Optimal. Leaf size=127 \[ \frac{4 c^5}{a^3 x^2}-\frac{2 c^5}{3 a^4 x^3}-\frac{3 c^5}{a^5 x^4}-\frac{2 c^5}{5 a^6 x^5}+\frac{4 c^5}{3 a^7 x^6}+\frac{3 c^5}{7 a^8 x^7}-\frac{c^5}{4 a^9 x^8}-\frac{c^5}{9 a^{10} x^9}+\frac{3 c^5}{a^2 x}+\frac{2 c^5 \log (x)}{a}+c^5 x \]
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Rubi [A] time = 0.175185, antiderivative size = 127, 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} \[ \frac{4 c^5}{a^3 x^2}-\frac{2 c^5}{3 a^4 x^3}-\frac{3 c^5}{a^5 x^4}-\frac{2 c^5}{5 a^6 x^5}+\frac{4 c^5}{3 a^7 x^6}+\frac{3 c^5}{7 a^8 x^7}-\frac{c^5}{4 a^9 x^8}-\frac{c^5}{9 a^{10} x^9}+\frac{3 c^5}{a^2 x}+\frac{2 c^5 \log (x)}{a}+c^5 x \]
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 )^5 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^5 \, dx\\ &=\frac{c^5 \int \frac{e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^5}{x^{10}} \, dx}{a^{10}}\\ &=\frac{c^5 \int \frac{(1-a x)^4 (1+a x)^6}{x^{10}} \, dx}{a^{10}}\\ &=\frac{c^5 \int \left (a^{10}+\frac{1}{x^{10}}+\frac{2 a}{x^9}-\frac{3 a^2}{x^8}-\frac{8 a^3}{x^7}+\frac{2 a^4}{x^6}+\frac{12 a^5}{x^5}+\frac{2 a^6}{x^4}-\frac{8 a^7}{x^3}-\frac{3 a^8}{x^2}+\frac{2 a^9}{x}\right ) \, dx}{a^{10}}\\ &=-\frac{c^5}{9 a^{10} x^9}-\frac{c^5}{4 a^9 x^8}+\frac{3 c^5}{7 a^8 x^7}+\frac{4 c^5}{3 a^7 x^6}-\frac{2 c^5}{5 a^6 x^5}-\frac{3 c^5}{a^5 x^4}-\frac{2 c^5}{3 a^4 x^3}+\frac{4 c^5}{a^3 x^2}+\frac{3 c^5}{a^2 x}+c^5 x+\frac{2 c^5 \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0429798, size = 127, normalized size = 1. \[ \frac{4 c^5}{a^3 x^2}-\frac{2 c^5}{3 a^4 x^3}-\frac{3 c^5}{a^5 x^4}-\frac{2 c^5}{5 a^6 x^5}+\frac{4 c^5}{3 a^7 x^6}+\frac{3 c^5}{7 a^8 x^7}-\frac{c^5}{4 a^9 x^8}-\frac{c^5}{9 a^{10} x^9}+\frac{3 c^5}{a^2 x}+\frac{2 c^5 \log (x)}{a}+c^5 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 116, normalized size = 0.9 \begin{align*} -{\frac{{c}^{5}}{9\,{a}^{10}{x}^{9}}}-{\frac{{c}^{5}}{4\,{a}^{9}{x}^{8}}}+{\frac{3\,{c}^{5}}{7\,{a}^{8}{x}^{7}}}+{\frac{4\,{c}^{5}}{3\,{a}^{7}{x}^{6}}}-{\frac{2\,{c}^{5}}{5\,{a}^{6}{x}^{5}}}-3\,{\frac{{c}^{5}}{{a}^{5}{x}^{4}}}-{\frac{2\,{c}^{5}}{3\,{a}^{4}{x}^{3}}}+4\,{\frac{{c}^{5}}{{x}^{2}{a}^{3}}}+3\,{\frac{{c}^{5}}{{a}^{2}x}}+{c}^{5}x+2\,{\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 = 1.0479, size = 154, normalized size = 1.21 \begin{align*} c^{5} x + \frac{2 \, c^{5} \log \left (x\right )}{a} + \frac{3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20419, size = 297, normalized size = 2.34 \begin{align*} \frac{1260 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \left (x\right ) + 3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.900577, size = 124, normalized size = 0.98 \begin{align*} \frac{a^{10} c^{5} x + 2 a^{9} c^{5} \log{\left (x \right )} + \frac{3780 a^{8} c^{5} x^{8} + 5040 a^{7} c^{5} x^{7} - 840 a^{6} c^{5} x^{6} - 3780 a^{5} c^{5} x^{5} - 504 a^{4} c^{5} x^{4} + 1680 a^{3} c^{5} x^{3} + 540 a^{2} c^{5} x^{2} - 315 a c^{5} x - 140 c^{5}}{1260 x^{9}}}{a^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1442, size = 155, normalized size = 1.22 \begin{align*} c^{5} x + \frac{2 \, c^{5} \log \left ({\left | x \right |}\right )}{a} + \frac{3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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