Optimal. Leaf size=16 \[ -\frac{e^{-\coth ^{-1}(a x)}}{a c} \]
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Rubi [A] time = 0.0301235, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {6183} \[ -\frac{e^{-\coth ^{-1}(a x)}}{a c} \]
Antiderivative was successfully verified.
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Rule 6183
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=-\frac{e^{-\coth ^{-1}(a x)}}{a c}\\ \end{align*}
Mathematica [A] time = 0.0446563, size = 16, normalized size = 1. \[ -\frac{e^{-\coth ^{-1}(a x)}}{a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 24, normalized size = 1.5 \begin{align*} -{\frac{1}{ac}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04444, size = 31, normalized size = 1.94 \begin{align*} -\frac{\sqrt{\frac{a x - 1}{a x + 1}}}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82908, size = 46, normalized size = 2.88 \begin{align*} -\frac{\sqrt{\frac{a x - 1}{a x + 1}}}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{2} x^{2} - 1}\, dx}{c} \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*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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