Optimal. Leaf size=40 \[ \frac{c^2}{a^2 x}+\frac{4 c^2 \log (x)}{a}-\frac{8 c^2 \log (a x+1)}{a}+c^2 x \]
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Rubi [A] time = 0.134509, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 6131, 6129, 88} \[ \frac{c^2}{a^2 x}+\frac{4 c^2 \log (x)}{a}-\frac{8 c^2 \log (a x+1)}{a}+c^2 x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6131
Rule 6129
Rule 88
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx\\ &=-\frac{c^2 \int \frac{e^{-2 \tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=-\frac{c^2 \int \frac{(1-a x)^3}{x^2 (1+a x)} \, dx}{a^2}\\ &=-\frac{c^2 \int \left (-a^2+\frac{1}{x^2}-\frac{4 a}{x}+\frac{8 a^2}{1+a x}\right ) \, dx}{a^2}\\ &=\frac{c^2}{a^2 x}+c^2 x+\frac{4 c^2 \log (x)}{a}-\frac{8 c^2 \log (1+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0998416, size = 42, normalized size = 1.05 \[ \frac{c^2}{a^2 x}+\frac{4 c^2 \log (a x)}{a}-\frac{8 c^2 \log (a x+1)}{a}+c^2 x \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 41, normalized size = 1. \begin{align*}{\frac{{c}^{2}}{{a}^{2}x}}+x{c}^{2}+4\,{\frac{{c}^{2}\ln \left ( x \right ) }{a}}-8\,{\frac{{c}^{2}\ln \left ( ax+1 \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04258, size = 54, normalized size = 1.35 \begin{align*} c^{2} x - \frac{8 \, c^{2} \log \left (a x + 1\right )}{a} + \frac{4 \, c^{2} \log \left (x\right )}{a} + \frac{c^{2}}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77615, size = 99, normalized size = 2.48 \begin{align*} \frac{a^{2} c^{2} x^{2} - 8 \, a c^{2} x \log \left (a x + 1\right ) + 4 \, a c^{2} x \log \left (x\right ) + c^{2}}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.470257, size = 31, normalized size = 0.78 \begin{align*} c^{2} x + \frac{4 c^{2} \left (\log{\left (x \right )} - 2 \log{\left (x + \frac{1}{a} \right )}\right )}{a} + \frac{c^{2}}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19342, size = 57, normalized size = 1.42 \begin{align*} c^{2} x - \frac{8 \, c^{2} \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac{4 \, c^{2} \log \left ({\left | x \right |}\right )}{a} + \frac{c^{2}}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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