Optimal. Leaf size=28 \[ \frac {a-\frac {1}{x}}{a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.10, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6175, 6178, 637} \[ \frac {a-\frac {1}{x}}{a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 637
Rule 6175
Rule 6178
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac {\int \frac {e^{-3 \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 {1-\frac {x}{a}}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{a^2 c^2}\\ &=\frac {a-\frac {1}{x}}{a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 26, normalized size = 0.93 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^2 (a x+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 22, normalized size = 0.79 \[ \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 35, normalized size = 1.25 \[ \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )}{\left (a x -1\right ) a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 22, normalized size = 0.79 \[ \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 22, normalized size = 0.79 \[ \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c^2} \]
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 \[ \frac {\int \left (- \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\right )\, dx + \int \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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