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 = 0.0059009, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 4847
Rule 260
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*}
Mathematica [A] time = 0.0027033, size = 24, normalized size = 1. \[ \frac{\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 23, normalized size = 1. \begin{align*} x{\rm arccot} \left (ax\right )+{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{2\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972293, size = 32, normalized size = 1.33 \begin{align*} \frac{2 \, a x \operatorname{arccot}\left (a x\right ) + \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02604, size = 62, normalized size = 2.58 \begin{align*} \frac{2 \, a x \operatorname{arccot}\left (a x\right ) + \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.257776, size = 24, normalized size = 1. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12187, size = 35, normalized size = 1.46 \begin{align*} x \arctan \left (\frac{1}{a x}\right ) + \frac{\log \left (a^{2} x^{2} + 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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