Optimal. Leaf size=16 \[ -\frac {1}{2} \log \left (1-x^2\right )+\log (\text {ArcSin}(x)) \]
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Rubi [F]
time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\sqrt {1-x^2}+x \text {ArcSin}(x)}{\text {ArcSin}(x)-x^2 \text {ArcSin}(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^2}+x \sin ^{-1}(x)}{\sin ^{-1}(x)-x^2 \sin ^{-1}(x)} \, dx &=\int \frac {\sqrt {1-x^2}+x \sin ^{-1}(x)}{\left (1-x^2\right ) \sin ^{-1}(x)} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \log \left (1-x^2\right )+\log (\text {ArcSin}(x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 17, normalized size = 1.06
method | result | size |
default | \(-\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}+\ln \left (\arcsin \left (x \right )\right )\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (-\arcsin \left (x\right )\right ) \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 {\sqrt {1 - x^{2}}}{x^{2} \operatorname {asin}{\left (x \right )} - \operatorname {asin}{\left (x \right )}}\, dx - \int \frac {x \operatorname {asin}{\left (x \right )}}{x^{2} \operatorname {asin}{\left (x \right )} - \operatorname {asin}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 20, normalized size = 1.25 \begin {gather*} -\log \left (2\right ) - \frac {1}{2} \, \log \left ({\left | -x^{2} + 1 \right |}\right ) + \log \left ({\left | \arcsin \left (x\right ) \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 12, normalized size = 0.75 \begin {gather*} \ln \left (\mathrm {asin}\left (x\right )\right )-\frac {\ln \left (x^2-1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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