Optimal. Leaf size=125 \[ \frac{\left (a+\frac{1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{11/2}}-\frac{10 \left (a+\frac{1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{79 \left (a+\frac{1}{x}\right )^6}{231 a^7 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{152 \left (a+\frac{1}{x}\right )^5}{1155 a^6 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}} \]
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Rubi [A] time = 0.389062, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6175, 6178, 852, 1635, 789, 651} \[ \frac{\left (a+\frac{1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{11/2}}-\frac{10 \left (a+\frac{1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{79 \left (a+\frac{1}{x}\right )^6}{231 a^7 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{152 \left (a+\frac{1}{x}\right )^5}{1155 a^6 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 852
Rule 1635
Rule 789
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx &=-\frac{\int \frac{e^{3 \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 \left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (1-\frac{x}{a}\right )^8} \, dx,x,\frac{1}{x}\right )}{a^5 c^5}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3 \left (1+\frac{x}{a}\right )^8}{\left (1-\frac{x^2}{a^2}\right )^{13/2}} \, dx,x,\frac{1}{x}\right )}{a^5 c^5}\\ &=\frac{\left (a+\frac{1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{11/2}}-\frac{\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^7 \left (8 a^3+11 a^2 x+11 a x^2\right )}{\left (1-\frac{x^2}{a^2}\right )^{11/2}} \, dx,x,\frac{1}{x}\right )}{11 a^5 c^5}\\ &=-\frac{10 \left (a+\frac{1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{\left (a+\frac{1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{11/2}}+\frac{\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^6 \left (138 a^3+99 a^2 x\right )}{\left (1-\frac{x^2}{a^2}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{99 a^5 c^5}\\ &=\frac{79 \left (a+\frac{1}{x}\right )^6}{231 a^7 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{10 \left (a+\frac{1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{\left (a+\frac{1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{11/2}}-\frac{152 \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^5}{\left (1-\frac{x^2}{a^2}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{231 a^2 c^5}\\ &=-\frac{152 \left (a+\frac{1}{x}\right )^5}{1155 a^6 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{79 \left (a+\frac{1}{x}\right )^6}{231 a^7 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{10 \left (a+\frac{1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{\left (a+\frac{1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.0680123, size = 58, normalized size = 0.46 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x+1)^2 \left (2 a^3 x^3-16 a^2 x^2+61 a x-152\right )}{1155 c^5 (a x-1)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 58, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,{x}^{3}{a}^{3}-16\,{a}^{2}{x}^{2}+61\,ax-152 \right ) \left ( ax+1 \right ) }{1155\,{c}^{5} \left ( ax-1 \right ) ^{4}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 = 1.03058, size = 96, normalized size = 0.77 \begin{align*} -\frac{\frac{385 \,{\left (a x - 1\right )}}{a x + 1} - \frac{495 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{231 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 105}{9240 \, a c^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54921, size = 296, normalized size = 2.37 \begin{align*} -\frac{{\left (2 \, a^{6} x^{6} - 10 \, a^{5} x^{5} + 19 \, a^{4} x^{4} - 15 \, a^{3} x^{3} - 289 \, a^{2} x^{2} - 395 \, a x - 152\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{1155 \,{\left (a^{7} c^{5} x^{6} - 6 \, a^{6} c^{5} x^{5} + 15 \, a^{5} c^{5} x^{4} - 20 \, a^{4} c^{5} x^{3} + 15 \, a^{3} c^{5} x^{2} - 6 \, 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 = 1.53506, size = 115, normalized size = 0.92 \begin{align*} -\frac{{\left (a x + 1\right )}^{5}{\left (\frac{385 \,{\left (a x - 1\right )}}{a x + 1} - \frac{495 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{231 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 105\right )}}{9240 \,{\left (a x - 1\right )}^{5} 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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