Optimal. Leaf size=143 \[ \frac{11}{64 a c^4 (1-a x)}-\frac{99}{32 a c^4 (a x+1)}-\frac{1}{64 a c^4 (1-a x)^2}+\frac{35}{32 a c^4 (a x+1)^2}-\frac{13}{48 a c^4 (a x+1)^3}+\frac{1}{32 a c^4 (a x+1)^4}+\frac{47 \log (1-a x)}{128 a c^4}-\frac{303 \log (a x+1)}{128 a c^4}+\frac{x}{c^4} \]
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Rubi [A] time = 0.231239, antiderivative size = 143, 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} \[ \frac{11}{64 a c^4 (1-a x)}-\frac{99}{32 a c^4 (a x+1)}-\frac{1}{64 a c^4 (1-a x)^2}+\frac{35}{32 a c^4 (a x+1)^2}-\frac{13}{48 a c^4 (a x+1)^3}+\frac{1}{32 a c^4 (a x+1)^4}+\frac{47 \log (1-a x)}{128 a c^4}-\frac{303 \log (a x+1)}{128 a c^4}+\frac{x}{c^4} \]
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 )^4} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^4} \, dx\\ &=-\frac{a^8 \int \frac{e^{-2 \tanh ^{-1}(a x)} x^8}{\left (1-a^2 x^2\right )^4} \, dx}{c^4}\\ &=-\frac{a^8 \int \frac{x^8}{(1-a x)^3 (1+a x)^5} \, dx}{c^4}\\ &=-\frac{a^8 \int \left (-\frac{1}{a^8}-\frac{1}{32 a^8 (-1+a x)^3}-\frac{11}{64 a^8 (-1+a x)^2}-\frac{47}{128 a^8 (-1+a x)}+\frac{1}{8 a^8 (1+a x)^5}-\frac{13}{16 a^8 (1+a x)^4}+\frac{35}{16 a^8 (1+a x)^3}-\frac{99}{32 a^8 (1+a x)^2}+\frac{303}{128 a^8 (1+a x)}\right ) \, dx}{c^4}\\ &=\frac{x}{c^4}-\frac{1}{64 a c^4 (1-a x)^2}+\frac{11}{64 a c^4 (1-a x)}+\frac{1}{32 a c^4 (1+a x)^4}-\frac{13}{48 a c^4 (1+a x)^3}+\frac{35}{32 a c^4 (1+a x)^2}-\frac{99}{32 a c^4 (1+a x)}+\frac{47 \log (1-a x)}{128 a c^4}-\frac{303 \log (1+a x)}{128 a c^4}\\ \end{align*}
Mathematica [A] time = 0.0946506, size = 124, normalized size = 0.87 \[ \frac{2 \left (192 a^7 x^7+384 a^6 x^6-819 a^5 x^5-1254 a^4 x^4+866 a^3 x^3+1258 a^2 x^2-275 a x-400\right )+141 (a x-1)^2 (a x+1)^4 \log (1-a x)-909 (a x-1)^2 (a x+1)^4 \log (a x+1)}{384 a (a x-1)^2 (a c x+c)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 125, normalized size = 0.9 \begin{align*}{\frac{x}{{c}^{4}}}+{\frac{1}{32\,a{c}^{4} \left ( ax+1 \right ) ^{4}}}-{\frac{13}{48\,a{c}^{4} \left ( ax+1 \right ) ^{3}}}+{\frac{35}{32\,a{c}^{4} \left ( ax+1 \right ) ^{2}}}-{\frac{99}{32\,a{c}^{4} \left ( ax+1 \right ) }}-{\frac{303\,\ln \left ( ax+1 \right ) }{128\,a{c}^{4}}}-{\frac{1}{64\,a{c}^{4} \left ( ax-1 \right ) ^{2}}}-{\frac{11}{64\,a{c}^{4} \left ( ax-1 \right ) }}+{\frac{47\,\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 = 1.05436, size = 196, normalized size = 1.37 \begin{align*} -\frac{627 \, a^{5} x^{5} + 486 \, a^{4} x^{4} - 1058 \, a^{3} x^{3} - 874 \, a^{2} x^{2} + 467 \, a x + 400}{192 \,{\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{x}{c^{4}} - \frac{303 \, \log \left (a x + 1\right )}{128 \, a c^{4}} + \frac{47 \, \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 [A] time = 1.33578, size = 509, normalized size = 3.56 \begin{align*} \frac{384 \, a^{7} x^{7} + 768 \, a^{6} x^{6} - 1638 \, a^{5} x^{5} - 2508 \, a^{4} x^{4} + 1732 \, a^{3} x^{3} + 2516 \, a^{2} x^{2} - 550 \, a x - 909 \,{\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 ) + 141 \,{\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 ) - 800}{384 \,{\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 = 1.34274, size = 156, normalized size = 1.09 \begin{align*} a^{8} \left (- \frac{627 a^{5} x^{5} + 486 a^{4} x^{4} - 1058 a^{3} x^{3} - 874 a^{2} x^{2} + 467 a x + 400}{192 a^{15} c^{4} x^{6} + 384 a^{14} c^{4} x^{5} - 192 a^{13} c^{4} x^{4} - 768 a^{12} c^{4} x^{3} - 192 a^{11} c^{4} x^{2} + 384 a^{10} c^{4} x + 192 a^{9} c^{4}} + \frac{x}{a^{8} c^{4}} + \frac{\frac{47 \log{\left (x - \frac{1}{a} \right )}}{128} - \frac{303 \log{\left (x + \frac{1}{a} \right )}}{128}}{a^{9} c^{4}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15428, size = 130, normalized size = 0.91 \begin{align*} \frac{x}{c^{4}} - \frac{303 \, \log \left ({\left | a x + 1 \right |}\right )}{128 \, a c^{4}} + \frac{47 \, \log \left ({\left | a x - 1 \right |}\right )}{128 \, a c^{4}} - \frac{627 \, a^{5} x^{5} + 486 \, a^{4} x^{4} - 1058 \, a^{3} x^{3} - 874 \, a^{2} x^{2} + 467 \, a x + 400}{192 \,{\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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