Optimal. Leaf size=13 \[ \frac {\log (a x+1)}{a c} \]
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Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6129, 31} \[ \frac {\log (a x+1)}{a c} \]
Antiderivative was successfully verified.
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Rule 31
Rule 6129
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{c-a c x} \, dx &=\frac {\int \frac {1}{1+a x} \, dx}{c}\\ &=\frac {\log (1+a x)}{a c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ \frac {\log (a x+1)}{a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 13, normalized size = 1.00 \[ \frac {\log \left (a x + 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.76, size = 27, normalized size = 2.08 \[ -\frac {\log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 14, normalized size = 1.08 \[ \frac {\ln \left (a x +1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 13, normalized size = 1.00 \[ \frac {\log \left (a x + 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 13, normalized size = 1.00 \[ \frac {\ln \left (a\,x+1\right )}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 10, normalized size = 0.77 \[ \frac {\log {\left (a c x + c \right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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