Optimal. Leaf size=18 \[ -\frac{e^{-3 \coth ^{-1}(a x)}}{3 a c} \]
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Rubi [A] time = 0.03191, 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 = 0.047077, 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 = 0.043, 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 = 1.04226, size = 31, normalized size = 1.72 \begin{align*} -\frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{3 \, a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51913, size = 78, normalized size = 4.33 \begin{align*} -\frac{{\left (a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (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{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{3} x^{3} + a^{2} x^{2} - a x - 1}\, dx + \int \frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{3} x^{3} + a^{2} x^{2} - a x - 1}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20713, size = 66, normalized size = 3.67 \begin{align*} \frac{2 \,{\left (3 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )}^{2} x^{2} + 1\right )}}{3 \,{\left ({\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x + 1\right )}^{3} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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