Optimal. Leaf size=45 \[ \frac{x}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2}{a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.0494055, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6177, 271, 191} \[ \frac{x}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2}{a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 271
Rule 191
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{x}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{a^2 c^3}\\ &=-\frac{2}{a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}} x}+\frac{x}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0220276, size = 33, normalized size = 0.73 \[ \frac{a^2 x^2-2}{a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 44, normalized size = 1. \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}-2 \right ) \left ( ax+1 \right ) }{a \left ( ax-1 \right ) ^{2}{c}^{3}} \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 [B] time = 1.02256, size = 124, normalized size = 2.76 \begin{align*} -\frac{1}{2} \, a{\left (\frac{\frac{5 \,{\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - a^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77781, size = 82, normalized size = 1.82 \begin{align*} \frac{{\left (a^{2} x^{2} - 2\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3} x - a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22081, size = 51, normalized size = 1.13 \begin{align*} \frac{{\left (\sqrt{a^{2} x^{2} - 1} - \frac{1}{\sqrt{a^{2} x^{2} - 1}}\right )} \mathrm{sgn}\left (a x + 1\right )}{a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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