Optimal. Leaf size=11 \[ \frac {c \log (x)}{a}+c x \]
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Rubi [A] time = 0.07, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6167, 6131, 6129, 43} \[ \frac {c \log (x)}{a}+c x \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx\\ &=\frac {c \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=\frac {c \int \frac {1+a x}{x} \, dx}{a}\\ &=\frac {c \int \left (a+\frac {1}{x}\right ) \, dx}{a}\\ &=c x+\frac {c \log (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 11, normalized size = 1.00 \[ \frac {c \log (x)}{a}+c x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 13, normalized size = 1.18 \[ \frac {a c x + c \log \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 12, normalized size = 1.09 \[ c x + \frac {c \log \left ({\left | x \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 12, normalized size = 1.09 \[ c x +\frac {c \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 11, normalized size = 1.00 \[ c x + \frac {c \log \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 11, normalized size = 1.00 \[ \frac {c\,\left (\ln \relax (x)+a\,x\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 10, normalized size = 0.91 \[ \frac {a c x + c \log {\relax (x )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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