Optimal. Leaf size=67 \[ \frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 c^3 \left (a-\frac{1}{x}\right )^6}-\frac{6 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{35 c^3 \left (a-\frac{1}{x}\right )^5} \]
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Rubi [A] time = 0.134002, antiderivative size = 67, 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} \[ \frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 c^3 \left (a-\frac{1}{x}\right )^6}-\frac{6 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{35 c^3 \left (a-\frac{1}{x}\right )^5} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 793
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=-\frac{\int \frac{e^{3 \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^2}{a^2}\right )^{3/2}}{\left (1-\frac{x}{a}\right )^6} \, dx,x,\frac{1}{x}\right )}{a^3 c^3}\\ &=\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 c^3 \left (a-\frac{1}{x}\right )^6}-\frac{6 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (1-\frac{x}{a}\right )^5} \, dx,x,\frac{1}{x}\right )}{7 a^2 c^3}\\ &=\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{7 c^3 \left (a-\frac{1}{x}\right )^6}-\frac{6 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{35 c^3 \left (a-\frac{1}{x}\right )^5}\\ \end{align*}
Mathematica [A] time = 0.0612204, size = 41, normalized size = 0.61 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x-6) (a x+1)^2}{35 c^3 (a x-1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 41, normalized size = 0.6 \begin{align*} -{\frac{ \left ( ax-6 \right ) \left ( ax+1 \right ) }{35\,{c}^{3} \left ( ax-1 \right ) ^{2}a} \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 = 1.03509, size = 53, normalized size = 0.79 \begin{align*} -\frac{\frac{7 \,{\left (a x - 1\right )}}{a x + 1} - 5}{70 \, a c^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63554, size = 201, normalized size = 3. \begin{align*} -\frac{{\left (a^{4} x^{4} - 3 \, a^{3} x^{3} - 15 \, a^{2} x^{2} - 17 \, a x - 6\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{35 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, 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(-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 = 1.30981, size = 72, normalized size = 1.07 \begin{align*} -\frac{{\left (a x + 1\right )}^{3}{\left (\frac{7 \,{\left (a x - 1\right )}}{a x + 1} - 5\right )}}{70 \,{\left (a x - 1\right )}^{3} 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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