Optimal. Leaf size=26 \[ \frac {x}{a}-\frac {\log \left (a+b e^{c+d x}\right )}{a d} \]
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Rubi [A] time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2282, 36, 29, 31} \[ \frac {x}{a}-\frac {\log \left (a+b e^{c+d x}\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{a+b e^{c+d x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{c+d x}\right )}{a d}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a+b x} \, dx,x,e^{c+d x}\right )}{a d}\\ &=\frac {x}{a}-\frac {\log \left (a+b e^{c+d x}\right )}{a d}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 26, normalized size = 1.00 \[ \frac {x}{a}-\frac {\log \left (a+b e^{c+d x}\right )}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 24, normalized size = 0.92 \[ \frac {d x - \log \left (b e^{\left (d x + c\right )} + a\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 31, normalized size = 1.19 \[ \frac {\frac {d x + c}{a} - \frac {\log \left ({\left | b e^{\left (d x + c\right )} + a \right |}\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 1.35 \[ -\frac {\ln \left (b \,{\mathrm e}^{d x +c}+a \right )}{a d}+\frac {\ln \left ({\mathrm e}^{d x +c}\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 32, normalized size = 1.23 \[ \frac {d x + c}{a d} - \frac {\log \left (b e^{\left (d x + c\right )} + a\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.50, size = 24, normalized size = 0.92 \[ -\frac {\ln \left (a+b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )-d\,x}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 17, normalized size = 0.65 \[ \frac {x}{a} - \frac {\log {\left (\frac {a}{b} + e^{c + d x} \right )}}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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