Optimal. Leaf size=39 \[ \frac{x}{a^2 c}+\frac{1}{a^3 c (1-a x)}+\frac{2 \log (1-a x)}{a^3 c} \]
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Rubi [A] time = 0.0955794, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 43} \[ \frac{x}{a^2 c}+\frac{1}{a^3 c (1-a x)}+\frac{2 \log (1-a x)}{a^3 c} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x^2}{c-a^2 c x^2} \, dx &=\frac{\int \frac{x^2}{(1-a x)^2} \, dx}{c}\\ &=\frac{\int \left (\frac{1}{a^2}+\frac{1}{a^2 (-1+a x)^2}+\frac{2}{a^2 (-1+a x)}\right ) \, dx}{c}\\ &=\frac{x}{a^2 c}+\frac{1}{a^3 c (1-a x)}+\frac{2 \log (1-a x)}{a^3 c}\\ \end{align*}
Mathematica [A] time = 0.024749, size = 28, normalized size = 0.72 \[ \frac{a x+\frac{1}{1-a x}+2 \log (1-a x)}{a^3 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 39, normalized size = 1. \begin{align*}{\frac{x}{{a}^{2}c}}-{\frac{1}{{a}^{3}c \left ( ax-1 \right ) }}+2\,{\frac{\ln \left ( ax-1 \right ) }{{a}^{3}c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953028, size = 54, normalized size = 1.38 \begin{align*} -\frac{1}{a^{4} c x - a^{3} c} + \frac{x}{a^{2} c} + \frac{2 \, \log \left (a x - 1\right )}{a^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02149, size = 89, normalized size = 2.28 \begin{align*} \frac{a^{2} x^{2} - a x + 2 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 1}{a^{4} c x - a^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.330464, size = 32, normalized size = 0.82 \begin{align*} - \frac{1}{a^{4} c x - a^{3} c} + \frac{x}{a^{2} c} + \frac{2 \log{\left (a x - 1 \right )}}{a^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13432, size = 53, normalized size = 1.36 \begin{align*} \frac{x}{a^{2} c} + \frac{2 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{3} c} - \frac{1}{{\left (a x - 1\right )} a^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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