Optimal. Leaf size=122 \[ \frac{5}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (a x+1)}+\frac{1}{16 a c^4 (1-a x)^2}+\frac{1}{16 a c^4 (1-a x)^3}+\frac{1}{16 a c^4 (1-a x)^4}+\frac{1}{20 a c^4 (1-a x)^5}+\frac{3 \tanh ^{-1}(a x)}{32 a c^4} \]
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Rubi [A] time = 0.117967, antiderivative size = 122, 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} \[ \frac{5}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (a x+1)}+\frac{1}{16 a c^4 (1-a x)^2}+\frac{1}{16 a c^4 (1-a x)^3}+\frac{1}{16 a c^4 (1-a x)^4}+\frac{1}{20 a c^4 (1-a x)^5}+\frac{3 \tanh ^{-1}(a x)}{32 a c^4} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6140
Rule 44
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx\\ &=\frac{\int \frac{1}{(1-a x)^6 (1+a x)^2} \, dx}{c^4}\\ &=\frac{\int \left (\frac{1}{4 (-1+a x)^6}-\frac{1}{4 (-1+a x)^5}+\frac{3}{16 (-1+a x)^4}-\frac{1}{8 (-1+a x)^3}+\frac{5}{64 (-1+a x)^2}+\frac{1}{64 (1+a x)^2}-\frac{3}{32 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^4}\\ &=\frac{1}{20 a c^4 (1-a x)^5}+\frac{1}{16 a c^4 (1-a x)^4}+\frac{1}{16 a c^4 (1-a x)^3}+\frac{1}{16 a c^4 (1-a x)^2}+\frac{5}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (1+a x)}-\frac{3 \int \frac{1}{-1+a^2 x^2} \, dx}{32 c^4}\\ &=\frac{1}{20 a c^4 (1-a x)^5}+\frac{1}{16 a c^4 (1-a x)^4}+\frac{1}{16 a c^4 (1-a x)^3}+\frac{1}{16 a c^4 (1-a x)^2}+\frac{5}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (1+a x)}+\frac{3 \tanh ^{-1}(a x)}{32 a c^4}\\ \end{align*}
Mathematica [A] time = 0.0611854, size = 80, normalized size = 0.66 \[ \frac{-15 a^5 x^5+60 a^4 x^4-80 a^3 x^3+20 a^2 x^2+47 a x+15 (a x-1)^5 (a x+1) \tanh ^{-1}(a x)-48}{160 a c^4 (a x-1)^5 (a x+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 120, normalized size = 1. \begin{align*} -{\frac{1}{64\,a{c}^{4} \left ( ax+1 \right ) }}+{\frac{3\,\ln \left ( ax+1 \right ) }{64\,a{c}^{4}}}-{\frac{1}{20\,a{c}^{4} \left ( ax-1 \right ) ^{5}}}+{\frac{1}{16\,a{c}^{4} \left ( ax-1 \right ) ^{4}}}-{\frac{1}{16\,a{c}^{4} \left ( ax-1 \right ) ^{3}}}+{\frac{1}{16\,a{c}^{4} \left ( ax-1 \right ) ^{2}}}-{\frac{5}{64\,a{c}^{4} \left ( ax-1 \right ) }}-{\frac{3\,\ln \left ( ax-1 \right ) }{64\,a{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08779, size = 176, normalized size = 1.44 \begin{align*} -\frac{15 \, a^{5} x^{5} - 60 \, a^{4} x^{4} + 80 \, a^{3} x^{3} - 20 \, a^{2} x^{2} - 47 \, a x + 48}{160 \,{\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} + \frac{3 \, \log \left (a x + 1\right )}{64 \, a c^{4}} - \frac{3 \, \log \left (a x - 1\right )}{64 \, a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48673, size = 421, normalized size = 3.45 \begin{align*} -\frac{30 \, a^{5} x^{5} - 120 \, a^{4} x^{4} + 160 \, a^{3} x^{3} - 40 \, a^{2} x^{2} - 94 \, a x - 15 \,{\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (a x + 1\right ) + 15 \,{\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (a x - 1\right ) + 96}{320 \,{\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, 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 = 0.957356, size = 129, normalized size = 1.06 \begin{align*} - \frac{15 a^{5} x^{5} - 60 a^{4} x^{4} + 80 a^{3} x^{3} - 20 a^{2} x^{2} - 47 a x + 48}{160 a^{7} c^{4} x^{6} - 640 a^{6} c^{4} x^{5} + 800 a^{5} c^{4} x^{4} - 800 a^{3} c^{4} x^{2} + 640 a^{2} c^{4} x - 160 a c^{4}} + \frac{- \frac{3 \log{\left (x - \frac{1}{a} \right )}}{64} + \frac{3 \log{\left (x + \frac{1}{a} \right )}}{64}}{a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1502, size = 171, normalized size = 1.4 \begin{align*} \frac{3 \, \log \left ({\left | -\frac{2}{a x - 1} - 1 \right |}\right )}{64 \, a c^{4}} + \frac{1}{128 \, a c^{4}{\left (\frac{2}{a x - 1} + 1\right )}} - \frac{\frac{25 \, a^{9} c^{16}}{a x - 1} - \frac{20 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{2}} + \frac{20 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{3}} - \frac{20 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{4}} + \frac{16 \, a^{9} c^{16}}{{\left (a x - 1\right )}^{5}}}{320 \, a^{10} c^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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