Optimal. Leaf size=67 \[ \frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{5 c^3 \left (a-\frac{1}{x}\right )^4}-\frac{4 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{15 c^3 \left (a-\frac{1}{x}\right )^3} \]
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Rubi [A] time = 0.124528, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6175, 6178, 793, 651} \[ \frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{5 c^3 \left (a-\frac{1}{x}\right )^4}-\frac{4 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{15 c^3 \left (a-\frac{1}{x}\right )^3} \]
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 \sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^4} \, dx,x,\frac{1}{x}\right )}{a^3 c^3}\\ &=\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{5 c^3 \left (a-\frac{1}{x}\right )^4}-\frac{4 \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )}{5 a^2 c^3}\\ &=\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{5 c^3 \left (a-\frac{1}{x}\right )^4}-\frac{4 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{15 c^3 \left (a-\frac{1}{x}\right )^3}\\ \end{align*}
Mathematica [A] time = 0.052161, size = 42, normalized size = 0.63 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (a^2 x^2-3 a x-4\right )}{15 c^3 (a x-1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 41, normalized size = 0.6 \begin{align*} -{\frac{ \left ( ax-4 \right ) \left ( ax+1 \right ) }{15\,{c}^{3} \left ( ax-1 \right ) ^{2}a}{\frac{1}{\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 = 1.0957, size = 53, normalized size = 0.79 \begin{align*} -\frac{\frac{5 \,{\left (a x - 1\right )}}{a x + 1} - 3}{30 \, a c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50954, size = 161, normalized size = 2.4 \begin{align*} -\frac{{\left (a^{3} x^{3} - 2 \, a^{2} x^{2} - 7 \, a x - 4\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, 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{1}{a^{3} x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} - 3 a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} + 3 a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} - \sqrt{\frac{a x}{a x + 1} - \frac{1}{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 = 1.17073, size = 72, normalized size = 1.07 \begin{align*} -\frac{{\left (a x + 1\right )}^{2}{\left (\frac{5 \,{\left (a x - 1\right )}}{a x + 1} - 3\right )}}{30 \,{\left (a x - 1\right )}^{2} a c^{3} \sqrt{\frac{a x - 1}{a x + 1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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