Optimal. Leaf size=27 \[ -\frac {c^2}{a^2 x}+\frac {2 c^2 \log (x)}{a}+c^2 x \]
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Rubi [A] time = 0.10, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6131, 6129, 43} \[ -\frac {c^2}{a^2 x}+\frac {2 c^2 \log (x)}{a}+c^2 x \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rule 6131
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx &=\frac {c^2 \int \frac {e^{4 \tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \frac {(1+a x)^2}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \left (a^2+\frac {1}{x^2}+\frac {2 a}{x}\right ) \, dx}{a^2}\\ &=-\frac {c^2}{a^2 x}+c^2 x+\frac {2 c^2 \log (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 29, normalized size = 1.07 \[ -\frac {c^2}{a^2 x}+\frac {2 c^2 \log (a x)}{a}+c^2 x \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.41, size = 32, normalized size = 1.19 \[ \frac {a^{2} c^{2} x^{2} + 2 \, a c^{2} x \log \relax (x) - c^{2}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 28, normalized size = 1.04 \[ c^{2} x + \frac {2 \, c^{2} \log \left ({\left | x \right |}\right )}{a} - \frac {c^{2}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 28, normalized size = 1.04 \[ -\frac {c^{2}}{a^{2} x}+c^{2} x +\frac {2 c^{2} \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 27, normalized size = 1.00 \[ c^{2} x + \frac {2 \, c^{2} \log \relax (x)}{a} - \frac {c^{2}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 25, normalized size = 0.93 \[ \frac {c^2\,\left (a^2\,x^2+2\,a\,x\,\ln \relax (x)-1\right )}{a^2\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 26, normalized size = 0.96 \[ \frac {a^{2} c^{2} x + 2 a c^{2} \log {\relax (x )} - \frac {c^{2}}{x}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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