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 = 0.01, 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 = {2320, 36, 29,
31} \begin {gather*} \frac {x}{a}-\frac {\log \left (a+b e^{m x}\right )}{a m} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2320
Rubi steps
\begin {align*} \int \frac {1}{a+b e^{m x}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,e^{m x}\right )}{m}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{m x}\right )}{a m}-\frac {b \text {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 = 0.02, size = 38, normalized size = 1.58 \begin {gather*} \frac {\log \left (e^{m x}\right )}{a m}-\frac {\log \left (a^2 m+a b e^{m x} m\right )}{a m} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 29, normalized size = 1.21
method | result | size |
norman | \(\frac {x}{a}-\frac {\ln \left (a +b \,{\mathrm e}^{m x}\right )}{a m}\) | \(24\) |
risch | \(\frac {x}{a}-\frac {\ln \left ({\mathrm e}^{m x}+\frac {a}{b}\right )}{a m}\) | \(26\) |
derivativedivides | \(\frac {-\frac {\ln \left (a +b \,{\mathrm e}^{m x}\right )}{a}+\frac {\ln \left ({\mathrm e}^{m x}\right )}{a}}{m}\) | \(29\) |
default | \(\frac {-\frac {\ln \left (a +b \,{\mathrm e}^{m x}\right )}{a}+\frac {\ln \left ({\mathrm e}^{m x}\right )}{a}}{m}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.64, size = 23, normalized size = 0.96 \begin {gather*} \frac {x}{a} - \frac {\log \left (b e^{\left (m x\right )} + a\right )}{a m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.62, size = 22, normalized size = 0.92 \begin {gather*} \frac {m x - \log \left (b e^{\left (m x\right )} + a\right )}{a m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 15, normalized size = 0.62 \begin {gather*} \frac {x}{a} - \frac {\log {\left (\frac {a}{b} + e^{m x} \right )}}{a m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.92, size = 26, normalized size = 1.08 \begin {gather*} \frac {\frac {m x}{a} - \frac {\log \left ({\left | b e^{\left (m x\right )} + a \right |}\right )}{a}}{m} \end {gather*}
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 \begin {gather*} -\frac {\ln \left (a+b\,{\mathrm {e}}^{m\,x}\right )-m\,x}{a\,m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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