Optimal. Leaf size=95 \[ -\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{5 c^4 \left (a-\frac{1}{x}\right )^3}+\frac{8 a \sqrt{1-\frac{1}{a^2 x^2}}}{15 c^4 \left (a-\frac{1}{x}\right )^2}-\frac{7 \sqrt{1-\frac{1}{a^2 x^2}}}{15 c^4 \left (a-\frac{1}{x}\right )} \]
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Rubi [A] time = 0.229206, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{5 c^4 \left (a-\frac{1}{x}\right )^3}+\frac{8 a \sqrt{1-\frac{1}{a^2 x^2}}}{15 c^4 \left (a-\frac{1}{x}\right )^2}-\frac{7 \sqrt{1-\frac{1}{a^2 x^2}}}{15 c^4 \left (a-\frac{1}{x}\right )} \]
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)^4} \, dx &=\frac{\int \frac{e^{-\coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right )^4 x^4} \, dx}{a^4 c^4}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (1-\frac{x}{a}\right )^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^4 c^4}\\ &=\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{c^4 \left (a-\frac{1}{x}\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{\frac{2}{a^2}-\frac{x}{a^3}}{\left (1-\frac{x}{a}\right )^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^4}\\ &=-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{5 c^4 \left (a-\frac{1}{x}\right )^3}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{c^4 \left (a-\frac{1}{x}\right )^2}-\frac{7 \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 )}{5 a^2 c^4}\\ &=-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{5 c^4 \left (a-\frac{1}{x}\right )^3}+\frac{8 a \sqrt{1-\frac{1}{a^2 x^2}}}{15 c^4 \left (a-\frac{1}{x}\right )^2}-\frac{7 \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 )}{15 a^2 c^4}\\ &=-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{5 c^4 \left (a-\frac{1}{x}\right )^3}+\frac{8 a \sqrt{1-\frac{1}{a^2 x^2}}}{15 c^4 \left (a-\frac{1}{x}\right )^2}-\frac{7 \sqrt{1-\frac{1}{a^2 x^2}}}{15 c^4 \left (a-\frac{1}{x}\right )}\\ \end{align*}
Mathematica [A] time = 0.0562102, size = 43, normalized size = 0.45 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^2 x^2-6 a x+7\right )}{15 c^4 (a x-1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 50, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,{a}^{2}{x}^{2}-6\,ax+7 \right ) \left ( ax+1 \right ) }{15\, \left ( ax-1 \right ) ^{3}{c}^{4}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 = 1.01394, size = 74, normalized size = 0.78 \begin{align*} \frac{\frac{10 \,{\left (a x - 1\right )}}{a x + 1} - \frac{15 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 3}{60 \, a c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56422, size = 161, normalized size = 1.69 \begin{align*} -\frac{{\left (2 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x + 7\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{4} x^{4} - 4 a^{3} x^{3} + 6 a^{2} x^{2} - 4 a x + 1}\, dx}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19153, size = 88, normalized size = 0.93 \begin{align*} -\frac{4 \,{\left (10 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )}^{2} x^{2} - 5 \,{\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x + 1\right )}}{15 \,{\left ({\left (a + \sqrt{a^{2} - \frac{1}{x^{2}}}\right )} x - 1\right )}^{5} a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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