Optimal. Leaf size=73 \[ -\frac{7}{4 a c^2 (a x+1)}+\frac{1}{4 a c^2 (a x+1)^2}+\frac{\log (1-a x)}{8 a c^2}-\frac{17 \log (a x+1)}{8 a c^2}+\frac{x}{c^2} \]
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Rubi [A] time = 0.17431, antiderivative size = 73, 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, 6157, 6150, 88} \[ -\frac{7}{4 a c^2 (a x+1)}+\frac{1}{4 a c^2 (a x+1)^2}+\frac{\log (1-a x)}{8 a c^2}-\frac{17 \log (a x+1)}{8 a c^2}+\frac{x}{c^2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6157
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^2} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^2} \, dx\\ &=-\frac{a^4 \int \frac{e^{-2 \tanh ^{-1}(a x)} x^4}{\left (1-a^2 x^2\right )^2} \, dx}{c^2}\\ &=-\frac{a^4 \int \frac{x^4}{(1-a x) (1+a x)^3} \, dx}{c^2}\\ &=-\frac{a^4 \int \left (-\frac{1}{a^4}-\frac{1}{8 a^4 (-1+a x)}+\frac{1}{2 a^4 (1+a x)^3}-\frac{7}{4 a^4 (1+a x)^2}+\frac{17}{8 a^4 (1+a x)}\right ) \, dx}{c^2}\\ &=\frac{x}{c^2}+\frac{1}{4 a c^2 (1+a x)^2}-\frac{7}{4 a c^2 (1+a x)}+\frac{\log (1-a x)}{8 a c^2}-\frac{17 \log (1+a x)}{8 a c^2}\\ \end{align*}
Mathematica [A] time = 0.0468059, size = 70, normalized size = 0.96 \[ \frac{2 \left (4 a^3 x^3+8 a^2 x^2-3 a x-6\right )+(a x+1)^2 \log (1-a x)-17 (a x+1)^2 \log (a x+1)}{8 a (a c x+c)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 65, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{2}}}+{\frac{1}{4\,a{c}^{2} \left ( ax+1 \right ) ^{2}}}-{\frac{7}{4\,a{c}^{2} \left ( ax+1 \right ) }}-{\frac{17\,\ln \left ( ax+1 \right ) }{8\,a{c}^{2}}}+{\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 = 1.04798, size = 93, normalized size = 1.27 \begin{align*} -\frac{7 \, a x + 6}{4 \,{\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}} + \frac{x}{c^{2}} - \frac{17 \, \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 = 1.38272, size = 211, normalized size = 2.89 \begin{align*} \frac{8 \, a^{3} x^{3} + 16 \, a^{2} x^{2} - 6 \, a x - 17 \,{\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 ) - 12}{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 = 0.599817, size = 73, normalized size = 1. \begin{align*} a^{4} \left (- \frac{7 a x + 6}{4 a^{7} c^{2} x^{2} + 8 a^{6} c^{2} x + 4 a^{5} c^{2}} + \frac{x}{a^{4} c^{2}} + \frac{\frac{\log{\left (x - \frac{1}{a} \right )}}{8} - \frac{17 \log{\left (x + \frac{1}{a} \right )}}{8}}{a^{5} c^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14497, size = 77, normalized size = 1.05 \begin{align*} \frac{x}{c^{2}} - \frac{17 \, \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{7 \, a x + 6}{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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