Optimal. Leaf size=13 \[ \frac{x}{c (1-a x)^2} \]
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Rubi [A] time = 0.0625637, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6167, 6140, 34} \[ \frac{x}{c (1-a x)^2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6140
Rule 34
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{c-a^2 c x^2} \, dx\\ &=\frac{\int \frac{1+a x}{(1-a x)^3} \, dx}{c}\\ &=\frac{x}{c (1-a x)^2}\\ \end{align*}
Mathematica [A] time = 0.0096785, size = 25, normalized size = 1.92 \[ \frac{(a x+1)^2}{4 a c (1-a x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 28, normalized size = 2.2 \begin{align*}{\frac{1}{c} \left ({\frac{1}{a \left ( ax-1 \right ) ^{2}}}+{\frac{1}{a \left ( ax-1 \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09274, size = 26, normalized size = 2. \begin{align*} \frac{x}{a^{2} c x^{2} - 2 \, a c x + c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48406, size = 39, normalized size = 3. \begin{align*} \frac{x}{a^{2} c x^{2} - 2 \, a c x + c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.343586, size = 17, normalized size = 1.31 \begin{align*} \frac{x}{a^{2} c x^{2} - 2 a c x + c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11978, size = 36, normalized size = 2.77 \begin{align*} \frac{\frac{1}{{\left (a x - 1\right )} a} + \frac{1}{{\left (a x - 1\right )}^{2} a}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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