Optimal. Leaf size=14 \[ \frac{\tan ^{-1}\left (\frac{b x}{a}\right )}{a b} \]
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Rubi [A] time = 0.0048183, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {205} \[ \frac{\tan ^{-1}\left (\frac{b x}{a}\right )}{a b} \]
Antiderivative was successfully verified.
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Rule 205
Rubi steps
\begin{align*} \int \frac{1}{a^2+b^2 x^2} \, dx &=\frac{\tan ^{-1}\left (\frac{b x}{a}\right )}{a b}\\ \end{align*}
Mathematica [A] time = 0.0030547, size = 14, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{b x}{a}\right )}{a b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 15, normalized size = 1.1 \begin{align*}{\frac{1}{ab}\arctan \left ({\frac{bx}{a}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40985, size = 19, normalized size = 1.36 \begin{align*} \frac{\arctan \left (\frac{b x}{a}\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95438, size = 28, normalized size = 2. \begin{align*} \frac{\arctan \left (\frac{b x}{a}\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.125087, size = 26, normalized size = 1.86 \begin{align*} \frac{- \frac{i \log{\left (- \frac{i a}{b} + x \right )}}{2} + \frac{i \log{\left (\frac{i a}{b} + x \right )}}{2}}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0527, size = 19, normalized size = 1.36 \begin{align*} \frac{\arctan \left (\frac{b x}{a}\right )}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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