Optimal. Leaf size=73 \[ \frac {9}{a c^3 (1-a x)}-\frac {7}{2 a c^3 (1-a x)^2}+\frac {2}{3 a c^3 (1-a x)^3}+\frac {5 \log (1-a x)}{a c^3}+\frac {x}{c^3} \]
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Rubi [A] time = 0.16, antiderivative size = 73, 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} \[ \frac {9}{a c^3 (1-a x)}-\frac {7}{2 a c^3 (1-a x)^2}+\frac {2}{3 a c^3 (1-a x)^3}+\frac {5 \log (1-a x)}{a c^3}+\frac {x}{c^3} \]
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 )^3} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx\\ &=\frac {a^3 \int \frac {e^{2 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=\frac {a^3 \int \frac {x^3 (1+a x)}{(1-a x)^4} \, dx}{c^3}\\ &=\frac {a^3 \int \left (\frac {1}{a^3}+\frac {2}{a^3 (-1+a x)^4}+\frac {7}{a^3 (-1+a x)^3}+\frac {9}{a^3 (-1+a x)^2}+\frac {5}{a^3 (-1+a x)}\right ) \, dx}{c^3}\\ &=\frac {x}{c^3}+\frac {2}{3 a c^3 (1-a x)^3}-\frac {7}{2 a c^3 (1-a x)^2}+\frac {9}{a c^3 (1-a x)}+\frac {5 \log (1-a x)}{a c^3}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 63, normalized size = 0.86 \[ \frac {6 a^4 x^4-18 a^3 x^3-36 a^2 x^2+81 a x+30 (a x-1)^3 \log (1-a x)-37}{6 a c^3 (a x-1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 100, normalized size = 1.37 \[ \frac {6 \, a^{4} x^{4} - 18 \, a^{3} x^{3} - 36 \, a^{2} x^{2} + 81 \, a x + 30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) - 37}{6 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 50, normalized size = 0.68 \[ \frac {x}{c^{3}} + \frac {5 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{3}} - \frac {54 \, a^{2} x^{2} - 87 \, a x + 37}{6 \, {\left (a x - 1\right )}^{3} a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 66, normalized size = 0.90 \[ \frac {x}{c^{3}}+\frac {5 \ln \left (a x -1\right )}{a \,c^{3}}-\frac {9}{a \,c^{3} \left (a x -1\right )}-\frac {2}{3 a \,c^{3} \left (a x -1\right )^{3}}-\frac {7}{2 a \,c^{3} \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 75, normalized size = 1.03 \[ -\frac {54 \, a^{2} x^{2} - 87 \, a x + 37}{6 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} + \frac {x}{c^{3}} + \frac {5 \, \log \left (a x - 1\right )}{a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 71, normalized size = 0.97 \[ \frac {9\,a\,x^2-\frac {29\,x}{2}+\frac {37}{6\,a}}{-a^3\,c^3\,x^3+3\,a^2\,c^3\,x^2-3\,a\,c^3\,x+c^3}+\frac {x}{c^3}+\frac {5\,\ln \left (a\,x-1\right )}{a\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 73, normalized size = 1.00 \[ \frac {- 54 a^{2} x^{2} + 87 a x - 37}{6 a^{4} c^{3} x^{3} - 18 a^{3} c^{3} x^{2} + 18 a^{2} c^{3} x - 6 a c^{3}} + \frac {x}{c^{3}} + \frac {5 \log {\left (a x - 1 \right )}}{a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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