Optimal. Leaf size=21 \[ -\frac{c}{a^2 x}+\frac{2 c \log (x)}{a}+c x \]
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Rubi [A] time = 0.0854281, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6167, 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 6167
Rule 6157
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right ) \, dx &=-\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 = 0.0136718, 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 = 0.042, 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 = 1.04415, 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 = 1.44905, size = 57, normalized size = 2.71 \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 = 0.264319, 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 = 1.13647, 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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