Optimal. Leaf size=63 \[ \frac{5 c^3}{3 a^4 x^3}+\frac{c^3}{a^5 x^4}+\frac{c^3}{5 a^6 x^5}-\frac{5 c^3}{a^2 x}+\frac{4 c^3 \log (x)}{a}+c^3 x \]
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Rubi [A] time = 0.151695, antiderivative size = 63, 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, 75} \[ \frac{5 c^3}{3 a^4 x^3}+\frac{c^3}{a^5 x^4}+\frac{c^3}{5 a^6 x^5}-\frac{5 c^3}{a^2 x}+\frac{4 c^3 \log (x)}{a}+c^3 x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6157
Rule 6150
Rule 75
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^3 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^3 \, dx\\ &=-\frac{c^3 \int \frac{e^{4 \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) (1+a x)^5}{x^6} \, dx}{a^6}\\ &=-\frac{c^3 \int \left (-a^6+\frac{1}{x^6}+\frac{4 a}{x^5}+\frac{5 a^2}{x^4}-\frac{5 a^4}{x^2}-\frac{4 a^5}{x}\right ) \, dx}{a^6}\\ &=\frac{c^3}{5 a^6 x^5}+\frac{c^3}{a^5 x^4}+\frac{5 c^3}{3 a^4 x^3}-\frac{5 c^3}{a^2 x}+c^3 x+\frac{4 c^3 \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0231702, size = 63, normalized size = 1. \[ \frac{5 c^3}{3 a^4 x^3}+\frac{c^3}{a^5 x^4}+\frac{c^3}{5 a^6 x^5}-\frac{5 c^3}{a^2 x}+\frac{4 c^3 \log (x)}{a}+c^3 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 60, normalized size = 1. \begin{align*}{\frac{{c}^{3}}{5\,{a}^{6}{x}^{5}}}+{\frac{{c}^{3}}{{a}^{5}{x}^{4}}}+{\frac{5\,{c}^{3}}{3\,{a}^{4}{x}^{3}}}-5\,{\frac{{c}^{3}}{{a}^{2}x}}+{c}^{3}x+4\,{\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 = 1.07938, size = 80, normalized size = 1.27 \begin{align*} c^{3} x + \frac{4 \, c^{3} \log \left (x\right )}{a} - \frac{75 \, a^{4} c^{3} x^{4} - 25 \, a^{2} c^{3} x^{2} - 15 \, a c^{3} x - 3 \, c^{3}}{15 \, a^{6} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57881, size = 151, normalized size = 2.4 \begin{align*} \frac{15 \, a^{6} c^{3} x^{6} + 60 \, a^{5} c^{3} x^{5} \log \left (x\right ) - 75 \, a^{4} c^{3} x^{4} + 25 \, a^{2} c^{3} x^{2} + 15 \, a c^{3} x + 3 \, c^{3}}{15 \, a^{6} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.498356, size = 65, normalized size = 1.03 \begin{align*} \frac{a^{6} c^{3} x + 4 a^{5} c^{3} \log{\left (x \right )} - \frac{75 a^{4} c^{3} x^{4} - 25 a^{2} c^{3} x^{2} - 15 a c^{3} x - 3 c^{3}}{15 x^{5}}}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12317, size = 184, normalized size = 2.92 \begin{align*} -\frac{4 \, c^{3} \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{4 \, c^{3} \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right )}{a} + \frac{{\left (15 \, c^{3} + \frac{107 \, c^{3}}{a x - 1} + \frac{235 \, c^{3}}{{\left (a x - 1\right )}^{2}} + \frac{170 \, c^{3}}{{\left (a x - 1\right )}^{3}} - \frac{30 \, c^{3}}{{\left (a x - 1\right )}^{4}} - \frac{60 \, c^{3}}{{\left (a x - 1\right )}^{5}}\right )}{\left (a x - 1\right )}}{15 \, 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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