2} + 1}}\right )} \arcsin \left (x\right ) - x - \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 57, normalized size = 1.58 \begin {gather*} -\frac {2 \, x^{3} - 2 \, {\left (x^{2} - 2\right )} \sqrt {-x^{2} + 1} \arcsin \left (x\right ) + {\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 8.57, size = 37, normalized size = 1.03 \begin {gather*} - x - \left (- \sqrt {1 - x^{2}} - \frac {1}{\sqrt {1 - x^{2}}}\right ) \operatorname {asin}{\left (x \right )} + \frac {\log {\left (x - 1 \right )}}{2} - \frac {\log {\left (x + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 45, normalized size = 1.25 \begin {gather*} -x+\frac {\ln \left |x-1\right |}{2}-\frac {\ln \left |x+1\right |}{2}+\left (\sqrt {-x^{2}+1}+\frac 1{\sqrt {-x^{2}+1}}\right ) \arcsin x \end {gather*}
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 \begin {gather*} \int \frac {x^3\,\mathrm {asin}\left (x\right )}{{\left (1-x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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