Optimal. Leaf size=40 \[ \frac {c^4}{3 a^4 x^3}-\frac {c^4}{a^3 x^2}-\frac {2 c^4 \log (x)}{a}+c^4 x \]
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Rubi [A] time = 0.13, antiderivative size = 40, 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, 75} \[ -\frac {c^4}{a^3 x^2}+\frac {c^4}{3 a^4 x^3}-\frac {2 c^4 \log (x)}{a}+c^4 x \]
Antiderivative was successfully verified.
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Rule 75
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx\\ &=-\frac {c^4 \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^4}{x^4} \, dx}{a^4}\\ &=-\frac {c^4 \int \frac {(1-a x)^3 (1+a x)}{x^4} \, dx}{a^4}\\ &=-\frac {c^4 \int \left (-a^4+\frac {1}{x^4}-\frac {2 a}{x^3}+\frac {2 a^3}{x}\right ) \, dx}{a^4}\\ &=\frac {c^4}{3 a^4 x^3}-\frac {c^4}{a^3 x^2}+c^4 x-\frac {2 c^4 \log (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 42, normalized size = 1.05 \[ \frac {c^4}{3 a^4 x^3}-\frac {c^4}{a^3 x^2}-\frac {2 c^4 \log (a x)}{a}+c^4 x \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.62, size = 43, normalized size = 1.08 \[ \frac {3 \, a^{4} c^{4} x^{4} - 6 \, a^{3} c^{4} x^{3} \log \relax (x) - 3 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 38, normalized size = 0.95 \[ c^{4} x - \frac {2 \, c^{4} \log \left ({\left | x \right |}\right )}{a} - \frac {3 \, a c^{4} x - c^{4}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 39, normalized size = 0.98 \[ \frac {c^{4}}{3 a^{4} x^{3}}-\frac {c^{4}}{x^{2} a^{3}}+c^{4} x -\frac {2 c^{4} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 37, normalized size = 0.92 \[ c^{4} x - \frac {2 \, c^{4} \log \relax (x)}{a} - \frac {3 \, a c^{4} x - c^{4}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 35, normalized size = 0.88 \[ -\frac {c^4\,\left (3\,a\,x-3\,a^4\,x^4+6\,a^3\,x^3\,\ln \relax (x)-1\right )}{3\,a^4\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 39, normalized size = 0.98 \[ \frac {a^{4} c^{4} x - 2 a^{3} c^{4} \log {\relax (x )} + \frac {- 3 a c^{4} x + c^{4}}{3 x^{3}}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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