Optimal. Leaf size=24 \[ \frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x) \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4847, 260} \[ \frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 260
Rule 4847
Rubi steps
\begin {align*} \int \cot ^{-1}(a x) \, dx &=x \cot ^{-1}(a x)+a \int \frac {x}{1+a^2 x^2} \, dx\\ &=x \cot ^{-1}(a x)+\frac {\log \left (1+a^2 x^2\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 24, normalized size = 1.00 \[ \frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 24, normalized size = 1.00 \[ \frac {2 \, a x \operatorname {arccot}\left (a x\right ) + \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 45, normalized size = 1.88 \[ \frac {1}{2} \, a {\left (\frac {2 \, x \arctan \left (\frac {1}{a x}\right )}{a} + \frac {\log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{2}} - \frac {\log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 23, normalized size = 0.96 \[ x \,\mathrm {arccot}\left (a x \right )+\frac {\ln \left (a^{2} x^{2}+1\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 24, normalized size = 1.00 \[ \frac {2 \, a x \operatorname {arccot}\left (a x\right ) + \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 22, normalized size = 0.92 \[ x\,\mathrm {acot}\left (a\,x\right )+\frac {\ln \left (a^2\,x^2+1\right )}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 24, normalized size = 1.00 \[ \begin {cases} x \operatorname {acot}{\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2 a} & \text {for}\: a \neq 0 \\\frac {\pi x}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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