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 = 0.23, antiderivative size = 143, normalized size of antiderivative = 1.00, 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 88
Rule 6150
Rule 6157
Rule 6167
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*}
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Mathematica [A] time = 0.13, 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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fricas [A] time = 0.59, size = 233, normalized size = 1.63 \[ \frac {384 \, a^{7} x^{7} + 768 \, a^{6} x^{6} - 1638 \, a^{5} x^{5} - 2508 \, a^{4} x^{4} + 1732 \, a^{3} x^{3} + 2516 \, a^{2} x^{2} - 550 \, a x - 909 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 141 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 800}{384 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 96, normalized size = 0.67 \[ \frac {x}{c^{4}} - \frac {303 \, \log \left ({\left | a x + 1 \right |}\right )}{128 \, a c^{4}} + \frac {47 \, \log \left ({\left | a x - 1 \right |}\right )}{128 \, a c^{4}} - \frac {627 \, a^{5} x^{5} + 486 \, a^{4} x^{4} - 1058 \, a^{3} x^{3} - 874 \, a^{2} x^{2} + 467 \, a x + 400}{192 \, {\left (a x + 1\right )}^{4} {\left (a x - 1\right )}^{2} a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 125, normalized size = 0.87 \[ \frac {x}{c^{4}}-\frac {1}{64 c^{4} a \left (a x -1\right )^{2}}-\frac {11}{64 a \,c^{4} \left (a x -1\right )}+\frac {47 \ln \left (a x -1\right )}{128 c^{4} a}+\frac {1}{32 a \,c^{4} \left (a x +1\right )^{4}}-\frac {13}{48 a \,c^{4} \left (a x +1\right )^{3}}+\frac {35}{32 a \,c^{4} \left (a x +1\right )^{2}}-\frac {99}{32 a \,c^{4} \left (a x +1\right )}-\frac {303 \ln \left (a x +1\right )}{128 a \,c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 145, normalized size = 1.01 \[ -\frac {627 \, a^{5} x^{5} + 486 \, a^{4} x^{4} - 1058 \, a^{3} x^{3} - 874 \, a^{2} x^{2} + 467 \, a x + 400}{192 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {x}{c^{4}} - \frac {303 \, \log \left (a x + 1\right )}{128 \, a c^{4}} + \frac {47 \, \log \left (a x - 1\right )}{128 \, a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 142, normalized size = 0.99 \[ \frac {x}{c^4}-\frac {\frac {467\,x}{192}-\frac {437\,a\,x^2}{96}+\frac {25}{12\,a}-\frac {529\,a^2\,x^3}{96}+\frac {81\,a^3\,x^4}{32}+\frac {209\,a^4\,x^5}{64}}{a^6\,c^4\,x^6+2\,a^5\,c^4\,x^5-a^4\,c^4\,x^4-4\,a^3\,c^4\,x^3-a^2\,c^4\,x^2+2\,a\,c^4\,x+c^4}+\frac {47\,\ln \left (a\,x-1\right )}{128\,a\,c^4}-\frac {303\,\ln \left (a\,x+1\right )}{128\,a\,c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 156, normalized size = 1.09 \[ a^{8} \left (\frac {- 627 a^{5} x^{5} - 486 a^{4} x^{4} + 1058 a^{3} x^{3} + 874 a^{2} x^{2} - 467 a x - 400}{192 a^{15} c^{4} x^{6} + 384 a^{14} c^{4} x^{5} - 192 a^{13} c^{4} x^{4} - 768 a^{12} c^{4} x^{3} - 192 a^{11} c^{4} x^{2} + 384 a^{10} c^{4} x + 192 a^{9} c^{4}} + \frac {x}{a^{8} c^{4}} + \frac {\frac {47 \log {\left (x - \frac {1}{a} \right )}}{128} - \frac {303 \log {\left (x + \frac {1}{a} \right )}}{128}}{a^{9} c^{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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