Optimal. Leaf size=119 \[ -\frac{5}{64 a c^4 (1-a x)}+\frac{5}{32 a c^4 (a x+1)}-\frac{1}{64 a c^4 (1-a x)^2}+\frac{3}{32 a c^4 (a x+1)^2}+\frac{1}{16 a c^4 (a x+1)^3}+\frac{1}{32 a c^4 (a x+1)^4}-\frac{15 \tanh ^{-1}(a x)}{64 a c^4} \]
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Rubi [A] time = 0.119755, antiderivative size = 119, 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} \[ -\frac{5}{64 a c^4 (1-a x)}+\frac{5}{32 a c^4 (a x+1)}-\frac{1}{64 a c^4 (1-a x)^2}+\frac{3}{32 a c^4 (a x+1)^2}+\frac{1}{16 a c^4 (a x+1)^3}+\frac{1}{32 a c^4 (a x+1)^4}-\frac{15 \tanh ^{-1}(a x)}{64 a c^4} \]
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 )^4} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx\\ &=-\frac{\int \frac{1}{(1-a x)^3 (1+a x)^5} \, dx}{c^4}\\ &=-\frac{\int \left (-\frac{1}{32 (-1+a x)^3}+\frac{5}{64 (-1+a x)^2}+\frac{1}{8 (1+a x)^5}+\frac{3}{16 (1+a x)^4}+\frac{3}{16 (1+a x)^3}+\frac{5}{32 (1+a x)^2}-\frac{15}{64 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^4}\\ &=-\frac{1}{64 a c^4 (1-a x)^2}-\frac{5}{64 a c^4 (1-a x)}+\frac{1}{32 a c^4 (1+a x)^4}+\frac{1}{16 a c^4 (1+a x)^3}+\frac{3}{32 a c^4 (1+a x)^2}+\frac{5}{32 a c^4 (1+a x)}+\frac{15 \int \frac{1}{-1+a^2 x^2} \, dx}{64 c^4}\\ &=-\frac{1}{64 a c^4 (1-a x)^2}-\frac{5}{64 a c^4 (1-a x)}+\frac{1}{32 a c^4 (1+a x)^4}+\frac{1}{16 a c^4 (1+a x)^3}+\frac{3}{32 a c^4 (1+a x)^2}+\frac{5}{32 a c^4 (1+a x)}-\frac{15 \tanh ^{-1}(a x)}{64 a c^4}\\ \end{align*}
Mathematica [A] time = 0.0617833, size = 80, normalized size = 0.67 \[ \frac{15 a^5 x^5+30 a^4 x^4-10 a^3 x^3-50 a^2 x^2-17 a x-15 (a x-1)^2 (a x+1)^4 \tanh ^{-1}(a x)+16}{64 a (a x-1)^2 (a c x+c)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 120, normalized size = 1. \begin{align*}{\frac{1}{32\,a{c}^{4} \left ( ax+1 \right ) ^{4}}}+{\frac{1}{16\,a{c}^{4} \left ( ax+1 \right ) ^{3}}}+{\frac{3}{32\,a{c}^{4} \left ( ax+1 \right ) ^{2}}}+{\frac{5}{32\,a{c}^{4} \left ( ax+1 \right ) }}-{\frac{15\,\ln \left ( ax+1 \right ) }{128\,a{c}^{4}}}-{\frac{1}{64\,a{c}^{4} \left ( ax-1 \right ) ^{2}}}+{\frac{5}{64\,a{c}^{4} \left ( ax-1 \right ) }}+{\frac{15\,\ln \left ( ax-1 \right ) }{128\,a{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03425, size = 189, normalized size = 1.59 \begin{align*} \frac{15 \, a^{5} x^{5} + 30 \, a^{4} x^{4} - 10 \, a^{3} x^{3} - 50 \, a^{2} x^{2} - 17 \, a x + 16}{64 \,{\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} - \frac{15 \, \log \left (a x + 1\right )}{128 \, a c^{4}} + \frac{15 \, \log \left (a x - 1\right )}{128 \, a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.64474, size = 456, normalized size = 3.83 \begin{align*} \frac{30 \, a^{5} x^{5} + 60 \, a^{4} x^{4} - 20 \, a^{3} x^{3} - 100 \, a^{2} x^{2} - 34 \, a x - 15 \,{\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 15 \,{\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 32}{128 \,{\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.974898, size = 141, normalized size = 1.18 \begin{align*} \frac{15 a^{5} x^{5} + 30 a^{4} x^{4} - 10 a^{3} x^{3} - 50 a^{2} x^{2} - 17 a x + 16}{64 a^{7} c^{4} x^{6} + 128 a^{6} c^{4} x^{5} - 64 a^{5} c^{4} x^{4} - 256 a^{4} c^{4} x^{3} - 64 a^{3} c^{4} x^{2} + 128 a^{2} c^{4} x + 64 a c^{4}} + \frac{\frac{15 \log{\left (x - \frac{1}{a} \right )}}{128} - \frac{15 \log{\left (x + \frac{1}{a} \right )}}{128}}{a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14136, size = 123, normalized size = 1.03 \begin{align*} -\frac{15 \, \log \left ({\left | a x + 1 \right |}\right )}{128 \, a c^{4}} + \frac{15 \, \log \left ({\left | a x - 1 \right |}\right )}{128 \, a c^{4}} + \frac{15 \, a^{5} x^{5} + 30 \, a^{4} x^{4} - 10 \, a^{3} x^{3} - 50 \, a^{2} x^{2} - 17 \, a x + 16}{64 \,{\left (a x + 1\right )}^{4}{\left (a x - 1\right )}^{2} a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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