Optimal. Leaf size=76 \[ -\frac{2 c^3}{a^3 x^2}+\frac{c^3}{3 a^4 x^3}+\frac{c^3}{2 a^5 x^4}-\frac{c^3}{5 a^6 x^5}+\frac{c^3}{a^2 x}-\frac{2 c^3 \log (x)}{a}+c^3 x \]
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Rubi [A] time = 0.155948, antiderivative size = 76, 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, 6157, 6150, 88} \[ -\frac{2 c^3}{a^3 x^2}+\frac{c^3}{3 a^4 x^3}+\frac{c^3}{2 a^5 x^4}-\frac{c^3}{5 a^6 x^5}+\frac{c^3}{a^2 x}-\frac{2 c^3 \log (x)}{a}+c^3 x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6157
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^3 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^3 \, dx\\ &=\frac{c^3 \int \frac{e^{-2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{x^6} \, dx}{a^6}\\ &=\frac{c^3 \int \frac{(1-a x)^4 (1+a x)^2}{x^6} \, dx}{a^6}\\ &=\frac{c^3 \int \left (a^6+\frac{1}{x^6}-\frac{2 a}{x^5}-\frac{a^2}{x^4}+\frac{4 a^3}{x^3}-\frac{a^4}{x^2}-\frac{2 a^5}{x}\right ) \, dx}{a^6}\\ &=-\frac{c^3}{5 a^6 x^5}+\frac{c^3}{2 a^5 x^4}+\frac{c^3}{3 a^4 x^3}-\frac{2 c^3}{a^3 x^2}+\frac{c^3}{a^2 x}+c^3 x-\frac{2 c^3 \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0247236, size = 76, normalized size = 1. \[ -\frac{2 c^3}{a^3 x^2}+\frac{c^3}{3 a^4 x^3}+\frac{c^3}{2 a^5 x^4}-\frac{c^3}{5 a^6 x^5}+\frac{c^3}{a^2 x}-\frac{2 c^3 \log (x)}{a}+c^3 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 71, normalized size = 0.9 \begin{align*} -{\frac{{c}^{3}}{5\,{a}^{6}{x}^{5}}}+{\frac{{c}^{3}}{2\,{a}^{5}{x}^{4}}}+{\frac{{c}^{3}}{3\,{a}^{4}{x}^{3}}}-2\,{\frac{{c}^{3}}{{x}^{2}{a}^{3}}}+{\frac{{c}^{3}}{{a}^{2}x}}+{c}^{3}x-2\,{\frac{{c}^{3}\ln \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04221, size = 95, normalized size = 1.25 \begin{align*} c^{3} x - \frac{2 \, c^{3} \log \left (x\right )}{a} + \frac{30 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15343, size = 174, normalized size = 2.29 \begin{align*} \frac{30 \, a^{6} c^{3} x^{6} - 60 \, a^{5} c^{3} x^{5} \log \left (x\right ) + 30 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.515016, size = 76, normalized size = 1. \begin{align*} \frac{a^{6} c^{3} x - 2 a^{5} c^{3} \log{\left (x \right )} + \frac{30 a^{4} c^{3} x^{4} - 60 a^{3} c^{3} x^{3} + 10 a^{2} c^{3} x^{2} + 15 a c^{3} x - 6 c^{3}}{30 x^{5}}}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1096, size = 96, normalized size = 1.26 \begin{align*} c^{3} x - \frac{2 \, c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac{30 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} + 10 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x - 6 \, c^{3}}{30 \, a^{6} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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