Optimal. Leaf size=13 \[ \frac{e^{\coth ^{-1}(a x)}}{a c} \]
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Rubi [A] time = 0.0288307, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, 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 = 0.0438475, size = 13, 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 = 23, normalized size = 1.8 \begin{align*}{\frac{1}{ac}{\frac{1}{\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 = 1.08499, size = 30, normalized size = 2.31 \begin{align*} \frac{1}{a c \sqrt{\frac{a x - 1}{a x + 1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54037, size = 72, normalized size = 5.54 \begin{align*} \frac{{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c x - 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{1}{a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} - \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12888, size = 30, normalized size = 2.31 \begin{align*} \frac{1}{a c \sqrt{\frac{a x - 1}{a x + 1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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