Optimal. Leaf size=25 \[ \frac {2 \sqrt {1-x^2}}{1-x}-\sin ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {677, 222}
\begin {gather*} \frac {2 \sqrt {1-x^2}}{1-x}-\text {ArcSin}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 677
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^2}}{(1-x)^2} \, dx &=\frac {2 \sqrt {1-x^2}}{1-x}-\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {2 \sqrt {1-x^2}}{1-x}-\sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 39, normalized size = 1.56 \begin {gather*} -\frac {2 \sqrt {1-x^2}}{-1+x}+2 \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.56, size = 40, normalized size = 1.60
method | result | size |
risch | \(\frac {2+2 x}{\sqrt {-x^{2}+1}}-\arcsin \left (x \right )\) | \(20\) |
default | \(\frac {\left (-\left (x -1\right )^{2}-2 x +2\right )^{\frac {3}{2}}}{\left (x -1\right )^{2}}+\sqrt {-\left (x -1\right )^{2}-2 x +2}-\arcsin \left (x \right )\) | \(40\) |
trager | \(-\frac {2 \sqrt {-x^{2}+1}}{x -1}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 21, normalized size = 0.84 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1}}{x - 1} - \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.42, size = 41, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left ({\left (x - 1\right )} \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) + x - \sqrt {-x^{2} + 1} - 1\right )}}{x - 1} \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 {- \left (x - 1\right ) \left (x + 1\right )}}{\left (x - 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 21, normalized size = 0.84 \begin {gather*} -\mathrm {asin}\left (x\right )-\frac {2\,\sqrt {1-x^2}}{x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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