Optimal. Leaf size=14 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b} \]
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Rubi [A] time = 0.0076569, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {208} \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b} \]
Antiderivative was successfully verified.
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Rule 208
Rubi steps
\begin{align*} \int \frac{1}{a^2-b^2 x^2} \, dx &=\frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b}\\ \end{align*}
Mathematica [A] time = 0.0032388, size = 14, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{a b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 32, normalized size = 2.3 \begin{align*} -{\frac{\ln \left ( bx-a \right ) }{2\,ab}}+{\frac{\ln \left ( bx+a \right ) }{2\,ab}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.931373, size = 42, normalized size = 3. \begin{align*} \frac{\log \left (b x + a\right )}{2 \, a b} - \frac{\log \left (b x - a\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99127, size = 55, normalized size = 3.93 \begin{align*} \frac{\log \left (b x + a\right ) - \log \left (b x - a\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.130621, size = 20, normalized size = 1.43 \begin{align*} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{2} - \frac{\log{\left (\frac{a}{b} + x \right )}}{2}}{a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05949, size = 45, normalized size = 3.21 \begin{align*} \frac{\log \left ({\left | b x + a \right |}\right )}{2 \, a b} - \frac{\log \left ({\left | b x - a \right |}\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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