Optimal. Leaf size=12 \[ -\frac {\tanh ^{-1}(a x)}{a c^2} \]
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Rubi [A] time = 0.05, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6167, 6129, 35, 206} \[ -\frac {\tanh ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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Rule 35
Rule 206
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx\\ &=-\frac {\int \frac {1}{(1-a x) (1+a x)} \, dx}{c^2}\\ &=-\frac {\int \frac {1}{1-a^2 x^2} \, dx}{c^2}\\ &=-\frac {\tanh ^{-1}(a x)}{a c^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ -\frac {\tanh ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 23, normalized size = 1.92 \[ -\frac {\log \left (a x + 1\right ) - \log \left (a x - 1\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 25, normalized size = 2.08 \[ -\frac {\log \left ({\left | -\frac {2 \, c}{a c x - c} - 1 \right |}\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 30, normalized size = 2.50 \[ \frac {\ln \left (a x -1\right )}{2 c^{2} a}-\frac {\ln \left (a x +1\right )}{2 a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 29, normalized size = 2.42 \[ -\frac {\log \left (a x + 1\right )}{2 \, a c^{2}} + \frac {\log \left (a x - 1\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 12, normalized size = 1.00 \[ -\frac {\mathrm {atanh}\left (a\,x\right )}{a\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 20, normalized size = 1.67 \[ \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{2} - \frac {\log {\left (x + \frac {1}{a} \right )}}{2}}{a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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