Optimal. Leaf size=19 \[ \frac {\log \left (a+b e^{c+d x}\right )}{b d} \]
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Rubi [A] time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2246, 31} \[ \frac {\log \left (a+b e^{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2246
Rubi steps
\begin {align*} \int \frac {e^{c+d x}}{a+b e^{c+d x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+b x} \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac {\log \left (a+b e^{c+d x}\right )}{b d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ \frac {\log \left (a+b e^{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 18, normalized size = 0.95 \[ \frac {\log \left (b e^{\left (d x + c\right )} + a\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 19, normalized size = 1.00 \[ \frac {\log \left ({\left | b e^{\left (d x + c\right )} + a \right |}\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 1.00 \[ \frac {\ln \left (b \,{\mathrm e}^{d x +c}+a \right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 18, normalized size = 0.95 \[ \frac {\log \left (b e^{\left (d x + c\right )} + a\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 18, normalized size = 0.95 \[ \frac {\ln \left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}{b\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 14, normalized size = 0.74 \[ \frac {\log {\left (\frac {a}{b} + e^{c + d x} \right )}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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