Optimal. Leaf size=31 \[ \frac {x}{2 a}+\frac {1}{2} x^2 \cot ^{-1}(a x)-\frac {\text {ArcTan}(a x)}{2 a^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4947, 327, 209}
\begin {gather*} -\frac {\text {ArcTan}(a x)}{2 a^2}+\frac {1}{2} x^2 \cot ^{-1}(a x)+\frac {x}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 327
Rule 4947
Rubi steps
\begin {align*} \int x \cot ^{-1}(a x) \, dx &=\frac {1}{2} x^2 \cot ^{-1}(a x)+\frac {1}{2} a \int \frac {x^2}{1+a^2 x^2} \, dx\\ &=\frac {x}{2 a}+\frac {1}{2} x^2 \cot ^{-1}(a x)-\frac {\int \frac {1}{1+a^2 x^2} \, dx}{2 a}\\ &=\frac {x}{2 a}+\frac {1}{2} x^2 \cot ^{-1}(a x)-\frac {\tan ^{-1}(a x)}{2 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 31, normalized size = 1.00 \begin {gather*} \frac {x}{2 a}+\frac {1}{2} x^2 \cot ^{-1}(a x)-\frac {\text {ArcTan}(a x)}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 28, normalized size = 0.90
method | result | size |
derivativedivides | \(\frac {\frac {\mathrm {arccot}\left (a x \right ) a^{2} x^{2}}{2}+\frac {a x}{2}-\frac {\arctan \left (a x \right )}{2}}{a^{2}}\) | \(28\) |
default | \(\frac {\frac {\mathrm {arccot}\left (a x \right ) a^{2} x^{2}}{2}+\frac {a x}{2}-\frac {\arctan \left (a x \right )}{2}}{a^{2}}\) | \(28\) |
risch | \(\frac {i x^{2} \ln \left (i a x +1\right )}{4}-\frac {i x^{2} \ln \left (-i a x +1\right )}{4}+\frac {\pi \,x^{2}}{4}+\frac {x}{2 a}-\frac {\arctan \left (a x \right )}{2 a^{2}}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 28, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, x^{2} \operatorname {arccot}\left (a x\right ) + \frac {1}{2} \, a {\left (\frac {x}{a^{2}} - \frac {\arctan \left (a x\right )}{a^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.56, size = 23, normalized size = 0.74 \begin {gather*} \frac {a x + {\left (a^{2} x^{2} + 1\right )} \operatorname {arccot}\left (a x\right )}{2 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 31, normalized size = 1.00 \begin {gather*} \begin {cases} \frac {x^{2} \operatorname {acot}{\left (a x \right )}}{2} + \frac {x}{2 a} + \frac {\operatorname {acot}{\left (a x \right )}}{2 a^{2}} & \text {for}\: a \neq 0 \\\frac {\pi x^{2}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 36, normalized size = 1.16 \begin {gather*} \frac {1}{2} \, {\left (\frac {x^{2} \arctan \left (\frac {1}{a x}\right )}{a} + \frac {x}{a^{2}} + \frac {\arctan \left (\frac {1}{a x}\right )}{a^{3}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.76, size = 39, normalized size = 1.26 \begin {gather*} \left \{\begin {array}{cl} \frac {\pi \,x^2}{4} & \text {\ if\ \ }a=0\\ \frac {x-\frac {\mathrm {atan}\left (a\,x\right )}{a}}{2\,a}+\frac {x^2\,\mathrm {acot}\left (a\,x\right )}{2} & \text {\ if\ \ }a\neq 0 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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