Optimal. Leaf size=17 \[ \frac {\cos ^{-1}(x)}{\sqrt {1-x^2}}+\tanh ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4678, 206} \[ \frac {\cos ^{-1}(x)}{\sqrt {1-x^2}}+\tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 206
Rule 4678
Rubi steps
\begin {align*} \int \frac {x \cos ^{-1}(x)}{\left (1-x^2\right )^{3/2}} \, dx &=\frac {\cos ^{-1}(x)}{\sqrt {1-x^2}}+\int \frac {1}{1-x^2} \, dx\\ &=\frac {\cos ^{-1}(x)}{\sqrt {1-x^2}}+\tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 32, normalized size = 1.88 \[ \frac {1}{2} \left (\frac {2 \cos ^{-1}(x)}{\sqrt {1-x^2}}-\log (1-x)+\log (x+1)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \cos ^{-1}(x)}{\left (1-x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.19, size = 44, normalized size = 2.59 \[ \frac {{\left (x^{2} - 1\right )} \log \left (x + 1\right ) - {\left (x^{2} - 1\right )} \log \left (x - 1\right ) - 2 \, \sqrt {-x^{2} + 1} \arccos \relax (x)}{2 \, {\left (x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 27, normalized size = 1.59 \[ \frac {\arccos \relax (x)}{\sqrt {-x^{2} + 1}} + \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.31, size = 47, normalized size = 2.76
method | result | size |
default | \(-\frac {\sqrt {-x^{2}+1}\, \arccos \relax (x )}{x^{2}-1}-\ln \left (\frac {1}{\sqrt {-x^{2}+1}}-\frac {x}{\sqrt {-x^{2}+1}}\right )\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 25, normalized size = 1.47 \[ \frac {\arccos \relax (x)}{\sqrt {-x^{2} + 1}} + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {x\,\mathrm {acos}\relax (x)}{{\left (1-x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.91, size = 20, normalized size = 1.18 \[ \begin {cases} \operatorname {acoth}{\relax (x )} & \text {for}\: x^{2} > 1 \\\operatorname {atanh}{\relax (x )} & \text {for}\: x^{2} < 1 \end {cases} + \frac {\operatorname {acos}{\relax (x )}}{\sqrt {1 - x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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