Optimal. Leaf size=100 \[ -\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{7 c^4 \left (a-\frac{1}{x}\right )^5}+\frac{12 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{35 c^4 \left (a-\frac{1}{x}\right )^4}-\frac{23 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 c^4 \left (a-\frac{1}{x}\right )^3} \]
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Rubi [A] time = 0.229582, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{7 c^4 \left (a-\frac{1}{x}\right )^5}+\frac{12 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{35 c^4 \left (a-\frac{1}{x}\right )^4}-\frac{23 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 c^4 \left (a-\frac{1}{x}\right )^3} \]
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)^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 \sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^5} \, dx,x,\frac{1}{x}\right )}{a^4 c^4}\\ &=\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^4 \left (a-\frac{1}{x}\right )^4}-\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{4}{a^2}-\frac{3 x}{a^3}\right ) \sqrt{1-\frac{x^2}{a^2}}}{\left (1-\frac{x}{a}\right )^5} \, dx,x,\frac{1}{x}\right )}{c^4}\\ &=-\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{7 c^4 \left (a-\frac{1}{x}\right )^5}+\frac{a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{c^4 \left (a-\frac{1}{x}\right )^4}-\frac{23 \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 )}{7 a^2 c^4}\\ &=-\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{7 c^4 \left (a-\frac{1}{x}\right )^5}+\frac{12 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{35 c^4 \left (a-\frac{1}{x}\right )^4}-\frac{23 \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 )}{35 a^2 c^4}\\ &=-\frac{a^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{7 c^4 \left (a-\frac{1}{x}\right )^5}+\frac{12 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{35 c^4 \left (a-\frac{1}{x}\right )^4}-\frac{23 a^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{105 c^4 \left (a-\frac{1}{x}\right )^3}\\ \end{align*}
Mathematica [A] time = 0.0562056, size = 51, normalized size = 0.51 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a^3 x^3-8 a^2 x^2+13 a x+23\right )}{105 c^4 (a x-1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 50, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,{a}^{2}{x}^{2}-10\,ax+23 \right ) \left ( ax+1 \right ) }{105\,{c}^{4} \left ( ax-1 \right ) ^{3}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 = 0.995723, size = 74, normalized size = 0.74 \begin{align*} \frac{\frac{42 \,{\left (a x - 1\right )}}{a x + 1} - \frac{35 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 15}{420 \, a c^{4} \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 = 1.61714, size = 205, normalized size = 2.05 \begin{align*} -\frac{{\left (2 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 5 \, a^{2} x^{2} + 36 \, a x + 23\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{105 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, 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{1}{a^{4} x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} - 4 a^{3} x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} + 6 a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} - 4 a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} + \sqrt{\frac{a x}{a x + 1} - \frac{1}{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 = 1.14222, size = 93, normalized size = 0.93 \begin{align*} \frac{{\left (a x + 1\right )}^{3}{\left (\frac{42 \,{\left (a x - 1\right )}}{a x + 1} - \frac{35 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 15\right )}}{420 \,{\left (a x - 1\right )}^{3} a c^{4} \sqrt{\frac{a x - 1}{a x + 1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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