Optimal. Leaf size=18 \[ \frac {x}{c^2}-\frac {\tanh ^{-1}(a x)}{a c^2} \]
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Rubi [A] time = 0.14, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6167, 6131, 6129, 72, 207} \[ \frac {x}{c^2}-\frac {\tanh ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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Rule 72
Rule 207
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx\\ &=-\frac {a^2 \int \frac {e^{-2 \tanh ^{-1}(a x)} x^2}{(1-a x)^2} \, dx}{c^2}\\ &=-\frac {a^2 \int \frac {x^2}{(1-a x) (1+a x)} \, dx}{c^2}\\ &=-\frac {a^2 \int \left (-\frac {1}{a^2}-\frac {1}{a^2 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^2}\\ &=\frac {x}{c^2}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{c^2}\\ &=\frac {x}{c^2}-\frac {\tanh ^{-1}(a x)}{a c^2}\\ \end {align*}
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Mathematica [B] time = 0.09, size = 39, normalized size = 2.17 \[ \frac {\log (1-a x)}{2 a c^2}-\frac {\log (a x+1)}{2 a c^2}+\frac {x}{c^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.57, size = 27, normalized size = 1.50 \[ \frac {2 \, a x - \log \left (a x + 1\right ) + \log \left (a x - 1\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 36, normalized size = 2.00 \[ \frac {x}{c^{2}} - \frac {\log \left ({\left | a x + 1 \right |}\right )}{2 \, a c^{2}} + \frac {\log \left ({\left | a x - 1 \right |}\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 35, normalized size = 1.94 \[ \frac {x}{c^{2}}+\frac {\ln \left (a x -1\right )}{2 a \,c^{2}}-\frac {\ln \left (a x +1\right )}{2 a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 34, normalized size = 1.89 \[ \frac {x}{c^{2}} - \frac {\log \left (a x + 1\right )}{2 \, a c^{2}} + \frac {\log \left (a x - 1\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 17, normalized size = 0.94 \[ -\frac {\mathrm {atanh}\left (a\,x\right )-a\,x}{a\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.15, size = 34, normalized size = 1.89 \[ a^{2} \left (\frac {x}{a^{2} c^{2}} + \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{2} - \frac {\log {\left (x + \frac {1}{a} \right )}}{2}}{a^{3} c^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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