Optimal. Leaf size=5 \[ -\log \left (\cot ^{-1}(x)\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4883} \[ -\log \left (\cot ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Rule 4883
Rubi steps
\begin {align*} \int \frac {1}{\left (1+x^2\right ) \cot ^{-1}(x)} \, dx &=-\log \left (\cot ^{-1}(x)\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 5, normalized size = 1.00 \[ -\log \left (\cot ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 5, normalized size = 1.00 \[ -\log \left (\operatorname {arccot}\relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 8, normalized size = 1.60 \[ -\log \left ({\left | \arctan \left (\frac {1}{x}\right ) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 6, normalized size = 1.20 \[ -\ln \left (\mathrm {arccot}\relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 5, normalized size = 1.00 \[ -\log \left (\operatorname {arccot}\relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 5, normalized size = 1.00 \[ -\ln \left (\mathrm {acot}\relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 5, normalized size = 1.00 \[ - \log {\left (\operatorname {acot}{\relax (x )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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