Optimal. Leaf size=133 \[ \frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac{1}{x}\right )^6}-\frac{8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac{1}{x}\right )^5}+\frac{47 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{58 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac{1}{x}\right )^3} \]
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Rubi [A] time = 0.34239, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6175, 6178, 1639, 793, 659, 651} \[ \frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac{1}{x}\right )^6}-\frac{8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac{1}{x}\right )^5}+\frac{47 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{58 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac{1}{x}\right )^3} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 1639
Rule 793
Rule 659
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx &=-\frac{\int \frac{e^{\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^5 x^5} \, dx}{a^5 c^5}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3 \sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^6} \, dx,x,\frac{1}{x}\right )}{a^5 c^5}\\ &=\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac{1}{x}\right )^4}-\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{4}{a^2}-\frac{7 x}{a^3}+\frac{2 x^2}{a^4}\right ) \sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^6} \, dx,x,\frac{1}{x}\right )}{c^5}\\ &=\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac{1}{x}\right )^5}+\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac{1}{x}\right )^4}-\frac{a^4 \operatorname{Subst}\left (\int \frac{\left (\frac{18}{a^6}-\frac{20 x}{a^7}\right ) \sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^6} \, dx,x,\frac{1}{x}\right )}{2 c^5}\\ &=\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac{1}{x}\right )^6}+\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac{1}{x}\right )^5}+\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac{1}{x}\right )^4}-\frac{29 \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^5} \, dx,x,\frac{1}{x}\right )}{3 a^2 c^5}\\ &=\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac{1}{x}\right )^6}-\frac{8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac{1}{x}\right )^5}+\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^5 \left (a-\frac{1}{x}\right )^4}-\frac{58 \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^4} \, dx,x,\frac{1}{x}\right )}{21 a^2 c^5}\\ &=\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac{1}{x}\right )^6}-\frac{8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac{1}{x}\right )^5}+\frac{47 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{58 \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 )}{105 a^2 c^5}\\ &=\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{9 c^5 \left (a-\frac{1}{x}\right )^6}-\frac{8 a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{21 c^5 \left (a-\frac{1}{x}\right )^5}+\frac{47 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{58 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{315 c^5 \left (a-\frac{1}{x}\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0616556, size = 59, normalized size = 0.44 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^4 x^4-10 a^3 x^3+21 a^2 x^2-25 a x-58\right )}{315 c^5 (a x-1)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 58, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 2\,{x}^{3}{a}^{3}-12\,{a}^{2}{x}^{2}+33\,ax-58 \right ) \left ( ax+1 \right ) }{315\,{c}^{5} \left ( ax-1 \right ) ^{4}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 = 1.02514, size = 96, normalized size = 0.72 \begin{align*} -\frac{\frac{135 \,{\left (a x - 1\right )}}{a x + 1} - \frac{189 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{105 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 35}{2520 \, a c^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59009, size = 247, normalized size = 1.86 \begin{align*} -\frac{{\left (2 \, a^{5} x^{5} - 8 \, a^{4} x^{4} + 11 \, a^{3} x^{3} - 4 \, a^{2} x^{2} - 83 \, a x - 58\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{315 \,{\left (a^{6} c^{5} x^{5} - 5 \, a^{5} c^{5} x^{4} + 10 \, a^{4} c^{5} x^{3} - 10 \, a^{3} c^{5} x^{2} + 5 \, a^{2} c^{5} x - a c^{5}\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 = 1.14683, size = 115, normalized size = 0.86 \begin{align*} -\frac{{\left (a x + 1\right )}^{4}{\left (\frac{135 \,{\left (a x - 1\right )}}{a x + 1} - \frac{189 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{105 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 35\right )}}{2520 \,{\left (a x - 1\right )}^{4} a c^{5} \sqrt{\frac{a x - 1}{a x + 1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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