Optimal. Leaf size=54 \[ -\frac{c^3}{2 a^3 x^2}+\frac{5 c^3}{a^2 x}+\frac{11 c^3 \log (x)}{a}-\frac{16 c^3 \log (a x+1)}{a}+c^3 x \]
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Rubi [A] time = 0.141078, antiderivative size = 54, 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} \[ -\frac{c^3}{2 a^3 x^2}+\frac{5 c^3}{a^2 x}+\frac{11 c^3 \log (x)}{a}-\frac{16 c^3 \log (a x+1)}{a}+c^3 x \]
Antiderivative was successfully verified.
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Rule 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 )^3 \, dx &=-\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 = 0.128238, size = 56, 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 = 0.046, size = 53, 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 = 1.09595, size = 69, normalized size = 1.28 \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 = 1.84189, size = 140, normalized size = 2.59 \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 = 0.56804, size = 42, normalized size = 0.78 \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 = 1.17912, size = 72, normalized size = 1.33 \begin{align*} c^{3} x - \frac{16 \, c^{3} \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac{11 \, c^{3} \log \left ({\left | x \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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