Optimal. Leaf size=128 \[ \frac{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}{7 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{18 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left (a-\frac{1}{x}\right )^3}+\frac{23 a \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left (a-\frac{1}{x}\right )^2}-\frac{12 \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left (a-\frac{1}{x}\right )} \]
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Rubi [A] time = 0.254514, antiderivative size = 128, 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, 1639, 793, 659, 651} \[ \frac{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}{7 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{18 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left (a-\frac{1}{x}\right )^3}+\frac{23 a \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left (a-\frac{1}{x}\right )^2}-\frac{12 \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left (a-\frac{1}{x}\right )} \]
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}{\left (1-\frac{x}{a}\right )^4 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^5 c^5}\\ &=\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{c^5 \left (a-\frac{1}{x}\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{\frac{2}{a^2}-\frac{3 x}{a^3}}{\left (1-\frac{x}{a}\right )^4 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^5}\\ &=\frac{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}{7 c^5 \left (a-\frac{1}{x}\right )^4}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{c^5 \left (a-\frac{1}{x}\right )^2}-\frac{18 \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right )^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{7 a^2 c^5}\\ &=\frac{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}{7 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{18 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left (a-\frac{1}{x}\right )^3}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{c^5 \left (a-\frac{1}{x}\right )^2}-\frac{36 \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right )^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{35 a^2 c^5}\\ &=\frac{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}{7 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{18 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left (a-\frac{1}{x}\right )^3}+\frac{23 a \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left (a-\frac{1}{x}\right )^2}-\frac{12 \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{35 a^2 c^5}\\ &=\frac{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}{7 c^5 \left (a-\frac{1}{x}\right )^4}-\frac{18 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left (a-\frac{1}{x}\right )^3}+\frac{23 a \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left (a-\frac{1}{x}\right )^2}-\frac{12 \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left (a-\frac{1}{x}\right )}\\ \end{align*}
Mathematica [A] time = 0.0655887, size = 51, normalized size = 0.4 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^3 x^3-8 a^2 x^2+13 a x-12\right )}{35 c^5 (a x-1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 58, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,{x}^{3}{a}^{3}-8\,{a}^{2}{x}^{2}+13\,ax-12 \right ) \left ( ax+1 \right ) }{35\, \left ( ax-1 \right ) ^{4}{c}^{5}a}\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 = 1.00579, size = 96, normalized size = 0.75 \begin{align*} -\frac{\frac{21 \,{\left (a x - 1\right )}}{a x + 1} - \frac{35 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{35 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{280 \, a c^{5} \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 = 1.56308, size = 200, normalized size = 1.56 \begin{align*} -\frac{{\left (2 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 5 \, a^{2} x^{2} + a x - 12\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{35 \,{\left (a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} + 6 \, a^{3} c^{5} x^{2} - 4 \, 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 = 1.22216, size = 115, normalized size = 0.9 \begin{align*} \frac{4 \,{\left (35 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )}^{3} x^{3} - 21 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )}^{2} x^{2} + 7 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x - 1\right )}}{35 \,{\left ({\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x - 1\right )}^{7} a c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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