Optimal. Leaf size=21 \[ -\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}+\log (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4771, 29}
\begin {gather*} \log (x)-\frac {\sqrt {1-x^2} \text {ArcSin}(x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 4771
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(x)}{x^2 \sqrt {1-x^2}} \, dx &=-\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}+\int \frac {1}{x} \, dx\\ &=-\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}+\log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 20, normalized size = 0.95
method | result | size |
default | \(\ln \left (x \right )-\frac {\arcsin \left (x \right ) \sqrt {-x^{2}+1}}{x}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.06, size = 19, normalized size = 0.90 \begin {gather*} -\frac {\sqrt {-x^{2} + 1} \arcsin \left (x\right )}{x} + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 22, normalized size = 1.05 \begin {gather*} \frac {x \log \left (x\right ) - \sqrt {-x^{2} + 1} \arcsin \left (x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asin}{\left (x \right )}}{x^{2} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs.
\(2 (19) = 38\).
time = 0.45, size = 40, normalized size = 1.90 \begin {gather*} \frac {1}{2} \, {\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )} \arcsin \left (x\right ) + \log \left ({\left | x \right |}\right ) \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 = -0.05 \begin {gather*} \int \frac {\mathrm {asin}\left (x\right )}{x^2\,\sqrt {1-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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