Optimal. Leaf size=33 \[ -\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 c^2 \left (a-\frac {1}{x}\right )^3} \]
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Rubi [A] time = 0.10, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6175, 6178, 651} \[ -\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 c^2 \left (a-\frac {1}{x}\right )^3} \]
Antiderivative was successfully verified.
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Rule 651
Rule 6175
Rule 6178
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac {\int \frac {e^{\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^2 x^2} \, dx}{a^2 c^2}\\ &=-\frac {\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 )}{a^2 c^2}\\ &=-\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 c^2 \left (a-\frac {1}{x}\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 34, normalized size = 1.03 \[ -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} (a x+1)}{3 c^2 (a x-1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 57, normalized size = 1.73 \[ -\frac {{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 35, normalized size = 1.06 \[ -\frac {a x + 1}{3 \, {\left (a x - 1\right )} a c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 36, normalized size = 1.09 \[ -\frac {a x +1}{3 \left (a x -1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 23, normalized size = 0.70 \[ -\frac {1}{3 \, a c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 23, normalized size = 0.70 \[ -\frac {1}{3\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 2 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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