Optimal. Leaf size=91 \[ \frac{3 c^4}{a^3 x^2}-\frac{3 c^4}{2 a^5 x^4}+\frac{2 c^4}{5 a^6 x^5}+\frac{c^4}{3 a^7 x^6}-\frac{c^4}{7 a^8 x^7}-\frac{2 c^4}{a^2 x}+\frac{2 c^4 \log (x)}{a}+c^4 (-x) \]
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Rubi [A] time = 0.139025, antiderivative size = 91, 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} \[ \frac{3 c^4}{a^3 x^2}-\frac{3 c^4}{2 a^5 x^4}+\frac{2 c^4}{5 a^6 x^5}+\frac{c^4}{3 a^7 x^6}-\frac{c^4}{7 a^8 x^7}-\frac{2 c^4}{a^2 x}+\frac{2 c^4 \log (x)}{a}+c^4 (-x) \]
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 )^4 \, dx &=\frac{c^4 \int \frac{e^{-2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8}\\ &=\frac{c^4 \int \frac{(1-a x)^5 (1+a x)^3}{x^8} \, dx}{a^8}\\ &=\frac{c^4 \int \left (-a^8+\frac{1}{x^8}-\frac{2 a}{x^7}-\frac{2 a^2}{x^6}+\frac{6 a^3}{x^5}-\frac{6 a^5}{x^3}+\frac{2 a^6}{x^2}+\frac{2 a^7}{x}\right ) \, dx}{a^8}\\ &=-\frac{c^4}{7 a^8 x^7}+\frac{c^4}{3 a^7 x^6}+\frac{2 c^4}{5 a^6 x^5}-\frac{3 c^4}{2 a^5 x^4}+\frac{3 c^4}{a^3 x^2}-\frac{2 c^4}{a^2 x}-c^4 x+\frac{2 c^4 \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0264886, size = 91, normalized size = 1. \[ \frac{3 c^4}{a^3 x^2}-\frac{3 c^4}{2 a^5 x^4}+\frac{2 c^4}{5 a^6 x^5}+\frac{c^4}{3 a^7 x^6}-\frac{c^4}{7 a^8 x^7}-\frac{2 c^4}{a^2 x}+\frac{2 c^4 \log (x)}{a}+c^4 (-x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 84, normalized size = 0.9 \begin{align*} -{\frac{{c}^{4}}{7\,{a}^{8}{x}^{7}}}+{\frac{{c}^{4}}{3\,{a}^{7}{x}^{6}}}+{\frac{2\,{c}^{4}}{5\,{a}^{6}{x}^{5}}}-{\frac{3\,{c}^{4}}{2\,{a}^{5}{x}^{4}}}+3\,{\frac{{c}^{4}}{{x}^{2}{a}^{3}}}-2\,{\frac{{c}^{4}}{{a}^{2}x}}-{c}^{4}x+2\,{\frac{{c}^{4}\ln \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961714, size = 111, normalized size = 1.22 \begin{align*} -c^{4} x + \frac{2 \, c^{4} \log \left (x\right )}{a} - \frac{420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92597, size = 208, normalized size = 2.29 \begin{align*} -\frac{210 \, a^{8} c^{4} x^{8} - 420 \, a^{7} c^{4} x^{7} \log \left (x\right ) + 420 \, a^{6} c^{4} x^{6} - 630 \, a^{5} c^{4} x^{5} + 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.56531, size = 88, normalized size = 0.97 \begin{align*} \frac{- a^{8} c^{4} x + 2 a^{7} c^{4} \log{\left (x \right )} - \frac{420 a^{6} c^{4} x^{6} - 630 a^{5} c^{4} x^{5} + 315 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} - 70 a c^{4} x + 30 c^{4}}{210 x^{7}}}{a^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17059, size = 216, normalized size = 2.37 \begin{align*} -\frac{2 \, c^{4} \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{2 \, c^{4} \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right )}{a} + \frac{{\left (210 \, c^{4} - \frac{719 \, c^{4}}{a x + 1} - \frac{427 \, c^{4}}{{\left (a x + 1\right )}^{2}} + \frac{5271 \, c^{4}}{{\left (a x + 1\right )}^{3}} - \frac{9485 \, c^{4}}{{\left (a x + 1\right )}^{4}} + \frac{7490 \, c^{4}}{{\left (a x + 1\right )}^{5}} - \frac{2730 \, c^{4}}{{\left (a x + 1\right )}^{6}} + \frac{420 \, c^{4}}{{\left (a x + 1\right )}^{7}}\right )}{\left (a x + 1\right )}}{210 \, a{\left (\frac{1}{a x + 1} - 1\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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