Optimal. Leaf size=116 \[ \frac{4 c^5}{a^3 x^2}+\frac{14 c^5}{3 a^4 x^3}-\frac{14 c^5}{5 a^6 x^5}-\frac{4 c^5}{3 a^7 x^6}+\frac{3 c^5}{7 a^8 x^7}+\frac{c^5}{2 a^9 x^8}+\frac{c^5}{9 a^{10} x^9}-\frac{3 c^5}{a^2 x}+\frac{4 c^5 \log (x)}{a}+c^5 x \]
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Rubi [A] time = 0.170696, antiderivative size = 116, 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} \[ \frac{4 c^5}{a^3 x^2}+\frac{14 c^5}{3 a^4 x^3}-\frac{14 c^5}{5 a^6 x^5}-\frac{4 c^5}{3 a^7 x^6}+\frac{3 c^5}{7 a^8 x^7}+\frac{c^5}{2 a^9 x^8}+\frac{c^5}{9 a^{10} x^9}-\frac{3 c^5}{a^2 x}+\frac{4 c^5 \log (x)}{a}+c^5 x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6157
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^5 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^5 \, dx\\ &=-\frac{c^5 \int \frac{e^{4 \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)^3 (1+a x)^7}{x^{10}} \, dx}{a^{10}}\\ &=-\frac{c^5 \int \left (-a^{10}+\frac{1}{x^{10}}+\frac{4 a}{x^9}+\frac{3 a^2}{x^8}-\frac{8 a^3}{x^7}-\frac{14 a^4}{x^6}+\frac{14 a^6}{x^4}+\frac{8 a^7}{x^3}-\frac{3 a^8}{x^2}-\frac{4 a^9}{x}\right ) \, dx}{a^{10}}\\ &=\frac{c^5}{9 a^{10} x^9}+\frac{c^5}{2 a^9 x^8}+\frac{3 c^5}{7 a^8 x^7}-\frac{4 c^5}{3 a^7 x^6}-\frac{14 c^5}{5 a^6 x^5}+\frac{14 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{4 c^5 \log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.041808, size = 116, normalized size = 1. \[ \frac{4 c^5}{a^3 x^2}+\frac{14 c^5}{3 a^4 x^3}-\frac{14 c^5}{5 a^6 x^5}-\frac{4 c^5}{3 a^7 x^6}+\frac{3 c^5}{7 a^8 x^7}+\frac{c^5}{2 a^9 x^8}+\frac{c^5}{9 a^{10} x^9}-\frac{3 c^5}{a^2 x}+\frac{4 c^5 \log (x)}{a}+c^5 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 105, normalized size = 0.9 \begin{align*}{\frac{{c}^{5}}{9\,{a}^{10}{x}^{9}}}+{\frac{{c}^{5}}{2\,{a}^{9}{x}^{8}}}+{\frac{3\,{c}^{5}}{7\,{a}^{8}{x}^{7}}}-{\frac{4\,{c}^{5}}{3\,{a}^{7}{x}^{6}}}-{\frac{14\,{c}^{5}}{5\,{a}^{6}{x}^{5}}}+{\frac{14\,{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+4\,{\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 = 1.00432, size = 139, normalized size = 1.2 \begin{align*} c^{5} x + \frac{4 \, c^{5} \log \left (x\right )}{a} - \frac{1890 \, a^{8} c^{5} x^{8} - 2520 \, a^{7} c^{5} x^{7} - 2940 \, a^{6} c^{5} x^{6} + 1764 \, a^{4} c^{5} x^{4} + 840 \, a^{3} c^{5} x^{3} - 270 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 70 \, c^{5}}{630 \, a^{10} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54395, size = 269, normalized size = 2.32 \begin{align*} \frac{630 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \left (x\right ) - 1890 \, a^{8} c^{5} x^{8} + 2520 \, a^{7} c^{5} x^{7} + 2940 \, a^{6} c^{5} x^{6} - 1764 \, a^{4} c^{5} x^{4} - 840 \, a^{3} c^{5} x^{3} + 270 \, a^{2} c^{5} x^{2} + 315 \, a c^{5} x + 70 \, c^{5}}{630 \, a^{10} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.900921, size = 112, normalized size = 0.97 \begin{align*} \frac{a^{10} c^{5} x + 4 a^{9} c^{5} \log{\left (x \right )} - \frac{1890 a^{8} c^{5} x^{8} - 2520 a^{7} c^{5} x^{7} - 2940 a^{6} c^{5} x^{6} + 1764 a^{4} c^{5} x^{4} + 840 a^{3} c^{5} x^{3} - 270 a^{2} c^{5} x^{2} - 315 a c^{5} x - 70 c^{5}}{630 x^{9}}}{a^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12725, size = 248, normalized size = 2.14 \begin{align*} -\frac{4 \, c^{5} \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} + \frac{4 \, c^{5} \log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right )}{a} + \frac{{\left (630 \, c^{5} + \frac{4049 \, c^{5}}{a x - 1} + \frac{6201 \, c^{5}}{{\left (a x - 1\right )}^{2}} - \frac{18036 \, c^{5}}{{\left (a x - 1\right )}^{3}} - \frac{89124 \, c^{5}}{{\left (a x - 1\right )}^{4}} - \frac{160146 \, c^{5}}{{\left (a x - 1\right )}^{5}} - \frac{153090 \, c^{5}}{{\left (a x - 1\right )}^{6}} - \frac{80220 \, c^{5}}{{\left (a x - 1\right )}^{7}} - \frac{21420 \, c^{5}}{{\left (a x - 1\right )}^{8}} - \frac{2520 \, c^{5}}{{\left (a x - 1\right )}^{9}}\right )}{\left (a x - 1\right )}}{630 \, a{\left (\frac{1}{a x - 1} + 1\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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