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 = 0.12, antiderivative size = 119, 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, 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 44
Rule 207
Rule 6140
Rule 6167
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*}
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Mathematica [A] time = 0.07, 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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fricas [B] time = 0.58, size = 217, normalized size = 1.82 \[ \frac {30 \, a^{5} x^{5} + 60 \, a^{4} x^{4} - 20 \, a^{3} x^{3} - 100 \, a^{2} x^{2} - 34 \, a x - 15 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x + 1\right ) + 15 \, {\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 32}{128 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 91, normalized size = 0.76 \[ -\frac {15 \, \log \left ({\left | a x + 1 \right |}\right )}{128 \, a c^{4}} + \frac {15 \, \log \left ({\left | a x - 1 \right |}\right )}{128 \, a c^{4}} + \frac {15 \, a^{5} x^{5} + 30 \, a^{4} x^{4} - 10 \, a^{3} x^{3} - 50 \, a^{2} x^{2} - 17 \, a x + 16}{64 \, {\left (a x + 1\right )}^{4} {\left (a x - 1\right )}^{2} a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 120, normalized size = 1.01 \[ -\frac {1}{64 c^{4} a \left (a x -1\right )^{2}}+\frac {5}{64 a \,c^{4} \left (a x -1\right )}+\frac {15 \ln \left (a x -1\right )}{128 c^{4} a}+\frac {1}{32 a \,c^{4} \left (a x +1\right )^{4}}+\frac {1}{16 a \,c^{4} \left (a x +1\right )^{3}}+\frac {3}{32 a \,c^{4} \left (a x +1\right )^{2}}+\frac {5}{32 a \,c^{4} \left (a x +1\right )}-\frac {15 \ln \left (a x +1\right )}{128 a \,c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 140, normalized size = 1.18 \[ \frac {15 \, a^{5} x^{5} + 30 \, a^{4} x^{4} - 10 \, a^{3} x^{3} - 50 \, a^{2} x^{2} - 17 \, a x + 16}{64 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} - \frac {15 \, \log \left (a x + 1\right )}{128 \, a c^{4}} + \frac {15 \, \log \left (a x - 1\right )}{128 \, a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 121, normalized size = 1.02 \[ -\frac {\frac {17\,x}{64}+\frac {25\,a\,x^2}{32}-\frac {1}{4\,a}+\frac {5\,a^2\,x^3}{32}-\frac {15\,a^3\,x^4}{32}-\frac {15\,a^4\,x^5}{64}}{a^6\,c^4\,x^6+2\,a^5\,c^4\,x^5-a^4\,c^4\,x^4-4\,a^3\,c^4\,x^3-a^2\,c^4\,x^2+2\,a\,c^4\,x+c^4}-\frac {15\,\mathrm {atanh}\left (a\,x\right )}{64\,a\,c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 141, normalized size = 1.18 \[ \frac {15 a^{5} x^{5} + 30 a^{4} x^{4} - 10 a^{3} x^{3} - 50 a^{2} x^{2} - 17 a x + 16}{64 a^{7} c^{4} x^{6} + 128 a^{6} c^{4} x^{5} - 64 a^{5} c^{4} x^{4} - 256 a^{4} c^{4} x^{3} - 64 a^{3} c^{4} x^{2} + 128 a^{2} c^{4} x + 64 a c^{4}} + \frac {\frac {15 \log {\left (x - \frac {1}{a} \right )}}{128} - \frac {15 \log {\left (x + \frac {1}{a} \right )}}{128}}{a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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