Optimal. Leaf size=87 \[ \frac {14}{a c^4 (1-a x)}-\frac {8}{a c^4 (1-a x)^2}+\frac {3}{a c^4 (1-a x)^3}-\frac {1}{2 a c^4 (1-a x)^4}+\frac {6 \log (1-a x)}{a c^4}+\frac {x}{c^4} \]
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Rubi [A] time = 0.17, antiderivative size = 87, 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, 6131, 6129, 77} \[ \frac {14}{a c^4 (1-a x)}-\frac {8}{a c^4 (1-a x)^2}+\frac {3}{a c^4 (1-a x)^3}-\frac {1}{2 a c^4 (1-a x)^4}+\frac {6 \log (1-a x)}{a c^4}+\frac {x}{c^4} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx\\ &=-\frac {a^4 \int \frac {e^{2 \tanh ^{-1}(a x)} x^4}{(1-a x)^4} \, dx}{c^4}\\ &=-\frac {a^4 \int \frac {x^4 (1+a x)}{(1-a x)^5} \, dx}{c^4}\\ &=-\frac {a^4 \int \left (-\frac {1}{a^4}-\frac {2}{a^4 (-1+a x)^5}-\frac {9}{a^4 (-1+a x)^4}-\frac {16}{a^4 (-1+a x)^3}-\frac {14}{a^4 (-1+a x)^2}-\frac {6}{a^4 (-1+a x)}\right ) \, dx}{c^4}\\ &=\frac {x}{c^4}-\frac {1}{2 a c^4 (1-a x)^4}+\frac {3}{a c^4 (1-a x)^3}-\frac {8}{a c^4 (1-a x)^2}+\frac {14}{a c^4 (1-a x)}+\frac {6 \log (1-a x)}{a c^4}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 71, normalized size = 0.82 \[ \frac {2 a^5 x^5-8 a^4 x^4-16 a^3 x^3+60 a^2 x^2-56 a x+12 (a x-1)^4 \log (1-a x)+17}{2 a c^4 (a x-1)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.50, size = 126, normalized size = 1.45 \[ \frac {2 \, a^{5} x^{5} - 8 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 60 \, a^{2} x^{2} - 56 \, a x + 12 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (a x - 1\right ) + 17}{2 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 58, normalized size = 0.67 \[ \frac {x}{c^{4}} + \frac {6 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{4}} - \frac {28 \, a^{3} x^{3} - 68 \, a^{2} x^{2} + 58 \, a x - 17}{2 \, {\left (a x - 1\right )}^{4} a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 81, normalized size = 0.93 \[ \frac {x}{c^{4}}+\frac {6 \ln \left (a x -1\right )}{a \,c^{4}}-\frac {14}{a \,c^{4} \left (a x -1\right )}-\frac {1}{2 a \,c^{4} \left (a x -1\right )^{4}}-\frac {3}{a \,c^{4} \left (a x -1\right )^{3}}-\frac {8}{a \,c^{4} \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 93, normalized size = 1.07 \[ -\frac {28 \, a^{3} x^{3} - 68 \, a^{2} x^{2} + 58 \, a x - 17}{2 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {x}{c^{4}} + \frac {6 \, \log \left (a x - 1\right )}{a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 90, normalized size = 1.03 \[ \frac {x}{c^4}-\frac {29\,x-34\,a\,x^2-\frac {17}{2\,a}+14\,a^2\,x^3}{a^4\,c^4\,x^4-4\,a^3\,c^4\,x^3+6\,a^2\,c^4\,x^2-4\,a\,c^4\,x+c^4}+\frac {6\,\ln \left (a\,x-1\right )}{a\,c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 94, normalized size = 1.08 \[ \frac {- 28 a^{3} x^{3} + 68 a^{2} x^{2} - 58 a x + 17}{2 a^{5} c^{4} x^{4} - 8 a^{4} c^{4} x^{3} + 12 a^{3} c^{4} x^{2} - 8 a^{2} c^{4} x + 2 a c^{4}} + \frac {x}{c^{4}} + \frac {6 \log {\left (a x - 1 \right )}}{a c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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