Optimal. Leaf size=100 \[ \frac{2 c^4}{a^3 x^2}+\frac{10 c^4}{3 a^4 x^3}+\frac{c^4}{a^5 x^4}-\frac{4 c^4}{5 a^6 x^5}-\frac{2 c^4}{3 a^7 x^6}-\frac{c^4}{7 a^8 x^7}-\frac{4 c^4}{a^2 x}+\frac{4 c^4 \log (x)}{a}+c^4 x \]
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Rubi [A] time = 0.166298, antiderivative size = 100, 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} \[ \frac{2 c^4}{a^3 x^2}+\frac{10 c^4}{3 a^4 x^3}+\frac{c^4}{a^5 x^4}-\frac{4 c^4}{5 a^6 x^5}-\frac{2 c^4}{3 a^7 x^6}-\frac{c^4}{7 a^8 x^7}-\frac{4 c^4}{a^2 x}+\frac{4 c^4 \log (x)}{a}+c^4 x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6157
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^4 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^4 \, dx\\ &=\frac{c^4 \int \frac{e^{4 \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)^2 (1+a x)^6}{x^8} \, dx}{a^8}\\ &=\frac{c^4 \int \left (a^8+\frac{1}{x^8}+\frac{4 a}{x^7}+\frac{4 a^2}{x^6}-\frac{4 a^3}{x^5}-\frac{10 a^4}{x^4}-\frac{4 a^5}{x^3}+\frac{4 a^6}{x^2}+\frac{4 a^7}{x}\right ) \, dx}{a^8}\\ &=-\frac{c^4}{7 a^8 x^7}-\frac{2 c^4}{3 a^7 x^6}-\frac{4 c^4}{5 a^6 x^5}+\frac{c^4}{a^5 x^4}+\frac{10 c^4}{3 a^4 x^3}+\frac{2 c^4}{a^3 x^2}-\frac{4 c^4}{a^2 x}+c^4 x+\frac{4 c^4 \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0329438, size = 100, normalized size = 1. \[ \frac{2 c^4}{a^3 x^2}+\frac{10 c^4}{3 a^4 x^3}+\frac{c^4}{a^5 x^4}-\frac{4 c^4}{5 a^6 x^5}-\frac{2 c^4}{3 a^7 x^6}-\frac{c^4}{7 a^8 x^7}-\frac{4 c^4}{a^2 x}+\frac{4 c^4 \log (x)}{a}+c^4 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 93, normalized size = 0.9 \begin{align*} -{\frac{{c}^{4}}{7\,{a}^{8}{x}^{7}}}-{\frac{2\,{c}^{4}}{3\,{a}^{7}{x}^{6}}}-{\frac{4\,{c}^{4}}{5\,{a}^{6}{x}^{5}}}+{\frac{{c}^{4}}{{a}^{5}{x}^{4}}}+{\frac{10\,{c}^{4}}{3\,{a}^{4}{x}^{3}}}+2\,{\frac{{c}^{4}}{{x}^{2}{a}^{3}}}-4\,{\frac{{c}^{4}}{{a}^{2}x}}+{c}^{4}x+4\,{\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 = 1.02203, size = 124, normalized size = 1.24 \begin{align*} c^{4} x + \frac{4 \, c^{4} \log \left (x\right )}{a} - \frac{420 \, a^{6} c^{4} x^{6} - 210 \, a^{5} c^{4} x^{5} - 350 \, a^{4} c^{4} x^{4} - 105 \, a^{3} c^{4} x^{3} + 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 15 \, c^{4}}{105 \, a^{8} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4738, size = 231, normalized size = 2.31 \begin{align*} \frac{105 \, a^{8} c^{4} x^{8} + 420 \, a^{7} c^{4} x^{7} \log \left (x\right ) - 420 \, a^{6} c^{4} x^{6} + 210 \, a^{5} c^{4} x^{5} + 350 \, a^{4} c^{4} x^{4} + 105 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} - 70 \, a c^{4} x - 15 \, c^{4}}{105 \, a^{8} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.721691, size = 100, normalized size = 1. \begin{align*} \frac{a^{8} c^{4} x + 4 a^{7} c^{4} \log{\left (x \right )} - \frac{420 a^{6} c^{4} x^{6} - 210 a^{5} c^{4} x^{5} - 350 a^{4} c^{4} x^{4} - 105 a^{3} c^{4} x^{3} + 84 a^{2} c^{4} x^{2} + 70 a c^{4} x + 15 c^{4}}{105 x^{7}}}{a^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12703, size = 216, normalized size = 2.16 \begin{align*} -\frac{4 \, c^{4} \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{4 \, c^{4} \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right )}{a} + \frac{{\left (105 \, c^{4} + \frac{659 \, c^{4}}{a x - 1} + \frac{1253 \, c^{4}}{{\left (a x - 1\right )}^{2}} - \frac{231 \, c^{4}}{{\left (a x - 1\right )}^{3}} - \frac{3885 \, c^{4}}{{\left (a x - 1\right )}^{4}} - \frac{5250 \, 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 )}}{105 \, 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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