Optimal. Leaf size=78 \[ -\frac{2 c^3}{a^3 x^2}-\frac{c^3}{3 a^4 x^3}+\frac{c^3}{2 a^5 x^4}+\frac{c^3}{5 a^6 x^5}-\frac{c^3}{a^2 x}-\frac{2 c^3 \log (x)}{a}+c^3 (-x) \]
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Rubi [A] time = 0.124213, antiderivative size = 78, 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 = {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 6157
Rule 6150
Rule 88
Rubi steps
\begin{align*} \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)^2 (1+a x)^4}{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 = 0.0224063, size = 78, 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 = 0.036, size = 73, 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 = 0.959085, size = 96, normalized size = 1.23 \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 = 1.95947, size = 176, normalized size = 2.26 \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 = 2.60921, size = 78, 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 = 1.20588, size = 97, normalized size = 1.24 \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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