Optimal. Leaf size=108 \[ \frac{1}{16 a c^3 (1-a x)}-\frac{39}{16 a c^3 (a x+1)}+\frac{5}{8 a c^3 (a x+1)^2}-\frac{1}{12 a c^3 (a x+1)^3}+\frac{\log (1-a x)}{4 a c^3}-\frac{9 \log (a x+1)}{4 a c^3}+\frac{x}{c^3} \]
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Rubi [A] time = 0.202373, antiderivative size = 108, 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} \[ \frac{1}{16 a c^3 (1-a x)}-\frac{39}{16 a c^3 (a x+1)}+\frac{5}{8 a c^3 (a x+1)^2}-\frac{1}{12 a c^3 (a x+1)^3}+\frac{\log (1-a x)}{4 a c^3}-\frac{9 \log (a x+1)}{4 a c^3}+\frac{x}{c^3} \]
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 )^3} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^3} \, dx\\ &=\frac{a^6 \int \frac{e^{-2 \tanh ^{-1}(a x)} x^6}{\left (1-a^2 x^2\right )^3} \, dx}{c^3}\\ &=\frac{a^6 \int \frac{x^6}{(1-a x)^2 (1+a x)^4} \, dx}{c^3}\\ &=\frac{a^6 \int \left (\frac{1}{a^6}+\frac{1}{16 a^6 (-1+a x)^2}+\frac{1}{4 a^6 (-1+a x)}+\frac{1}{4 a^6 (1+a x)^4}-\frac{5}{4 a^6 (1+a x)^3}+\frac{39}{16 a^6 (1+a x)^2}-\frac{9}{4 a^6 (1+a x)}\right ) \, dx}{c^3}\\ &=\frac{x}{c^3}+\frac{1}{16 a c^3 (1-a x)}-\frac{1}{12 a c^3 (1+a x)^3}+\frac{5}{8 a c^3 (1+a x)^2}-\frac{39}{16 a c^3 (1+a x)}+\frac{\log (1-a x)}{4 a c^3}-\frac{9 \log (1+a x)}{4 a c^3}\\ \end{align*}
Mathematica [A] time = 0.0662567, size = 104, normalized size = 0.96 \[ \frac{2 \left (6 a^5 x^5+12 a^4 x^4-15 a^3 x^3-24 a^2 x^2+7 a x+11\right )+3 (a x-1) (a x+1)^3 \log (1-a x)-27 (a x-1) (a x+1)^3 \log (a x+1)}{12 a (a x-1) (a c x+c)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 95, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{3}}}-{\frac{1}{12\,a{c}^{3} \left ( ax+1 \right ) ^{3}}}+{\frac{5}{8\,a{c}^{3} \left ( ax+1 \right ) ^{2}}}-{\frac{39}{16\,a{c}^{3} \left ( ax+1 \right ) }}-{\frac{9\,\ln \left ( ax+1 \right ) }{4\,a{c}^{3}}}-{\frac{1}{16\,a{c}^{3} \left ( ax-1 \right ) }}+{\frac{\ln \left ( ax-1 \right ) }{4\,a{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02748, size = 131, normalized size = 1.21 \begin{align*} -\frac{15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 13 \, a x - 11}{6 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} + \frac{x}{c^{3}} - \frac{9 \, \log \left (a x + 1\right )}{4 \, a c^{3}} + \frac{\log \left (a x - 1\right )}{4 \, a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.17249, size = 306, normalized size = 2.83 \begin{align*} \frac{12 \, a^{5} x^{5} + 24 \, a^{4} x^{4} - 30 \, a^{3} x^{3} - 48 \, a^{2} x^{2} + 14 \, a x - 27 \,{\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (a x - 1\right ) + 22}{12 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.887411, size = 102, normalized size = 0.94 \begin{align*} a^{6} \left (- \frac{15 a^{3} x^{3} + 12 a^{2} x^{2} - 13 a x - 11}{6 a^{11} c^{3} x^{4} + 12 a^{10} c^{3} x^{3} - 12 a^{8} c^{3} x - 6 a^{7} c^{3}} + \frac{x}{a^{6} c^{3}} + \frac{\frac{\log{\left (x - \frac{1}{a} \right )}}{4} - \frac{9 \log{\left (x + \frac{1}{a} \right )}}{4}}{a^{7} c^{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11636, size = 108, normalized size = 1. \begin{align*} \frac{x}{c^{3}} - \frac{9 \, \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{15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 13 \, a x - 11}{6 \,{\left (a x + 1\right )}^{3}{\left (a x - 1\right )} a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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