Optimal. Leaf size=20 \[ \frac {\log (a x+1)}{a c}-\frac {x}{c} \]
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Rubi [A] time = 0.09, antiderivative size = 20, 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} \[ \frac {\log (a x+1)}{a c}-\frac {x}{c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=-\frac {a \int \frac {e^{-2 \tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=-\frac {a \int \frac {x}{1+a x} \, dx}{c}\\ &=-\frac {a \int \left (\frac {1}{a}-\frac {1}{a (1+a x)}\right ) \, dx}{c}\\ &=-\frac {x}{c}+\frac {\log (1+a x)}{a c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 18, normalized size = 0.90 \[ \frac {\log (a x+1)-a x}{a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 20, normalized size = 1.00 \[ -\frac {a x - \log \left (a x + 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.77, size = 41, normalized size = 2.05 \[ -\frac {a x + 1}{a c} - \frac {\log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 21, normalized size = 1.05 \[ -\frac {x}{c}+\frac {\ln \left (a x +1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 20, normalized size = 1.00 \[ -\frac {x}{c} + \frac {\log \left (a x + 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 18, normalized size = 0.90 \[ \frac {\ln \left (a\,x+1\right )-a\,x}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 19, normalized size = 0.95 \[ - a \left (\frac {x}{a c} - \frac {\log {\left (a x + 1 \right )}}{a^{2} c}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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