Optimal. Leaf size=90 \[ \frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+\frac {2 c^4 \log (x)}{a}+c^4 x \]
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Rubi [A] time = 0.16, antiderivative size = 90, 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} \[ \frac {3 c^4}{a^3 x^2}-\frac {3 c^4}{2 a^5 x^4}-\frac {2 c^4}{5 a^6 x^5}+\frac {c^4}{3 a^7 x^6}+\frac {c^4}{7 a^8 x^7}+\frac {2 c^4}{a^2 x}+\frac {2 c^4 \log (x)}{a}+c^4 x \]
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 )^4 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx\\ &=-\frac {c^4 \int \frac {e^{2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8}\\ &=-\frac {c^4 \int \frac {(1-a x)^3 (1+a x)^5}{x^8} \, dx}{a^8}\\ &=-\frac {c^4 \int \left (-a^8+\frac {1}{x^8}+\frac {2 a}{x^7}-\frac {2 a^2}{x^6}-\frac {6 a^3}{x^5}+\frac {6 a^5}{x^3}+\frac {2 a^6}{x^2}-\frac {2 a^7}{x}\right ) \, dx}{a^8}\\ &=\frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+c^4 x+\frac {2 c^4 \log (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 90, normalized size = 1.00 \[ \frac {c^4}{7 a^8 x^7}+\frac {c^4}{3 a^7 x^6}-\frac {2 c^4}{5 a^6 x^5}-\frac {3 c^4}{2 a^5 x^4}+\frac {3 c^4}{a^3 x^2}+\frac {2 c^4}{a^2 x}+\frac {2 c^4 \log (x)}{a}+c^4 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 89, normalized size = 0.99 \[ \frac {210 \, a^{8} c^{4} x^{8} + 420 \, a^{7} c^{4} x^{7} \log \relax (x) + 420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 82, normalized size = 0.91 \[ c^{4} x + \frac {2 \, c^{4} \log \left ({\left | x \right |}\right )}{a} + \frac {420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 83, normalized size = 0.92 \[ \frac {c^{4}}{7 a^{8} x^{7}}+\frac {c^{4}}{3 a^{7} x^{6}}-\frac {2 c^{4}}{5 a^{6} x^{5}}-\frac {3 c^{4}}{2 a^{5} x^{4}}+\frac {3 c^{4}}{x^{2} a^{3}}+\frac {2 c^{4}}{a^{2} x}+c^{4} x +\frac {2 c^{4} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 81, normalized size = 0.90 \[ c^{4} x + \frac {2 \, c^{4} \log \relax (x)}{a} + \frac {420 \, a^{6} c^{4} x^{6} + 630 \, a^{5} c^{4} x^{5} - 315 \, a^{3} c^{4} x^{3} - 84 \, a^{2} c^{4} x^{2} + 70 \, a c^{4} x + 30 \, c^{4}}{210 \, a^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 65, normalized size = 0.72 \[ \frac {c^4\,\left (\frac {a\,x}{3}-\frac {2\,a^2\,x^2}{5}-\frac {3\,a^3\,x^3}{2}+3\,a^5\,x^5+2\,a^6\,x^6+a^8\,x^8+2\,a^7\,x^7\,\ln \relax (x)+\frac {1}{7}\right )}{a^8\,x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 88, normalized size = 0.98 \[ \frac {a^{8} c^{4} x + 2 a^{7} c^{4} \log {\relax (x )} + \frac {420 a^{6} c^{4} x^{6} + 630 a^{5} c^{4} x^{5} - 315 a^{3} c^{4} x^{3} - 84 a^{2} c^{4} x^{2} + 70 a c^{4} x + 30 c^{4}}{210 x^{7}}}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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