Optimal. Leaf size=53 \[ \frac {5}{a c^2 (1-a x)}-\frac {1}{a c^2 (1-a x)^2}+\frac {4 \log (1-a x)}{a c^2}+\frac {x}{c^2} \]
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Rubi [A] time = 0.15, antiderivative size = 53, 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} \[ \frac {5}{a c^2 (1-a x)}-\frac {1}{a c^2 (1-a x)^2}+\frac {4 \log (1-a x)}{a c^2}+\frac {x}{c^2} \]
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 )^2} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx\\ &=-\frac {a^2 \int \frac {e^{2 \tanh ^{-1}(a x)} x^2}{(1-a x)^2} \, dx}{c^2}\\ &=-\frac {a^2 \int \frac {x^2 (1+a x)}{(1-a x)^3} \, dx}{c^2}\\ &=-\frac {a^2 \int \left (-\frac {1}{a^2}-\frac {2}{a^2 (-1+a x)^3}-\frac {5}{a^2 (-1+a x)^2}-\frac {4}{a^2 (-1+a x)}\right ) \, dx}{c^2}\\ &=\frac {x}{c^2}-\frac {1}{a c^2 (1-a x)^2}+\frac {5}{a c^2 (1-a x)}+\frac {4 \log (1-a x)}{a c^2}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 51, normalized size = 0.96 \[ -\frac {5}{a c^2 (a x-1)}-\frac {1}{a c^2 (a x-1)^2}+\frac {4 \log (1-a x)}{a c^2}+\frac {x}{c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 70, normalized size = 1.32 \[ \frac {a^{3} x^{3} - 2 \, a^{2} x^{2} - 4 \, a x + 4 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 4}{a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 42, normalized size = 0.79 \[ \frac {x}{c^{2}} + \frac {4 \, \log \left ({\left | a x - 1 \right |}\right )}{a c^{2}} - \frac {5 \, a x - 4}{{\left (a x - 1\right )}^{2} a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 51, normalized size = 0.96 \[ \frac {x}{c^{2}}+\frac {4 \ln \left (a x -1\right )}{a \,c^{2}}-\frac {5}{a \,c^{2} \left (a x -1\right )}-\frac {1}{a \,c^{2} \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 55, normalized size = 1.04 \[ -\frac {5 \, a x - 4}{a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}} + \frac {x}{c^{2}} + \frac {4 \, \log \left (a x - 1\right )}{a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 54, normalized size = 1.02 \[ \frac {x}{c^2}-\frac {5\,x-\frac {4}{a}}{a^2\,c^2\,x^2-2\,a\,c^2\,x+c^2}+\frac {4\,\ln \left (a\,x-1\right )}{a\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 49, normalized size = 0.92 \[ \frac {- 5 a x + 4}{a^{3} c^{2} x^{2} - 2 a^{2} c^{2} x + a c^{2}} + \frac {x}{c^{2}} + \frac {4 \log {\left (a x - 1 \right )}}{a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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