Optimal. Leaf size=30 \[ \frac {1}{2} \log \left (x^2+1\right )-\log (x)+\frac {1}{2} \cot ^{-1}(x)^2-\frac {\cot ^{-1}(x)}{x} \]
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Rubi [A] time = 0.06, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {4919, 4853, 266, 36, 29, 31, 4885} \[ \frac {1}{2} \log \left (x^2+1\right )-\log (x)+\frac {1}{2} \cot ^{-1}(x)^2-\frac {\cot ^{-1}(x)}{x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 4853
Rule 4885
Rule 4919
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(x)}{x^2 \left (1+x^2\right )} \, dx &=\int \frac {\cot ^{-1}(x)}{x^2} \, dx-\int \frac {\cot ^{-1}(x)}{1+x^2} \, dx\\ &=-\frac {\cot ^{-1}(x)}{x}+\frac {1}{2} \cot ^{-1}(x)^2-\int \frac {1}{x \left (1+x^2\right )} \, dx\\ &=-\frac {\cot ^{-1}(x)}{x}+\frac {1}{2} \cot ^{-1}(x)^2-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,x^2\right )\\ &=-\frac {\cot ^{-1}(x)}{x}+\frac {1}{2} \cot ^{-1}(x)^2-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )\\ &=-\frac {\cot ^{-1}(x)}{x}+\frac {1}{2} \cot ^{-1}(x)^2-\log (x)+\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ \frac {1}{2} \log \left (x^2+1\right )-\log (x)+\frac {1}{2} \cot ^{-1}(x)^2-\frac {\cot ^{-1}(x)}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 29, normalized size = 0.97 \[ \frac {x \operatorname {arccot}\relax (x)^{2} + x \log \left (x^{2} + 1\right ) - 2 \, x \log \relax (x) - 2 \, \operatorname {arccot}\relax (x)}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 26, normalized size = 0.87 \[ \frac {1}{2} \, \arctan \left (\frac {1}{x}\right )^{2} - \frac {\arctan \left (\frac {1}{x}\right )}{x} + \frac {1}{2} \, \log \left (\frac {1}{x^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 33, normalized size = 1.10 \[ -\frac {\mathrm {arccot}\relax (x )}{x}-\mathrm {arccot}\relax (x ) \arctan \relax (x )-\ln \relax (x )+\frac {\ln \left (x^{2}+1\right )}{2}-\frac {\arctan \relax (x )^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 29, normalized size = 0.97 \[ -{\left (\frac {1}{x} + \arctan \relax (x)\right )} \operatorname {arccot}\relax (x) - \frac {1}{2} \, \arctan \relax (x)^{2} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 26, normalized size = 0.87 \[ \frac {\ln \left (x^2+1\right )}{2}-\ln \relax (x)-\frac {\mathrm {acot}\relax (x)}{x}+\frac {{\mathrm {acot}\relax (x)}^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 22, normalized size = 0.73 \[ - \log {\relax (x )} + \frac {\log {\left (x^{2} + 1 \right )}}{2} + \frac {\operatorname {acot}^{2}{\relax (x )}}{2} - \frac {\operatorname {acot}{\relax (x )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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