Optimal. Leaf size=24 \[ \frac {x}{a}-\frac {\log \left (a+b e^{m x}\right )}{a m} \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2282, 36, 29, 31} \[ \frac {x}{a}-\frac {\log \left (a+b e^{m x}\right )}{a m} \]
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^{m x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,e^{m x}\right )}{m}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{m x}\right )}{a m}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a+b x} \, dx,x,e^{m x}\right )}{a m}\\ &=\frac {x}{a}-\frac {\log \left (a+b e^{m x}\right )}{a m}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ \frac {x}{a}-\frac {\log \left (a+b e^{m x}\right )}{a m} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 22, normalized size = 0.92 \[ \frac {m x - \log \left (b e^{\left (m x\right )} + a\right )}{a m} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 26, normalized size = 1.08 \[ \frac {\frac {m x}{a} - \frac {\log \left ({\left | b e^{\left (m x\right )} + a \right |}\right )}{a}}{m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 31, normalized size = 1.29 \[ -\frac {\ln \left (b \,{\mathrm e}^{m x}+a \right )}{a m}+\frac {\ln \left ({\mathrm e}^{m x}\right )}{a m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 23, normalized size = 0.96 \[ \frac {x}{a} - \frac {\log \left (b e^{\left (m x\right )} + a\right )}{a m} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 22, normalized size = 0.92 \[ -\frac {\ln \left (a+b\,{\mathrm {e}}^{m\,x}\right )-m\,x}{a\,m} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 15, normalized size = 0.62 \[ \frac {x}{a} - \frac {\log {\left (\frac {a}{b} + e^{m x} \right )}}{a m} \]
Verification of antiderivative is not currently implemented for this CAS.
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