Optimal. Leaf size=53 \[ \frac{x^2}{2 a^2 c}+\frac{2 x}{a^3 c}+\frac{1}{a^4 c (1-a x)}+\frac{3 \log (1-a x)}{a^4 c} \]
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Rubi [A] time = 0.10409, antiderivative size = 53, 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} \[ \frac{x^2}{2 a^2 c}+\frac{2 x}{a^3 c}+\frac{1}{a^4 c (1-a x)}+\frac{3 \log (1-a x)}{a^4 c} \]
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^3}{c-a^2 c x^2} \, dx &=\frac{\int \frac{x^3}{(1-a x)^2} \, dx}{c}\\ &=\frac{\int \left (\frac{2}{a^3}+\frac{x}{a^2}+\frac{1}{a^3 (-1+a x)^2}+\frac{3}{a^3 (-1+a x)}\right ) \, dx}{c}\\ &=\frac{2 x}{a^3 c}+\frac{x^2}{2 a^2 c}+\frac{1}{a^4 c (1-a x)}+\frac{3 \log (1-a x)}{a^4 c}\\ \end{align*}
Mathematica [A] time = 0.0299233, size = 41, normalized size = 0.77 \[ \frac{a^2 x^2+4 a x+\frac{2}{1-a x}+6 \log (1-a x)}{2 a^4 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 51, normalized size = 1. \begin{align*}{\frac{{x}^{2}}{2\,{a}^{2}c}}+2\,{\frac{x}{{a}^{3}c}}-{\frac{1}{c{a}^{4} \left ( ax-1 \right ) }}+3\,{\frac{\ln \left ( ax-1 \right ) }{c{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965163, size = 66, normalized size = 1.25 \begin{align*} -\frac{1}{a^{5} c x - a^{4} c} + \frac{a x^{2} + 4 \, x}{2 \, a^{3} c} + \frac{3 \, \log \left (a x - 1\right )}{a^{4} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90143, size = 113, normalized size = 2.13 \begin{align*} \frac{a^{3} x^{3} + 3 \, a^{2} x^{2} - 4 \, a x + 6 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{2 \,{\left (a^{5} c x - a^{4} c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.35384, size = 44, normalized size = 0.83 \begin{align*} - \frac{1}{a^{5} c x - a^{4} c} + \frac{x^{2}}{2 a^{2} c} + \frac{2 x}{a^{3} c} + \frac{3 \log{\left (a x - 1 \right )}}{a^{4} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16419, size = 72, normalized size = 1.36 \begin{align*} \frac{3 \, \log \left ({\left | a x - 1 \right |}\right )}{a^{4} c} + \frac{a^{2} c x^{2} + 4 \, a c x}{2 \, a^{4} c^{2}} - \frac{1}{{\left (a x - 1\right )} a^{4} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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