Optimal. Leaf size=21 \[ \frac{c}{a^2 x}-\frac{2 c \log (x)}{a}+c (-x) \]
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Rubi [A] time = 0.0679866, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6157, 6150, 43} \[ \frac{c}{a^2 x}-\frac{2 c \log (x)}{a}+c (-x) \]
Antiderivative was successfully verified.
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Rule 6157
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right ) \, dx &=-\frac{c \int \frac{e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )}{x^2} \, dx}{a^2}\\ &=-\frac{c \int \frac{(1+a x)^2}{x^2} \, dx}{a^2}\\ &=-\frac{c \int \left (a^2+\frac{1}{x^2}+\frac{2 a}{x}\right ) \, dx}{a^2}\\ &=\frac{c}{a^2 x}-c x-\frac{2 c \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.011725, size = 21, normalized size = 1. \[ \frac{c}{a^2 x}-\frac{2 c \log (x)}{a}+c (-x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 22, normalized size = 1.1 \begin{align*}{\frac{c}{{a}^{2}x}}-cx-2\,{\frac{c\ln \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965781, size = 28, normalized size = 1.33 \begin{align*} -c x - \frac{2 \, c \log \left (x\right )}{a} + \frac{c}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99785, size = 58, normalized size = 2.76 \begin{align*} -\frac{a^{2} c x^{2} + 2 \, a c x \log \left (x\right ) - c}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.448533, size = 20, normalized size = 0.95 \begin{align*} \frac{- a^{2} c x - 2 a c \log{\left (x \right )} + \frac{c}{x}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17002, size = 30, normalized size = 1.43 \begin{align*} -c x - \frac{2 \, c \log \left ({\left | x \right |}\right )}{a} + \frac{c}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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