Optimal. Leaf size=62 \[ \frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )^2}-\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )} \]
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Rubi [A] time = 0.125238, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6175, 6178, 793, 651} \[ \frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )^2}-\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 793
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=-\frac{\int \frac{e^{-\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^3 x^3} \, dx}{a^3 c^3}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{\left (1-\frac{x}{a}\right )^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^3 c^3}\\ &=\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )^2}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 a^2 c^3}\\ &=\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )^2}-\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left (a-\frac{1}{x}\right )}\\ \end{align*}
Mathematica [A] time = 0.0549729, size = 34, normalized size = 0.55 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x-2)}{3 c^3 (a x-1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 41, normalized size = 0.7 \begin{align*} -{\frac{ \left ( ax-2 \right ) \left ( ax+1 \right ) }{3\, \left ( ax-1 \right ) ^{2}{c}^{3}a}\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 = 1.04444, size = 53, normalized size = 0.85 \begin{align*} -\frac{\frac{3 \,{\left (a x - 1\right )}}{a x + 1} - 1}{6 \, a c^{3} \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 [A] time = 1.54921, size = 119, normalized size = 1.92 \begin{align*} -\frac{{\left (a^{2} x^{2} - a x - 2\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\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} - 3 a^{2} x^{2} + 3 a x - 1}\, dx}{c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19694, size = 61, normalized size = 0.98 \begin{align*} \frac{2 \,{\left (3 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x - 1\right )}}{3 \,{\left ({\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x - 1\right )}^{3} a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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