Optimal. Leaf size=25 \[ \frac{\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x) \]
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Rubi [A] time = 0.0068101, antiderivative size = 25, 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 = {5911, 260} \[ \frac{\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 5911
Rule 260
Rubi steps
\begin{align*} \int \coth ^{-1}(a x) \, dx &=x \coth ^{-1}(a x)-a \int \frac{x}{1-a^2 x^2} \, dx\\ &=x \coth ^{-1}(a x)+\frac{\log \left (1-a^2 x^2\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0030681, size = 25, normalized size = 1. \[ \frac{\log \left (1-a^2 x^2\right )}{2 a}+x \coth ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 23, normalized size = 0.9 \begin{align*} x{\rm arccoth} \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 = 0.986804, size = 34, normalized size = 1.36 \begin{align*} \frac{2 \, a x \operatorname{arcoth}\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 = 1.68022, size = 77, normalized size = 3.08 \begin{align*} \frac{a x \log \left (\frac{a x + 1}{a x - 1}\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 = 0.609134, size = 27, normalized size = 1.08 \begin{align*} \begin{cases} x \operatorname{acoth}{\left (a x \right )} + \frac{\log{\left (a x + 1 \right )}}{a} - \frac{\operatorname{acoth}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\\frac{i \pi x}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arcoth}\left (a x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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