Optimal. Leaf size=14 \[ -\frac {x}{c^2 (1-a x)^2} \]
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Rubi [A] time = 0.05, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6167, 6129, 34} \[ -\frac {x}{c^2 (1-a x)^2} \]
Antiderivative was successfully verified.
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Rule 34
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx\\ &=-\frac {\int \frac {1+a x}{(1-a x)^3} \, dx}{c^2}\\ &=-\frac {x}{c^2 (1-a x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.79 \[ -\frac {(a x+1)^2}{4 a c^2 (1-a x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 26, normalized size = 1.86 \[ -\frac {x}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 34, normalized size = 2.43 \[ -\frac {1}{{\left (a c x - c\right )}^{2} a} - \frac {1}{{\left (a c x - c\right )} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 30, normalized size = 2.14 \[ \frac {-\frac {1}{a \left (a x -1\right )^{2}}-\frac {1}{a \left (a x -1\right )}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 26, normalized size = 1.86 \[ -\frac {x}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 13, normalized size = 0.93 \[ -\frac {x}{c^2\,{\left (a\,x-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 24, normalized size = 1.71 \[ - \frac {x}{a^{2} c^{2} x^{2} - 2 a c^{2} x + c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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