Optimal. Leaf size=32 \[ -\frac {x}{4 \left (x^2+1\right )}-\frac {\cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {1}{4} \tan ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4931, 199, 203} \[ -\frac {x}{4 \left (x^2+1\right )}-\frac {\cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {1}{4} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 199
Rule 203
Rule 4931
Rubi steps
\begin {align*} \int \frac {x \cot ^{-1}(x)}{\left (1+x^2\right )^2} \, dx &=-\frac {\cot ^{-1}(x)}{2 \left (1+x^2\right )}-\frac {1}{2} \int \frac {1}{\left (1+x^2\right )^2} \, dx\\ &=-\frac {x}{4 \left (1+x^2\right )}-\frac {\cot ^{-1}(x)}{2 \left (1+x^2\right )}-\frac {1}{4} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {x}{4 \left (1+x^2\right )}-\frac {\cot ^{-1}(x)}{2 \left (1+x^2\right )}-\frac {1}{4} \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 0.78 \[ -\frac {x^2 \tan ^{-1}(x)+x+\tan ^{-1}(x)+2 \cot ^{-1}(x)}{4 x^2+4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 21, normalized size = 0.66 \[ \frac {{\left (x^{2} - 1\right )} \operatorname {arccot}\relax (x) - x}{4 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 32, normalized size = 1.00 \[ -\frac {\arctan \left (\frac {1}{x}\right )}{2 \, {\left (x^{2} + 1\right )}} - \frac {1}{4 \, x {\left (\frac {1}{x^{2}} + 1\right )}} + \frac {1}{4} \, \arctan \left (\frac {1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 27, normalized size = 0.84 \[ -\frac {x}{4 \left (x^{2}+1\right )}-\frac {\mathrm {arccot}\relax (x )}{2 \left (x^{2}+1\right )}-\frac {\arctan \relax (x )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 26, normalized size = 0.81 \[ -\frac {x}{4 \, {\left (x^{2} + 1\right )}} - \frac {\operatorname {arccot}\relax (x)}{2 \, {\left (x^{2} + 1\right )}} - \frac {1}{4} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 22, normalized size = 0.69 \[ \frac {\mathrm {acot}\relax (x)}{4}-\frac {\frac {x}{4}+\frac {\mathrm {acot}\relax (x)}{2}}{x^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.50, size = 31, normalized size = 0.97 \[ \frac {x^{2} \operatorname {acot}{\relax (x )}}{4 x^{2} + 4} - \frac {x}{4 x^{2} + 4} - \frac {\operatorname {acot}{\relax (x )}}{4 x^{2} + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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