Optimal. Leaf size=18 \[ -\frac{e^{-3 \tanh ^{-1}(a x)}}{3 a c} \]
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Rubi [A] time = 0.0332538, 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 = {6137} \[ -\frac{e^{-3 \tanh ^{-1}(a x)}}{3 a c} \]
Antiderivative was successfully verified.
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Rule 6137
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=-\frac{e^{-3 \tanh ^{-1}(a x)}}{3 a c}\\ \end{align*}
Mathematica [A] time = 0.0147526, size = 29, normalized size = 1.61 \[ -\frac{(1-a x)^{3/2}}{3 a c (a x+1)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.029, size = 28, normalized size = 1.6 \begin{align*} -{\frac{1}{3\, \left ( ax+1 \right ) ^{3}ca} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a^{2} c x^{2} - c\right )}{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.69853, size = 120, normalized size = 6.67 \begin{align*} -\frac{a^{2} x^{2} + 2 \, a x - \sqrt{-a^{2} x^{2} + 1}{\left (a x - 1\right )} + 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{\sqrt{- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1546, size = 89, normalized size = 4.94 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + 1\right )}}{3 \, c{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}^{3}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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