Optimal. Leaf size=21 \[ \frac{1}{2} a^2 c x^2+2 a c x+c \log (x) \]
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Rubi [A] time = 0.0493203, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {6150, 43} \[ \frac{1}{2} a^2 c x^2+2 a c x+c \log (x) \]
Antiderivative was successfully verified.
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Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )}{x} \, dx &=c \int \frac{(1+a x)^2}{x} \, dx\\ &=c \int \left (2 a+\frac{1}{x}+a^2 x\right ) \, dx\\ &=2 a c x+\frac{1}{2} a^2 c x^2+c \log (x)\\ \end{align*}
Mathematica [A] time = 0.0147296, size = 19, normalized size = 0.9 \[ c \left (\frac{a^2 x^2}{2}+2 a x+\log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 20, normalized size = 1. \begin{align*} 2\,acx+{\frac{{a}^{2}c{x}^{2}}{2}}+c\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.944807, size = 26, normalized size = 1.24 \begin{align*} \frac{1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01512, size = 49, normalized size = 2.33 \begin{align*} \frac{1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.10553, size = 20, normalized size = 0.95 \begin{align*} \frac{a^{2} c x^{2}}{2} + 2 a c x + c \log{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16146, size = 27, normalized size = 1.29 \begin{align*} \frac{1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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