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 = 0.128178, 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)^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 = 0.0217449, 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 = 0.037, 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 = 0.967265, 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 = 1.85926, 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 = 2.21957, size = 76, normalized size = 0.97 \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 = 1.23353, size = 184, normalized size = 2.36 \begin{align*} -\frac{2 \, c^{3} \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{2 \, c^{3} \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right )}{a} + \frac{{\left (30 \, c^{3} - \frac{71 \, c^{3}}{a x + 1} - \frac{65 \, c^{3}}{{\left (a x + 1\right )}^{2}} + \frac{310 \, c^{3}}{{\left (a x + 1\right )}^{3}} - \frac{270 \, c^{3}}{{\left (a x + 1\right )}^{4}} + \frac{60 \, c^{3}}{{\left (a x + 1\right )}^{5}}\right )}{\left (a x + 1\right )}}{30 \, a{\left (\frac{1}{a x + 1} - 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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