Optimal. Leaf size=39 \[ -\frac {c^3}{2 a^3 x^2}+\frac {c^3}{a^2 x}-\frac {c^3 \log (x)}{a}+c^3 x \]
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Rubi [A] time = 0.13, antiderivative size = 39, 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} \[ -\frac {c^3}{2 a^3 x^2}+\frac {c^3}{a^2 x}-\frac {c^3 \log (x)}{a}+c^3 x \]
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 )^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx\\ &=\frac {c^3 \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^3}{x^3} \, dx}{a^3}\\ &=\frac {c^3 \int \frac {(1-a x)^2 (1+a x)}{x^3} \, dx}{a^3}\\ &=\frac {c^3 \int \left (a^3+\frac {1}{x^3}-\frac {a}{x^2}-\frac {a^2}{x}\right ) \, dx}{a^3}\\ &=-\frac {c^3}{2 a^3 x^2}+\frac {c^3}{a^2 x}+c^3 x-\frac {c^3 \log (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 41, normalized size = 1.05 \[ -\frac {c^3}{2 a^3 x^2}+\frac {c^3}{a^2 x}-\frac {c^3 \log (a x)}{a}+c^3 x \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.73, size = 45, normalized size = 1.15 \[ \frac {2 \, a^{3} c^{3} x^{3} - 2 \, a^{2} c^{3} x^{2} \log \relax (x) + 2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 38, normalized size = 0.97 \[ c^{3} x - \frac {c^{3} \log \left ({\left | x \right |}\right )}{a} + \frac {2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 38, normalized size = 0.97 \[ -\frac {c^{3}}{2 x^{2} a^{3}}+\frac {c^{3}}{a^{2} x}+c^{3} x -\frac {c^{3} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 37, normalized size = 0.95 \[ c^{3} x - \frac {c^{3} \log \relax (x)}{a} + \frac {2 \, a c^{3} x - c^{3}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 35, normalized size = 0.90 \[ \frac {c^3\,\left (2\,a\,x+2\,a^3\,x^3-2\,a^2\,x^2\,\ln \relax (x)-1\right )}{2\,a^3\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 37, normalized size = 0.95 \[ \frac {a^{3} c^{3} x - a^{2} c^{3} \log {\relax (x )} + \frac {2 a c^{3} x - c^{3}}{2 x^{2}}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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