Optimal. Leaf size=64 \[ \frac {c^5}{4 a^5 x^4}-\frac {c^5}{3 a^4 x^3}-\frac {c^5}{a^3 x^2}+\frac {2 c^5}{a^2 x}-\frac {c^5 \log (x)}{a}+c^5 x \]
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Rubi [A] time = 0.14, antiderivative size = 64, 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, 6131, 6129, 88} \[ -\frac {c^5}{a^3 x^2}-\frac {c^5}{3 a^4 x^3}+\frac {c^5}{4 a^5 x^4}+\frac {2 c^5}{a^2 x}-\frac {c^5 \log (x)}{a}+c^5 x \]
Antiderivative was successfully verified.
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Rule 88
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^5 \, dx\\ &=-\frac {c^5 \int \frac {e^{4 \tanh ^{-1}(a x)} (1-a x)^5}{x^5} \, dx}{a^5}\\ &=-\frac {c^5 \int \frac {(1-a x)^3 (1+a x)^2}{x^5} \, dx}{a^5}\\ &=-\frac {c^5 \int \left (-a^5+\frac {1}{x^5}-\frac {a}{x^4}-\frac {2 a^2}{x^3}+\frac {2 a^3}{x^2}+\frac {a^4}{x}\right ) \, dx}{a^5}\\ &=\frac {c^5}{4 a^5 x^4}-\frac {c^5}{3 a^4 x^3}-\frac {c^5}{a^3 x^2}+\frac {2 c^5}{a^2 x}+c^5 x-\frac {c^5 \log (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 66, normalized size = 1.03 \[ \frac {c^5}{4 a^5 x^4}-\frac {c^5}{3 a^4 x^3}-\frac {c^5}{a^3 x^2}+\frac {2 c^5}{a^2 x}-\frac {c^5 \log (a x)}{a}+c^5 x \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 67, normalized size = 1.05 \[ \frac {12 \, a^{5} c^{5} x^{5} - 12 \, a^{4} c^{5} x^{4} \log \relax (x) + 24 \, a^{3} c^{5} x^{3} - 12 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + 3 \, c^{5}}{12 \, a^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 123, normalized size = 1.92 \[ \frac {c^{5} \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} - \frac {c^{5} \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {{\left (12 \, c^{5} + \frac {37 \, c^{5}}{a x - 1} + \frac {52 \, c^{5}}{{\left (a x - 1\right )}^{2}} + \frac {42 \, c^{5}}{{\left (a x - 1\right )}^{3}} + \frac {12 \, c^{5}}{{\left (a x - 1\right )}^{4}}\right )} {\left (a x - 1\right )}}{12 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 61, normalized size = 0.95 \[ \frac {c^{5}}{4 a^{5} x^{4}}-\frac {c^{5}}{3 a^{4} x^{3}}-\frac {c^{5}}{x^{2} a^{3}}+\frac {2 c^{5}}{a^{2} x}+c^{5} x -\frac {c^{5} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 59, normalized size = 0.92 \[ c^{5} x - \frac {c^{5} \log \relax (x)}{a} + \frac {24 \, a^{3} c^{5} x^{3} - 12 \, a^{2} c^{5} x^{2} - 4 \, a c^{5} x + 3 \, c^{5}}{12 \, a^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 51, normalized size = 0.80 \[ -\frac {c^5\,\left (4\,a\,x+12\,a^2\,x^2-24\,a^3\,x^3-12\,a^5\,x^5+12\,a^4\,x^4\,\ln \relax (x)-3\right )}{12\,a^5\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 63, normalized size = 0.98 \[ \frac {a^{5} c^{5} x - a^{4} c^{5} \log {\relax (x )} + \frac {24 a^{3} c^{5} x^{3} - 12 a^{2} c^{5} x^{2} - 4 a c^{5} x + 3 c^{5}}{12 x^{4}}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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