Optimal. Leaf size=94 \[ \frac{\left (a+\frac{1}{x}\right )^2}{5 a^3 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{5 a+\frac{2}{x}}{5 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.312766, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6175, 6178, 852, 1635, 637} \[ \frac{\left (a+\frac{1}{x}\right )^2}{5 a^3 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{5 a+\frac{2}{x}}{5 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 852
Rule 1635
Rule 637
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}{a}\right )^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{a^5 c^5}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3 \left (1+\frac{x}{a}\right )^2}{\left (1-\frac{x^2}{a^2}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{a^5 c^5}\\ &=\frac{\left (a+\frac{1}{x}\right )^2}{5 a^3 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right ) \left (2 a^3+5 a^2 x+5 a x^2\right )}{\left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 a^5 c^5}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{\left (a+\frac{1}{x}\right )^2}{5 a^3 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{6 a^3+15 a^2 x}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{15 a^5 c^5}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{\left (a+\frac{1}{x}\right )^2}{5 a^3 c^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{5 a+\frac{2}{x}}{5 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0632543, size = 57, normalized size = 0.61 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^3 x^3-4 a^2 x^2+a x+2\right )}{5 c^5 (a x-1)^3 (a x+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 57, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,{x}^{3}{a}^{3}-4\,{a}^{2}{x}^{2}+ax+2 \right ) \left ( ax+1 \right ) }{5\, \left ( ax-1 \right ) ^{4}{c}^{5}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 = 1.03963, size = 111, normalized size = 1.18 \begin{align*} \frac{1}{40} \, a{\left (\frac{5 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{5}} - \frac{\frac{5 \,{\left (a x - 1\right )}}{a x + 1} - \frac{15 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 1}{a^{2} c^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58291, size = 158, normalized size = 1.68 \begin{align*} \frac{{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 2\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{5 \,{\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (a c x - c\right )}^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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