Optimal. Leaf size=19 \[ a^2 c x+2 a c \log (x)-\frac{c}{x} \]
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Rubi [A] time = 0.0494398, antiderivative size = 19, 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} \[ a^2 c x+2 a c \log (x)-\frac{c}{x} \]
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^2} \, dx &=c \int \frac{(1+a x)^2}{x^2} \, dx\\ &=c \int \left (a^2+\frac{1}{x^2}+\frac{2 a}{x}\right ) \, dx\\ &=-\frac{c}{x}+a^2 c x+2 a c \log (x)\\ \end{align*}
Mathematica [A] time = 0.0125687, size = 18, normalized size = 0.95 \[ c \left (a^2 x+2 a \log (x)-\frac{1}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 20, normalized size = 1.1 \begin{align*} -{\frac{c}{x}}+cx{a}^{2}+2\,ac\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94102, size = 26, normalized size = 1.37 \begin{align*} a^{2} c x + 2 \, a c \log \left (x\right ) - \frac{c}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71466, size = 49, normalized size = 2.58 \begin{align*} \frac{a^{2} c x^{2} + 2 \, a c x \log \left (x\right ) - c}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.277468, size = 17, normalized size = 0.89 \begin{align*} a^{2} c x + 2 a c \log{\left (x \right )} - \frac{c}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15575, size = 27, normalized size = 1.42 \begin{align*} a^{2} c x + 2 \, a c \log \left ({\left | x \right |}\right ) - \frac{c}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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