Optimal. Leaf size=44 \[ -\frac {3 x}{32 \left (x^2+1\right )}-\frac {x}{16 \left (x^2+1\right )^2}-\frac {\cot ^{-1}(x)}{4 \left (x^2+1\right )^2}-\frac {3}{32} \tan ^{-1}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4931, 199, 203} \[ -\frac {3 x}{32 \left (x^2+1\right )}-\frac {x}{16 \left (x^2+1\right )^2}-\frac {\cot ^{-1}(x)}{4 \left (x^2+1\right )^2}-\frac {3}{32} \tan ^{-1}(x) \]
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 )^3} \, dx &=-\frac {\cot ^{-1}(x)}{4 \left (1+x^2\right )^2}-\frac {1}{4} \int \frac {1}{\left (1+x^2\right )^3} \, dx\\ &=-\frac {x}{16 \left (1+x^2\right )^2}-\frac {\cot ^{-1}(x)}{4 \left (1+x^2\right )^2}-\frac {3}{16} \int \frac {1}{\left (1+x^2\right )^2} \, dx\\ &=-\frac {x}{16 \left (1+x^2\right )^2}-\frac {3 x}{32 \left (1+x^2\right )}-\frac {\cot ^{-1}(x)}{4 \left (1+x^2\right )^2}-\frac {3}{32} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {x}{16 \left (1+x^2\right )^2}-\frac {3 x}{32 \left (1+x^2\right )}-\frac {\cot ^{-1}(x)}{4 \left (1+x^2\right )^2}-\frac {3}{32} \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 36, normalized size = 0.82 \[ -\frac {x \left (3 x^2+5\right )+3 \left (x^2+1\right )^2 \tan ^{-1}(x)+8 \cot ^{-1}(x)}{32 \left (x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.97, size = 39, normalized size = 0.89 \[ -\frac {3 \, x^{3} - {\left (3 \, x^{4} + 6 \, x^{2} - 5\right )} \operatorname {arccot}\relax (x) + 5 \, x}{32 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 40, normalized size = 0.91 \[ -\frac {\frac {3}{x} + \frac {5}{x^{3}}}{32 \, {\left (\frac {1}{x^{2}} + 1\right )}^{2}} - \frac {\arctan \left (\frac {1}{x}\right )}{4 \, {\left (x^{2} + 1\right )}^{2}} + \frac {3}{32} \, \arctan \left (\frac {1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 37, normalized size = 0.84 \[ -\frac {x}{16 \left (x^{2}+1\right )^{2}}-\frac {3 x}{32 \left (x^{2}+1\right )}-\frac {\mathrm {arccot}\relax (x )}{4 \left (x^{2}+1\right )^{2}}-\frac {3 \arctan \relax (x )}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 39, normalized size = 0.89 \[ -\frac {3 \, x^{3} + 5 \, x}{32 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} - \frac {\operatorname {arccot}\relax (x)}{4 \, {\left (x^{2} + 1\right )}^{2}} - \frac {3}{32} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 27, normalized size = 0.61 \[ -\frac {3\,\mathrm {atan}\relax (x)}{32}-\frac {\frac {5\,x}{32}+\frac {\mathrm {acot}\relax (x)}{4}+\frac {3\,x^3}{32}}{{\left (x^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.80, size = 88, normalized size = 2.00 \[ \frac {3 x^{4} \operatorname {acot}{\relax (x )}}{32 x^{4} + 64 x^{2} + 32} - \frac {3 x^{3}}{32 x^{4} + 64 x^{2} + 32} + \frac {6 x^{2} \operatorname {acot}{\relax (x )}}{32 x^{4} + 64 x^{2} + 32} - \frac {5 x}{32 x^{4} + 64 x^{2} + 32} - \frac {5 \operatorname {acot}{\relax (x )}}{32 x^{4} + 64 x^{2} + 32} \]
Verification of antiderivative is not currently implemented for this CAS.
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