Optimal. Leaf size=127 \[ -\frac {c^5}{9 a^{10} x^9}-\frac {c^5}{4 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}+\frac {4 c^5}{3 a^7 x^6}-\frac {2 c^5}{5 a^6 x^5}-\frac {3 c^5}{a^5 x^4}-\frac {2 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}+\frac {3 c^5}{a^2 x}+\frac {2 c^5 \log (x)}{a}+c^5 x \]
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Rubi [A] time = 0.18, antiderivative size = 127, 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} \[ \frac {4 c^5}{a^3 x^2}-\frac {2 c^5}{3 a^4 x^3}-\frac {3 c^5}{a^5 x^4}-\frac {2 c^5}{5 a^6 x^5}+\frac {4 c^5}{3 a^7 x^6}+\frac {3 c^5}{7 a^8 x^7}-\frac {c^5}{4 a^9 x^8}-\frac {c^5}{9 a^{10} x^9}+\frac {3 c^5}{a^2 x}+\frac {2 c^5 \log (x)}{a}+c^5 x \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6157
Rule 6167
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*}
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Mathematica [A] time = 0.05, size = 127, normalized size = 1.00 \[ -\frac {c^5}{9 a^{10} x^9}-\frac {c^5}{4 a^9 x^8}+\frac {3 c^5}{7 a^8 x^7}+\frac {4 c^5}{3 a^7 x^6}-\frac {2 c^5}{5 a^6 x^5}-\frac {3 c^5}{a^5 x^4}-\frac {2 c^5}{3 a^4 x^3}+\frac {4 c^5}{a^3 x^2}+\frac {3 c^5}{a^2 x}+\frac {2 c^5 \log (x)}{a}+c^5 x \]
Antiderivative was successfully verified.
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fricas [A] time = 1.52, size = 122, normalized size = 0.96 \[ \frac {1260 \, a^{10} c^{5} x^{10} + 2520 \, a^{9} c^{5} x^{9} \log \relax (x) + 3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 115, normalized size = 0.91 \[ c^{5} x + \frac {2 \, c^{5} \log \left ({\left | x \right |}\right )}{a} + \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 116, normalized size = 0.91 \[ -\frac {c^{5}}{9 a^{10} x^{9}}-\frac {c^{5}}{4 a^{9} x^{8}}+\frac {3 c^{5}}{7 a^{8} x^{7}}+\frac {4 c^{5}}{3 a^{7} x^{6}}-\frac {2 c^{5}}{5 a^{6} x^{5}}-\frac {3 c^{5}}{a^{5} x^{4}}-\frac {2 c^{5}}{3 a^{4} x^{3}}+\frac {4 c^{5}}{x^{2} a^{3}}+\frac {3 c^{5}}{a^{2} x}+c^{5} x +\frac {2 c^{5} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 114, normalized size = 0.90 \[ c^{5} x + \frac {2 \, c^{5} \log \relax (x)}{a} + \frac {3780 \, a^{8} c^{5} x^{8} + 5040 \, a^{7} c^{5} x^{7} - 840 \, a^{6} c^{5} x^{6} - 3780 \, a^{5} c^{5} x^{5} - 504 \, a^{4} c^{5} x^{4} + 1680 \, a^{3} c^{5} x^{3} + 540 \, a^{2} c^{5} x^{2} - 315 \, a c^{5} x - 140 \, c^{5}}{1260 \, a^{10} x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 89, normalized size = 0.70 \[ \frac {c^5\,\left (\frac {3\,a^2\,x^2}{7}-\frac {a\,x}{4}+\frac {4\,a^3\,x^3}{3}-\frac {2\,a^4\,x^4}{5}-3\,a^5\,x^5-\frac {2\,a^6\,x^6}{3}+4\,a^7\,x^7+3\,a^8\,x^8+a^{10}\,x^{10}+2\,a^9\,x^9\,\ln \relax (x)-\frac {1}{9}\right )}{a^{10}\,x^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 124, normalized size = 0.98 \[ \frac {a^{10} c^{5} x + 2 a^{9} c^{5} \log {\relax (x )} + \frac {3780 a^{8} c^{5} x^{8} + 5040 a^{7} c^{5} x^{7} - 840 a^{6} c^{5} x^{6} - 3780 a^{5} c^{5} x^{5} - 504 a^{4} c^{5} x^{4} + 1680 a^{3} c^{5} x^{3} + 540 a^{2} c^{5} x^{2} - 315 a c^{5} x - 140 c^{5}}{1260 x^{9}}}{a^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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