Optimal. Leaf size=18 \[ \frac{e^{3 \coth ^{-1}(a x)}}{3 a c} \]
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Rubi [A] time = 0.0313877, antiderivative size = 18, 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} \[ \frac{e^{3 \coth ^{-1}(a x)}}{3 a c} \]
Antiderivative was successfully verified.
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Rule 6183
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=\frac{e^{3 \coth ^{-1}(a x)}}{3 a c}\\ \end{align*}
Mathematica [A] time = 0.0464494, size = 18, normalized size = 1. \[ \frac{e^{3 \coth ^{-1}(a x)}}{3 a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.123, size = 24, normalized size = 1.3 \begin{align*}{\frac{1}{3\,ac} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0772, size = 31, normalized size = 1.72 \begin{align*} \frac{1}{3 \, a c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.58278, size = 112, normalized size = 6.22 \begin{align*} \frac{{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \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}{\frac{a^{3} x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} + \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 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 = 1.17372, size = 47, normalized size = 2.61 \begin{align*} \frac{a x + 1}{3 \,{\left (a x - 1\right )} 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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