Optimal. Leaf size=21 \[ \frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} \tan ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 21, 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 = {4947, 327, 209}
\begin {gather*} -\frac {\text {ArcTan}(x)}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)+\frac {x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 327
Rule 4947
Rubi steps
\begin {align*} \int x \cot ^{-1}(x) \, dx &=\frac {1}{2} x^2 \cot ^{-1}(x)+\frac {1}{2} \int \frac {x^2}{1+x^2} \, dx\\ &=\frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} \int \frac {1}{1+x^2} \, dx\\ &=\frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} \frac {x}{2}+\frac {1}{2} x^2 \cot ^{-1}(x)-\frac {1}{2} \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 16, normalized size = 0.76
method | result | size |
default | \(\frac {x}{2}+\frac {x^{2} \mathrm {arccot}\left (x \right )}{2}-\frac {\arctan \left (x \right )}{2}\) | \(16\) |
risch | \(\frac {i x^{2} \ln \left (i x +1\right )}{4}-\frac {i x^{2} \ln \left (-i x +1\right )}{4}+\frac {\pi \,x^{2}}{4}+\frac {x}{2}-\frac {\arctan \left (x \right )}{2}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.68, size = 15, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {arccot}\left (x\right ) + \frac {1}{2} \, x - \frac {1}{2} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.03, size = 13, normalized size = 0.62 \begin {gather*} \frac {1}{2} \, {\left (x^{2} + 1\right )} \operatorname {arccot}\left (x\right ) + \frac {1}{2} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 15, normalized size = 0.71 \begin {gather*} \frac {x^{2} \operatorname {acot}{\left (x \right )}}{2} + \frac {x}{2} + \frac {\operatorname {acot}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 19, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, x^{2} \arctan \left (\frac {1}{x}\right ) + \frac {1}{2} \, x + \frac {1}{2} \, \arctan \left (\frac {1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 15, normalized size = 0.71 \begin {gather*} \frac {x}{2}-\frac {\mathrm {atan}\left (x\right )}{2}+\frac {x^2\,\mathrm {acot}\left (x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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