Optimal. Leaf size=19 \[ \frac{1}{2} \log \left (a^2+x^2\right )+\tan ^{-1}\left (\frac{x}{a}\right ) \]
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Rubi [A] time = 0.0062446, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {635, 203, 260} \[ \frac{1}{2} \log \left (a^2+x^2\right )+\tan ^{-1}\left (\frac{x}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{a+x}{a^2+x^2} \, dx &=a \int \frac{1}{a^2+x^2} \, dx+\int \frac{x}{a^2+x^2} \, dx\\ &=\tan ^{-1}\left (\frac{x}{a}\right )+\frac{1}{2} \log \left (a^2+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0042114, size = 19, normalized size = 1. \[ \frac{1}{2} \log \left (a^2+x^2\right )+\tan ^{-1}\left (\frac{x}{a}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 18, normalized size = 1. \begin{align*} \arctan \left ({\frac{x}{a}} \right ) +{\frac{\ln \left ({a}^{2}+{x}^{2} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42015, size = 23, normalized size = 1.21 \begin{align*} \arctan \left (\frac{x}{a}\right ) + \frac{1}{2} \, \log \left (a^{2} + x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09241, size = 46, normalized size = 2.42 \begin{align*} \arctan \left (\frac{x}{a}\right ) + \frac{1}{2} \, \log \left (a^{2} + x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.101728, size = 42, normalized size = 2.21 \begin{align*} \left (\frac{1}{2} - \frac{i}{2}\right ) \log{\left (- a + 2 a \left (\frac{1}{2} - \frac{i}{2}\right ) + x \right )} + \left (\frac{1}{2} + \frac{i}{2}\right ) \log{\left (- a + 2 a \left (\frac{1}{2} + \frac{i}{2}\right ) + x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05321, size = 23, normalized size = 1.21 \begin{align*} \arctan \left (\frac{x}{a}\right ) + \frac{1}{2} \, \log \left (a^{2} + x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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