Optimal. Leaf size=71 \[ \frac {13}{a c^2 (1-a x)}-\frac {6}{a c^2 (1-a x)^2}+\frac {4}{3 a c^2 (1-a x)^3}+\frac {6 \log (1-a x)}{a c^2}+\frac {x}{c^2} \]
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Rubi [A] time = 0.16, antiderivative size = 71, 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, 88} \[ \frac {13}{a c^2 (1-a x)}-\frac {6}{a c^2 (1-a x)^2}+\frac {4}{3 a c^2 (1-a x)^3}+\frac {6 \log (1-a x)}{a c^2}+\frac {x}{c^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{4 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx\\ &=\frac {a^2 \int \frac {e^{4 \tanh ^{-1}(a x)} x^2}{(1-a x)^2} \, dx}{c^2}\\ &=\frac {a^2 \int \frac {x^2 (1+a x)^2}{(1-a x)^4} \, dx}{c^2}\\ &=\frac {a^2 \int \left (\frac {1}{a^2}+\frac {4}{a^2 (-1+a x)^4}+\frac {12}{a^2 (-1+a x)^3}+\frac {13}{a^2 (-1+a x)^2}+\frac {6}{a^2 (-1+a x)}\right ) \, dx}{c^2}\\ &=\frac {x}{c^2}+\frac {4}{3 a c^2 (1-a x)^3}-\frac {6}{a c^2 (1-a x)^2}+\frac {13}{a c^2 (1-a x)}+\frac {6 \log (1-a x)}{a c^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 63, normalized size = 0.89 \[ \frac {3 a^4 x^4-9 a^3 x^3-30 a^2 x^2+57 a x+18 (a x-1)^3 \log (1-a x)-25}{3 a c^2 (a x-1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 100, normalized size = 1.41 \[ \frac {3 \, a^{4} x^{4} - 9 \, a^{3} x^{3} - 30 \, a^{2} x^{2} + 57 \, a x + 18 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (a x - 1\right ) - 25}{3 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 94, normalized size = 1.32 \[ \frac {a x - 1}{a c^{2}} - \frac {6 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a c^{2}} - \frac {\frac {39 \, a^{5} c^{4}}{a x - 1} + \frac {18 \, a^{5} c^{4}}{{\left (a x - 1\right )}^{2}} + \frac {4 \, a^{5} c^{4}}{{\left (a x - 1\right )}^{3}}}{3 \, a^{6} c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 66, normalized size = 0.93 \[ \frac {x}{c^{2}}+\frac {6 \ln \left (a x -1\right )}{a \,c^{2}}-\frac {13}{a \,c^{2} \left (a x -1\right )}-\frac {4}{3 a \,c^{2} \left (a x -1\right )^{3}}-\frac {6}{a \,c^{2} \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 75, normalized size = 1.06 \[ -\frac {39 \, a^{2} x^{2} - 60 \, a x + 25}{3 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} + \frac {x}{c^{2}} + \frac {6 \, \log \left (a x - 1\right )}{a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 71, normalized size = 1.00 \[ \frac {13\,a\,x^2-20\,x+\frac {25}{3\,a}}{-a^3\,c^2\,x^3+3\,a^2\,c^2\,x^2-3\,a\,c^2\,x+c^2}+\frac {x}{c^2}+\frac {6\,\ln \left (a\,x-1\right )}{a\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 73, normalized size = 1.03 \[ \frac {- 39 a^{2} x^{2} + 60 a x - 25}{3 a^{4} c^{2} x^{3} - 9 a^{3} c^{2} x^{2} + 9 a^{2} c^{2} x - 3 a c^{2}} + \frac {x}{c^{2}} + \frac {6 \log {\left (a x - 1 \right )}}{a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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