Optimal. Leaf size=13 \[ \frac{x}{c^2 (1-a x)^2} \]
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Rubi [A] time = 0.0261538, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6129, 34} \[ \frac{x}{c^2 (1-a x)^2} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 34
Rubi steps
\begin{align*} \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*}
Mathematica [A] time = 0.0069741, size = 25, normalized size = 1.92 \[ \frac{(a x+1)^2}{4 a c^2 (1-a x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 28, normalized size = 2.2 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{1}{a \left ( ax-1 \right ) }}+{\frac{1}{a \left ( ax-1 \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.938602, size = 34, normalized size = 2.62 \begin{align*} \frac{x}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68245, size = 47, normalized size = 3.62 \begin{align*} \frac{x}{a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.377952, size = 22, normalized size = 1.69 \begin{align*} \frac{x}{a^{2} c^{2} x^{2} - 2 a c^{2} x + c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2299, size = 43, normalized size = 3.31 \begin{align*} \frac{1}{{\left (a c x - c\right )}^{2} a} + \frac{1}{{\left (a c x - c\right )} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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