Optimal. Leaf size=17 \[ x \cot ^{-1}\left (\frac {1}{x}\right )-\frac {1}{2} \log \left (x^2+1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5028, 263, 260} \[ x \cot ^{-1}\left (\frac {1}{x}\right )-\frac {1}{2} \log \left (x^2+1\right ) \]
Antiderivative was successfully verified.
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Rule 260
Rule 263
Rule 5028
Rubi steps
\begin {align*} \int \cot ^{-1}\left (\frac {1}{x}\right ) \, dx &=x \cot ^{-1}\left (\frac {1}{x}\right )-\int \frac {1}{\left (1+\frac {1}{x^2}\right ) x} \, dx\\ &=x \cot ^{-1}\left (\frac {1}{x}\right )-\int \frac {x}{1+x^2} \, dx\\ &=x \cot ^{-1}\left (\frac {1}{x}\right )-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \[ x \cot ^{-1}\left (\frac {1}{x}\right )-\frac {1}{2} \log \left (x^2+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 15, normalized size = 0.88 \[ x \operatorname {arccot}\left (\frac {1}{x}\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 13, normalized size = 0.76 \[ x \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 20, normalized size = 1.18 \[ x \,\mathrm {arccot}\left (\frac {1}{x}\right )+\ln \left (\frac {1}{x}\right )-\frac {\ln \left (\frac {1}{x^{2}}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 15, normalized size = 0.88 \[ x \operatorname {arccot}\left (\frac {1}{x}\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 15, normalized size = 0.88 \[ x\,\mathrm {acot}\left (\frac {1}{x}\right )-\frac {\ln \left (x^2+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 14, normalized size = 0.82 \[ x \operatorname {acot}{\left (\frac {1}{x} \right )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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