Optimal. Leaf size=90 \[ -\frac{3 c^4}{a^3 x^2}+\frac{3 c^4}{2 a^5 x^4}-\frac{2 c^4}{5 a^6 x^5}-\frac{c^4}{3 a^7 x^6}+\frac{c^4}{7 a^8 x^7}+\frac{2 c^4}{a^2 x}-\frac{2 c^4 \log (x)}{a}+c^4 x \]
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Rubi [A] time = 0.153422, antiderivative size = 90, 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} \[ -\frac{3 c^4}{a^3 x^2}+\frac{3 c^4}{2 a^5 x^4}-\frac{2 c^4}{5 a^6 x^5}-\frac{c^4}{3 a^7 x^6}+\frac{c^4}{7 a^8 x^7}+\frac{2 c^4}{a^2 x}-\frac{2 c^4 \log (x)}{a}+c^4 x \]
Antiderivative was successfully verified.
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Rule 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 )^4 \, dx &=-\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 = 0.0329902, size = 90, 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 = 0.049, size = 83, 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 = 1.03559, size = 109, normalized size = 1.21 \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 = 1.34407, size = 207, normalized size = 2.3 \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 = 0.653665, size = 88, normalized size = 0.98 \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 = 1.12806, size = 111, normalized size = 1.23 \begin{align*} c^{4} x - \frac{2 \, c^{4} \log \left ({\left | x \right |}\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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