Optimal. Leaf size=146 \[ \frac {501}{64 a c^4 (1-a x)}-\frac {1}{64 a c^4 (a x+1)}-\frac {67}{16 a c^4 (1-a x)^2}+\frac {83}{48 a c^4 (1-a x)^3}-\frac {7}{16 a c^4 (1-a x)^4}+\frac {1}{20 a c^4 (1-a x)^5}+\frac {261 \log (1-a x)}{64 a c^4}-\frac {5 \log (a x+1)}{64 a c^4}+\frac {x}{c^4} \]
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Rubi [A] time = 0.19, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6157, 6150, 88} \[ \frac {501}{64 a c^4 (1-a x)}-\frac {1}{64 a c^4 (a x+1)}-\frac {67}{16 a c^4 (1-a x)^2}+\frac {83}{48 a c^4 (1-a x)^3}-\frac {7}{16 a c^4 (1-a x)^4}+\frac {1}{20 a c^4 (1-a x)^5}+\frac {261 \log (1-a x)}{64 a c^4}-\frac {5 \log (a x+1)}{64 a c^4}+\frac {x}{c^4} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6157
Rubi steps
\begin {align*} \int \frac {e^{4 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx &=\frac {a^8 \int \frac {e^{4 \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)^6 (1+a x)^2} \, dx}{c^4}\\ &=\frac {a^8 \int \left (\frac {1}{a^8}+\frac {1}{4 a^8 (-1+a x)^6}+\frac {7}{4 a^8 (-1+a x)^5}+\frac {83}{16 a^8 (-1+a x)^4}+\frac {67}{8 a^8 (-1+a x)^3}+\frac {501}{64 a^8 (-1+a x)^2}+\frac {261}{64 a^8 (-1+a x)}+\frac {1}{64 a^8 (1+a x)^2}-\frac {5}{64 a^8 (1+a x)}\right ) \, dx}{c^4}\\ &=\frac {x}{c^4}+\frac {1}{20 a c^4 (1-a x)^5}-\frac {7}{16 a c^4 (1-a x)^4}+\frac {83}{48 a c^4 (1-a x)^3}-\frac {67}{16 a c^4 (1-a x)^2}+\frac {501}{64 a c^4 (1-a x)}-\frac {1}{64 a c^4 (1+a x)}+\frac {261 \log (1-a x)}{64 a c^4}-\frac {5 \log (1+a x)}{64 a c^4}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 98, normalized size = 0.67 \[ \frac {\frac {2 \left (480 a^7 x^7-1920 a^6 x^6-1365 a^5 x^5+9300 a^4 x^4-6800 a^3 x^3-4900 a^2 x^2+7541 a x-2384\right )}{(a x-1)^5 (a x+1)}+3915 \log (1-a x)-75 \log (a x+1)}{960 a c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 207, normalized size = 1.42 \[ \frac {960 \, a^{7} x^{7} - 3840 \, a^{6} x^{6} - 2730 \, a^{5} x^{5} + 18600 \, a^{4} x^{4} - 13600 \, a^{3} x^{3} - 9800 \, a^{2} x^{2} + 15082 \, a x - 75 \, {\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (a x + 1\right ) + 3915 \, {\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \log \left (a x - 1\right ) - 4768}{960 \, {\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 96, normalized size = 0.66 \[ \frac {x}{c^{4}} - \frac {5 \, \log \left ({\left | a x + 1 \right |}\right )}{64 \, a c^{4}} + \frac {261 \, \log \left ({\left | a x - 1 \right |}\right )}{64 \, a c^{4}} - \frac {3765 \, a^{5} x^{5} - 9300 \, a^{4} x^{4} + 4400 \, a^{3} x^{3} + 6820 \, a^{2} x^{2} - 8021 \, a x + 2384}{480 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{5} a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 125, normalized size = 0.86 \[ \frac {x}{c^{4}}-\frac {1}{20 a \,c^{4} \left (a x -1\right )^{5}}-\frac {7}{16 a \,c^{4} \left (a x -1\right )^{4}}-\frac {83}{48 a \,c^{4} \left (a x -1\right )^{3}}-\frac {67}{16 a \,c^{4} \left (a x -1\right )^{2}}-\frac {501}{64 a \,c^{4} \left (a x -1\right )}+\frac {261 \ln \left (a x -1\right )}{64 a \,c^{4}}-\frac {1}{64 a \,c^{4} \left (a x +1\right )}-\frac {5 \ln \left (a x +1\right )}{64 a \,c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 135, normalized size = 0.92 \[ -\frac {3765 \, a^{5} x^{5} - 9300 \, a^{4} x^{4} + 4400 \, a^{3} x^{3} + 6820 \, a^{2} x^{2} - 8021 \, a x + 2384}{480 \, {\left (a^{7} c^{4} x^{6} - 4 \, a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{3} c^{4} x^{2} + 4 \, a^{2} c^{4} x - a c^{4}\right )}} + \frac {x}{c^{4}} - \frac {5 \, \log \left (a x + 1\right )}{64 \, a c^{4}} + \frac {261 \, \log \left (a x - 1\right )}{64 \, a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 131, normalized size = 0.90 \[ \frac {\frac {341\,a\,x^2}{24}-\frac {8021\,x}{480}+\frac {149}{30\,a}+\frac {55\,a^2\,x^3}{6}-\frac {155\,a^3\,x^4}{8}+\frac {251\,a^4\,x^5}{32}}{-a^6\,c^4\,x^6+4\,a^5\,c^4\,x^5-5\,a^4\,c^4\,x^4+5\,a^2\,c^4\,x^2-4\,a\,c^4\,x+c^4}+\frac {x}{c^4}+\frac {261\,\ln \left (a\,x-1\right )}{64\,a\,c^4}-\frac {5\,\ln \left (a\,x+1\right )}{64\,a\,c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.90, size = 144, normalized size = 0.99 \[ a^{8} \left (\frac {- 3765 a^{5} x^{5} + 9300 a^{4} x^{4} - 4400 a^{3} x^{3} - 6820 a^{2} x^{2} + 8021 a x - 2384}{480 a^{15} c^{4} x^{6} - 1920 a^{14} c^{4} x^{5} + 2400 a^{13} c^{4} x^{4} - 2400 a^{11} c^{4} x^{2} + 1920 a^{10} c^{4} x - 480 a^{9} c^{4}} + \frac {x}{a^{8} c^{4}} + \frac {\frac {261 \log {\left (x - \frac {1}{a} \right )}}{64} - \frac {5 \log {\left (x + \frac {1}{a} \right )}}{64}}{a^{9} c^{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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