Optimal. Leaf size=55 \[ \frac{c^3}{2 a^3 x^2}-\frac{5 c^3}{a^2 x}-\frac{11 c^3 \log (x)}{a}+\frac{16 c^3 \log (a x+1)}{a}+c^3 (-x) \]
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Rubi [A] time = 0.116294, antiderivative size = 55, 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} \[ \frac{c^3}{2 a^3 x^2}-\frac{5 c^3}{a^2 x}-\frac{11 c^3 \log (x)}{a}+\frac{16 c^3 \log (a x+1)}{a}+c^3 (-x) \]
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 )^3 \, dx &=-\frac{c^3 \int \frac{e^{-2 \tanh ^{-1}(a x)} (1-a x)^3}{x^3} \, dx}{a^3}\\ &=-\frac{c^3 \int \frac{(1-a x)^4}{x^3 (1+a x)} \, dx}{a^3}\\ &=-\frac{c^3 \int \left (a^3+\frac{1}{x^3}-\frac{5 a}{x^2}+\frac{11 a^2}{x}-\frac{16 a^3}{1+a x}\right ) \, dx}{a^3}\\ &=\frac{c^3}{2 a^3 x^2}-\frac{5 c^3}{a^2 x}-c^3 x-\frac{11 c^3 \log (x)}{a}+\frac{16 c^3 \log (1+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.124412, size = 57, normalized size = 1.04 \[ \frac{c^3}{2 a^3 x^2}-\frac{5 c^3}{a^2 x}-\frac{11 c^3 \log (a x)}{a}+\frac{16 c^3 \log (a x+1)}{a}+c^3 (-x) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.037, size = 54, normalized size = 1. \begin{align*}{\frac{{c}^{3}}{2\,{x}^{2}{a}^{3}}}-5\,{\frac{{c}^{3}}{{a}^{2}x}}-{c}^{3}x-11\,{\frac{{c}^{3}\ln \left ( x \right ) }{a}}+16\,{\frac{{c}^{3}\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 = 0.966151, size = 70, normalized size = 1.27 \begin{align*} -c^{3} x + \frac{16 \, c^{3} \log \left (a x + 1\right )}{a} - \frac{11 \, c^{3} \log \left (x\right )}{a} - \frac{10 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22177, size = 142, normalized size = 2.58 \begin{align*} -\frac{2 \, a^{3} c^{3} x^{3} - 32 \, a^{2} c^{3} x^{2} \log \left (a x + 1\right ) + 22 \, a^{2} c^{3} x^{2} \log \left (x\right ) + 10 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.629576, size = 44, normalized size = 0.8 \begin{align*} - c^{3} x - \frac{c^{3} \left (11 \log{\left (x \right )} - 16 \log{\left (x + \frac{1}{a} \right )}\right )}{a} - \frac{10 a c^{3} x - c^{3}}{2 a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16233, size = 135, normalized size = 2.45 \begin{align*} -\frac{5 \, c^{3} \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} - \frac{11 \, c^{3} \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right )}{a} - \frac{{\left (2 \, c^{3} + \frac{7 \, c^{3}}{a x + 1} - \frac{10 \, c^{3}}{{\left (a x + 1\right )}^{2}}\right )}{\left (a x + 1\right )}}{2 \, a{\left (\frac{1}{a x + 1} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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