Optimal. Leaf size=51 \[ -\frac{1}{4 a c^2 (1-a x)}-\frac{1}{4 a c^2 (1-a x)^2}-\frac{\tanh ^{-1}(a x)}{4 a c^2} \]
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Rubi [A] time = 0.0787354, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6140, 44, 207} \[ -\frac{1}{4 a c^2 (1-a x)}-\frac{1}{4 a c^2 (1-a x)^2}-\frac{\tanh ^{-1}(a x)}{4 a c^2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6140
Rule 44
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx\\ &=-\frac{\int \frac{1}{(1-a x)^3 (1+a x)} \, dx}{c^2}\\ &=-\frac{\int \left (-\frac{1}{2 (-1+a x)^3}+\frac{1}{4 (-1+a x)^2}-\frac{1}{4 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^2}\\ &=-\frac{1}{4 a c^2 (1-a x)^2}-\frac{1}{4 a c^2 (1-a x)}+\frac{\int \frac{1}{-1+a^2 x^2} \, dx}{4 c^2}\\ &=-\frac{1}{4 a c^2 (1-a x)^2}-\frac{1}{4 a c^2 (1-a x)}-\frac{\tanh ^{-1}(a x)}{4 a c^2}\\ \end{align*}
Mathematica [A] time = 0.0245763, size = 35, normalized size = 0.69 \[ \frac{a x+(a x-1)^2 \left (-\tanh ^{-1}(a x)\right )-2}{4 a c^2 (a x-1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 60, normalized size = 1.2 \begin{align*} -{\frac{\ln \left ( ax+1 \right ) }{8\,a{c}^{2}}}-{\frac{1}{4\,a{c}^{2} \left ( ax-1 \right ) ^{2}}}+{\frac{1}{4\,a{c}^{2} \left ( ax-1 \right ) }}+{\frac{\ln \left ( ax-1 \right ) }{8\,a{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0614, size = 85, normalized size = 1.67 \begin{align*} \frac{a x - 2}{4 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} - \frac{\log \left (a x + 1\right )}{8 \, a c^{2}} + \frac{\log \left (a x - 1\right )}{8 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57383, size = 171, normalized size = 3.35 \begin{align*} \frac{2 \, a x -{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x + 1\right ) +{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 4}{8 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.464657, size = 54, normalized size = 1.06 \begin{align*} \frac{a x - 2}{4 a^{3} c^{2} x^{2} - 8 a^{2} c^{2} x + 4 a c^{2}} + \frac{\frac{\log{\left (x - \frac{1}{a} \right )}}{8} - \frac{\log{\left (x + \frac{1}{a} \right )}}{8}}{a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11966, size = 69, normalized size = 1.35 \begin{align*} -\frac{\log \left ({\left | a x + 1 \right |}\right )}{8 \, a c^{2}} + \frac{\log \left ({\left | a x - 1 \right |}\right )}{8 \, a c^{2}} + \frac{a x - 2}{4 \,{\left (a x - 1\right )}^{2} a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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