Optimal. Leaf size=61 \[ -\frac{c^5}{a^3 x^2}+\frac{c^5}{a^4 x^3}-\frac{c^5}{4 a^5 x^4}-\frac{2 c^5}{a^2 x}-\frac{3 c^5 \log (x)}{a}+c^5 x \]
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Rubi [A] time = 0.13671, antiderivative size = 61, 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, 6131, 6129, 75} \[ -\frac{c^5}{a^3 x^2}+\frac{c^5}{a^4 x^3}-\frac{c^5}{4 a^5 x^4}-\frac{2 c^5}{a^2 x}-\frac{3 c^5 \log (x)}{a}+c^5 x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6131
Rule 6129
Rule 75
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^5 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^5 \, dx\\ &=\frac{c^5 \int \frac{e^{2 \tanh ^{-1}(a x)} (1-a x)^5}{x^5} \, dx}{a^5}\\ &=\frac{c^5 \int \frac{(1-a x)^4 (1+a x)}{x^5} \, dx}{a^5}\\ &=\frac{c^5 \int \left (a^5+\frac{1}{x^5}-\frac{3 a}{x^4}+\frac{2 a^2}{x^3}+\frac{2 a^3}{x^2}-\frac{3 a^4}{x}\right ) \, dx}{a^5}\\ &=-\frac{c^5}{4 a^5 x^4}+\frac{c^5}{a^4 x^3}-\frac{c^5}{a^3 x^2}-\frac{2 c^5}{a^2 x}+c^5 x-\frac{3 c^5 \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.219055, size = 63, normalized size = 1.03 \[ -\frac{c^5}{a^3 x^2}+\frac{c^5}{a^4 x^3}-\frac{c^5}{4 a^5 x^4}-\frac{2 c^5}{a^2 x}-\frac{3 c^5 \log (a x)}{a}+c^5 x \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 60, normalized size = 1. \begin{align*} -{\frac{{c}^{5}}{4\,{a}^{5}{x}^{4}}}+{\frac{{c}^{5}}{{a}^{4}{x}^{3}}}-{\frac{{c}^{5}}{{x}^{2}{a}^{3}}}-2\,{\frac{{c}^{5}}{{a}^{2}x}}+{c}^{5}x-3\,{\frac{{c}^{5}\ln \left ( x \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01436, size = 77, normalized size = 1.26 \begin{align*} c^{5} x - \frac{3 \, c^{5} \log \left (x\right )}{a} - \frac{8 \, a^{3} c^{5} x^{3} + 4 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + c^{5}}{4 \, a^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57395, size = 142, normalized size = 2.33 \begin{align*} \frac{4 \, a^{5} c^{5} x^{5} - 12 \, a^{4} c^{5} x^{4} \log \left (x\right ) - 8 \, a^{3} c^{5} x^{3} - 4 \, a^{2} c^{5} x^{2} + 4 \, a c^{5} x - c^{5}}{4 \, a^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.46256, size = 63, normalized size = 1.03 \begin{align*} \frac{a^{5} c^{5} x - 3 a^{4} c^{5} \log{\left (x \right )} - \frac{8 a^{3} c^{5} x^{3} + 4 a^{2} c^{5} x^{2} - 4 a c^{5} x + c^{5}}{4 x^{4}}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11383, size = 78, normalized size = 1.28 \begin{align*} c^{5} x - \frac{3 \, c^{5} \log \left ({\left | x \right |}\right )}{a} - \frac{8 \, a^{3} c^{5} x^{3} + 4 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + c^{5}}{4 \, a^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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