Optimal. Leaf size=40 \[ \frac {c^2}{a^2 x}+\frac {4 c^2 \log (x)}{a}-\frac {8 c^2 \log (a x+1)}{a}+c^2 x \]
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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, 88} \[ \frac {c^2}{a^2 x}+\frac {4 c^2 \log (x)}{a}-\frac {8 c^2 \log (a x+1)}{a}+c^2 x \]
Antiderivative was successfully verified.
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Rule 88
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx\\ &=-\frac {c^2 \int \frac {e^{-2 \tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=-\frac {c^2 \int \frac {(1-a x)^3}{x^2 (1+a x)} \, dx}{a^2}\\ &=-\frac {c^2 \int \left (-a^2+\frac {1}{x^2}-\frac {4 a}{x}+\frac {8 a^2}{1+a x}\right ) \, dx}{a^2}\\ &=\frac {c^2}{a^2 x}+c^2 x+\frac {4 c^2 \log (x)}{a}-\frac {8 c^2 \log (1+a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 42, normalized size = 1.05 \[ \frac {c^2}{a^2 x}+\frac {4 c^2 \log (a x)}{a}-\frac {8 c^2 \log (a x+1)}{a}+c^2 x \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.66, size = 43, normalized size = 1.08 \[ \frac {a^{2} c^{2} x^{2} - 8 \, a c^{2} x \log \left (a x + 1\right ) + 4 \, a c^{2} x \log \relax (x) + c^{2}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 42, normalized size = 1.05 \[ c^{2} x - \frac {8 \, c^{2} \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac {4 \, c^{2} \log \left ({\left | x \right |}\right )}{a} + \frac {c^{2}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 41, normalized size = 1.02 \[ \frac {c^{2}}{a^{2} x}+c^{2} x +\frac {4 c^{2} \ln \relax (x )}{a}-\frac {8 c^{2} \ln \left (a x +1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 40, normalized size = 1.00 \[ c^{2} x - \frac {8 \, c^{2} \log \left (a x + 1\right )}{a} + \frac {4 \, c^{2} \log \relax (x)}{a} + \frac {c^{2}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 40, normalized size = 1.00 \[ c^2\,x+\frac {c^2}{a^2\,x}+\frac {4\,c^2\,\ln \relax (x)}{a}-\frac {8\,c^2\,\ln \left (a\,x+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 31, normalized size = 0.78 \[ c^{2} x + \frac {4 c^{2} \left (\log {\relax (x )} - 2 \log {\left (x + \frac {1}{a} \right )}\right )}{a} + \frac {c^{2}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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