Optimal. Leaf size=34 \[ -\frac {1}{4 \left (1+x^2\right )}+\frac {x \cot ^{-1}(x)}{2 \left (1+x^2\right )}-\frac {1}{4} \cot ^{-1}(x)^2 \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5013, 267}
\begin {gather*} -\frac {1}{4 \left (x^2+1\right )}+\frac {x \cot ^{-1}(x)}{2 \left (x^2+1\right )}-\frac {1}{4} \cot ^{-1}(x)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5013
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(x)}{\left (1+x^2\right )^2} \, dx &=\frac {x \cot ^{-1}(x)}{2 \left (1+x^2\right )}-\frac {1}{4} \cot ^{-1}(x)^2+\frac {1}{2} \int \frac {x}{\left (1+x^2\right )^2} \, dx\\ &=-\frac {1}{4 \left (1+x^2\right )}+\frac {x \cot ^{-1}(x)}{2 \left (1+x^2\right )}-\frac {1}{4} \cot ^{-1}(x)^2\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.82 \begin {gather*} -\frac {1-2 x \cot ^{-1}(x)+\left (1+x^2\right ) \cot ^{-1}(x)^2}{4 \left (1+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 35, normalized size = 1.03
method | result | size |
default | \(\frac {x \,\mathrm {arccot}\left (x \right )}{2 x^{2}+2}+\frac {\mathrm {arccot}\left (x \right ) \arctan \left (x \right )}{2}-\frac {1}{4 \left (x^{2}+1\right )}+\frac {\arctan \left (x \right )^{2}}{4}\) | \(35\) |
risch | \(\frac {\ln \left (i x +1\right )^{2}}{16}-\frac {\left (x^{2} \ln \left (-i x +1\right )-2 i x +\ln \left (-i x +1\right )\right ) \ln \left (i x +1\right )}{8 \left (x^{2}+1\right )}+\frac {x^{2} \ln \left (-i x +1\right )^{2}+\ln \left (-i x +1\right )^{2}+2 i \pi \ln \left (i+x \right ) x^{2}+2 i \ln \left (i+x \right ) \pi -2 i \pi \ln \left (x -i\right ) x^{2}-2 i \pi \ln \left (x -i\right )+4 \pi x -4-4 i \ln \left (-i x +1\right ) x}{16 \left (i+x \right ) \left (x -i\right )}\) | \(147\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 38, normalized size = 1.12 \begin {gather*} \frac {1}{2} \, {\left (\frac {x}{x^{2} + 1} + \arctan \left (x\right )\right )} \operatorname {arccot}\left (x\right ) + \frac {{\left (x^{2} + 1\right )} \arctan \left (x\right )^{2} - 1}{4 \, {\left (x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 26, normalized size = 0.76 \begin {gather*} -\frac {{\left (x^{2} + 1\right )} \operatorname {arccot}\left (x\right )^{2} - 2 \, x \operatorname {arccot}\left (x\right ) + 1}{4 \, {\left (x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RecursionError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.61, size = 22, normalized size = 0.65 \begin {gather*} \frac {\frac {x\,\mathrm {acot}\left (x\right )}{2}-\frac {1}{4}}{x^2+1}-\frac {{\mathrm {acot}\left (x\right )}^2}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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