Optimal. Leaf size=14 \[ -\frac {\log (1+a x)}{a c} \]
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Rubi [A]
time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6302, 6264, 31}
\begin {gather*} -\frac {\log (a x+1)}{a c} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 6264
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{c-a c x} \, dx &=-\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 = 14, normalized size = 1.00 \begin {gather*} -\frac {\log (1+a x)}{a c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 15, normalized size = 1.07
method | result | size |
default | \(-\frac {\ln \left (a x +1\right )}{a c}\) | \(15\) |
norman | \(-\frac {\ln \left (a x +1\right )}{a c}\) | \(15\) |
risch | \(-\frac {\ln \left (a x +1\right )}{a c}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 14, normalized size = 1.00 \begin {gather*} -\frac {\log \left (a x + 1\right )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 14, normalized size = 1.00 \begin {gather*} -\frac {\log \left (a x + 1\right )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 12, normalized size = 0.86 \begin {gather*} - \frac {\log {\left (a c x + c \right )}}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 15, normalized size = 1.07 \begin {gather*} -\frac {\log \left ({\left | a x + 1 \right |}\right )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 14, normalized size = 1.00 \begin {gather*} -\frac {\ln \left (a\,x+1\right )}{a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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