Optimal. Leaf size=33 \[ -\frac {1}{2 a c^3 (1-a x)}-\frac {\tanh ^{-1}(a x)}{2 a c^3} \]
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Rubi [A] time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6167, 6129, 44, 207} \[ -\frac {1}{2 a c^3 (1-a x)}-\frac {\tanh ^{-1}(a x)}{2 a c^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 207
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx\\ &=-\frac {\int \frac {1}{(1-a x)^2 (1+a x)} \, dx}{c^3}\\ &=-\frac {\int \left (\frac {1}{2 (-1+a x)^2}-\frac {1}{2 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3}\\ &=-\frac {1}{2 a c^3 (1-a x)}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{2 c^3}\\ &=-\frac {1}{2 a c^3 (1-a x)}-\frac {\tanh ^{-1}(a x)}{2 a c^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.97 \[ -\frac {\frac {1}{2 a (1-a x)}+\frac {\tanh ^{-1}(a x)}{2 a}}{c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 46, normalized size = 1.39 \[ -\frac {{\left (a x - 1\right )} \log \left (a x + 1\right ) - {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \, {\left (a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 46, normalized size = 1.39 \[ -\frac {\log \left ({\left | a x + 1 \right |}\right )}{4 \, a c^{3}} + \frac {\log \left ({\left | a x - 1 \right |}\right )}{4 \, a c^{3}} + \frac {1}{2 \, {\left (a x - 1\right )} a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 45, normalized size = 1.36 \[ \frac {1}{2 c^{3} a \left (a x -1\right )}+\frac {\ln \left (a x -1\right )}{4 c^{3} a}-\frac {\ln \left (a x +1\right )}{4 a \,c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 48, normalized size = 1.45 \[ \frac {1}{2 \, {\left (a^{2} c^{3} x - a c^{3}\right )}} - \frac {\log \left (a x + 1\right )}{4 \, a c^{3}} + \frac {\log \left (a x - 1\right )}{4 \, a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 31, normalized size = 0.94 \[ -\frac {1}{2\,a\,\left (c^3-a\,c^3\,x\right )}-\frac {\mathrm {atanh}\left (a\,x\right )}{2\,a\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 39, normalized size = 1.18 \[ \frac {1}{2 a^{2} c^{3} x - 2 a c^{3}} - \frac {- \frac {\log {\left (x - \frac {1}{a} \right )}}{4} + \frac {\log {\left (x + \frac {1}{a} \right )}}{4}}{a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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