Optimal. Leaf size=5 \[ -\log \left (\cot ^{-1}(x)\right ) \]
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Rubi [A]
time = 0.01, 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 = {5003}
\begin {gather*} -\log \left (\cot ^{-1}(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 5003
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 = 0.02, size = 5, normalized size = 1.00 \begin {gather*} -\log \left (\cot ^{-1}(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 6, normalized size = 1.20
method | result | size |
derivativedivides | \(-\ln \left (\mathrm {arccot}\left (x \right )\right )\) | \(6\) |
default | \(-\ln \left (\mathrm {arccot}\left (x \right )\right )\) | \(6\) |
risch | \(-\ln \left (\ln \left (i x +1\right )+i \left (i \ln \left (-i x +1\right )-\pi \right )\right )\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 5, normalized size = 1.00 \begin {gather*} -\log \left (\operatorname {arccot}\left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 6.54, size = 5, normalized size = 1.00 \begin {gather*} -\log \left (\operatorname {arccot}\left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 5, normalized size = 1.00 \begin {gather*} - \log {\left (\operatorname {acot}{\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 8, normalized size = 1.60 \begin {gather*} -\log \left ({\left | \arctan \left (\frac {1}{x}\right ) \right |}\right ) \end {gather*}
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 \begin {gather*} -\ln \left (\mathrm {acot}\left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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