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 = 0.052897, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {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 6140
Rule 44
Rule 207
Rubi steps
\begin{align*} \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 = 0.0194994, size = 35, normalized size = 0.69 \[ \frac{-a x+(a x-1)^2 \tanh ^{-1}(a x)+2}{4 a c^2 (a x-1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, 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 = 0.972146, 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 = 2.28199, size = 173, normalized size = 3.39 \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 = 0.474745, size = 56, normalized size = 1.1 \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 = 1.14465, 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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