Optimal. Leaf size=116 \[ \frac {c^5}{9 a^{10} x^9}+\frac {c^5}{2 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}-\frac {14 c^5}{5 a^6 x^5}+\frac {14 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}+\frac {4 c^5 \log (x)}{a}+c^5 x \]
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Rubi [A] time = 0.17, antiderivative size = 116, normalized size of antiderivative = 1.00, 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 88
Rule 6150
Rule 6157
Rule 6167
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*}
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Mathematica [A] time = 0.05, size = 116, normalized size = 1.00 \[ \frac {c^5}{9 a^{10} x^9}+\frac {c^5}{2 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}-\frac {4 c^5}{3 a^7 x^6}-\frac {14 c^5}{5 a^6 x^5}+\frac {14 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}-\frac {3 c^5}{a^2 x}+\frac {4 c^5 \log (x)}{a}+c^5 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 111, normalized size = 0.96 \[ \frac {630 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \relax (x) - 1890 \, a^{8} c^{5} x^{8} + 2520 \, a^{7} c^{5} x^{7} + 2940 \, a^{6} c^{5} x^{6} - 1764 \, a^{4} c^{5} x^{4} - 840 \, a^{3} c^{5} x^{3} + 270 \, a^{2} c^{5} x^{2} + 315 \, a c^{5} x + 70 \, c^{5}}{630 \, a^{10} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 184, normalized size = 1.59 \[ -\frac {4 \, c^{5} \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {4 \, c^{5} \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {{\left (630 \, c^{5} + \frac {4049 \, c^{5}}{a x - 1} + \frac {6201 \, c^{5}}{{\left (a x - 1\right )}^{2}} - \frac {18036 \, c^{5}}{{\left (a x - 1\right )}^{3}} - \frac {89124 \, c^{5}}{{\left (a x - 1\right )}^{4}} - \frac {160146 \, c^{5}}{{\left (a x - 1\right )}^{5}} - \frac {153090 \, c^{5}}{{\left (a x - 1\right )}^{6}} - \frac {80220 \, c^{5}}{{\left (a x - 1\right )}^{7}} - \frac {21420 \, c^{5}}{{\left (a x - 1\right )}^{8}} - \frac {2520 \, c^{5}}{{\left (a x - 1\right )}^{9}}\right )} {\left (a x - 1\right )}}{630 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 105, normalized size = 0.91 \[ \frac {c^{5}}{9 a^{10} x^{9}}+\frac {c^{5}}{2 a^{9} x^{8}}+\frac {3 c^{5}}{7 a^{8} x^{7}}-\frac {4 c^{5}}{3 a^{7} x^{6}}-\frac {14 c^{5}}{5 a^{6} x^{5}}+\frac {14 c^{5}}{3 a^{4} x^{3}}+\frac {4 c^{5}}{x^{2} a^{3}}-\frac {3 c^{5}}{a^{2} x}+c^{5} x +\frac {4 c^{5} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 103, normalized size = 0.89 \[ c^{5} x + \frac {4 \, c^{5} \log \relax (x)}{a} - \frac {1890 \, a^{8} c^{5} x^{8} - 2520 \, a^{7} c^{5} x^{7} - 2940 \, a^{6} c^{5} x^{6} + 1764 \, a^{4} c^{5} x^{4} + 840 \, a^{3} c^{5} x^{3} - 270 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 70 \, c^{5}}{630 \, a^{10} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 81, normalized size = 0.70 \[ \frac {c^5\,\left (\frac {a\,x}{2}+\frac {3\,a^2\,x^2}{7}-\frac {4\,a^3\,x^3}{3}-\frac {14\,a^4\,x^4}{5}+\frac {14\,a^6\,x^6}{3}+4\,a^7\,x^7-3\,a^8\,x^8+a^{10}\,x^{10}+4\,a^9\,x^9\,\ln \relax (x)+\frac {1}{9}\right )}{a^{10}\,x^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 112, normalized size = 0.97 \[ \frac {a^{10} c^{5} x + 4 a^{9} c^{5} \log {\relax (x )} + \frac {- 1890 a^{8} c^{5} x^{8} + 2520 a^{7} c^{5} x^{7} + 2940 a^{6} c^{5} x^{6} - 1764 a^{4} c^{5} x^{4} - 840 a^{3} c^{5} x^{3} + 270 a^{2} c^{5} x^{2} + 315 a c^{5} x + 70 c^{5}}{630 x^{9}}}{a^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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