Optimal. Leaf size=23 \[ a^2 c \log (x)-\frac{2 a c}{x}-\frac{c}{2 x^2} \]
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Rubi [A] time = 0.0521579, antiderivative size = 23, 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} \[ a^2 c \log (x)-\frac{2 a c}{x}-\frac{c}{2 x^2} \]
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^3} \, dx &=c \int \frac{(1+a x)^2}{x^3} \, dx\\ &=c \int \left (\frac{1}{x^3}+\frac{2 a}{x^2}+\frac{a^2}{x}\right ) \, dx\\ &=-\frac{c}{2 x^2}-\frac{2 a c}{x}+a^2 c \log (x)\\ \end{align*}
Mathematica [A] time = 0.0166373, size = 22, normalized size = 0.96 \[ c \left (a^2 \log (x)-\frac{2 a}{x}-\frac{1}{2 x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 22, normalized size = 1. \begin{align*} -{\frac{c}{2\,{x}^{2}}}-2\,{\frac{ac}{x}}+{a}^{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 = 0.952203, size = 27, normalized size = 1.17 \begin{align*} a^{2} c \log \left (x\right ) - \frac{4 \, a c x + c}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71111, size = 59, normalized size = 2.57 \begin{align*} \frac{2 \, a^{2} c x^{2} \log \left (x\right ) - 4 \, a c x - c}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.321967, size = 20, normalized size = 0.87 \begin{align*} a^{2} c \log{\left (x \right )} - \frac{4 a c x + c}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13781, size = 28, normalized size = 1.22 \begin{align*} a^{2} c \log \left ({\left | x \right |}\right ) - \frac{4 \, a c x + c}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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