Optimal. Leaf size=42 \[ -\frac{c^2}{a^2 x}-\frac{4 c^2 \log (x)}{a}+\frac{8 c^2 \log (a x+1)}{a}+c^2 (-x) \]
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Rubi [A] time = 0.110625, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {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 6131
Rule 6129
Rule 88
Rubi steps
\begin{align*} \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 = 0.0974576, size = 44, 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 = 0.036, size = 43, 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 = 0.959153, size = 57, normalized size = 1.36 \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 = 2.24187, size = 100, normalized size = 2.38 \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 = 0.522319, size = 32, normalized size = 0.76 \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 = 1.20323, size = 97, normalized size = 2.31 \begin{align*} -\frac{4 \, c^{2} \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} - \frac{4 \, c^{2} \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right )}{a} + \frac{{\left (a x + 1\right )} c^{2}}{a{\left (\frac{1}{a x + 1} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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