Optimal. Leaf size=24 \[ \frac {x}{b}-\frac {\log \left (b+a e^{n x}\right )}{b n} \]
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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}{b}-\frac {\log \left (a e^{n x}+b\right )}{b n} \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}{b+a e^{n x}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x (b+a x)} \, dx,x,e^{n x}\right )}{n}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{n x}\right )}{b n}-\frac {a \text {Subst}\left (\int \frac {1}{b+a x} \, dx,x,e^{n x}\right )}{b n}\\ &=\frac {x}{b}-\frac {\log \left (b+a e^{n x}\right )}{b n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 38, normalized size = 1.58 \begin {gather*} \frac {\log \left (e^{n x}\right )}{b n}-\frac {\log \left (b^2 n+a b e^{n x} n\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 29, normalized size = 1.21
method | result | size |
norman | \(\frac {x}{b}-\frac {\ln \left (b +a \,{\mathrm e}^{n x}\right )}{b n}\) | \(24\) |
risch | \(\frac {x}{b}-\frac {\ln \left ({\mathrm e}^{n x}+\frac {b}{a}\right )}{b n}\) | \(26\) |
derivativedivides | \(\frac {-\frac {\ln \left (b +a \,{\mathrm e}^{n x}\right )}{b}+\frac {\ln \left ({\mathrm e}^{n x}\right )}{b}}{n}\) | \(29\) |
default | \(\frac {-\frac {\ln \left (b +a \,{\mathrm e}^{n x}\right )}{b}+\frac {\ln \left ({\mathrm e}^{n x}\right )}{b}}{n}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.52, size = 23, normalized size = 0.96 \begin {gather*} \frac {x}{b} - \frac {\log \left (a e^{\left (n x\right )} + b\right )}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 22, normalized size = 0.92 \begin {gather*} \frac {n x - \log \left (a e^{\left (n x\right )} + b\right )}{b n} \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}{b} - \frac {\log {\left (e^{n x} + \frac {b}{a} \right )}}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.39, size = 26, normalized size = 1.08 \begin {gather*} \frac {\frac {n x}{b} - \frac {\log \left ({\left | a e^{\left (n x\right )} + b \right |}\right )}{b}}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 22, normalized size = 0.92 \begin {gather*} -\frac {\ln \left (b+a\,{\mathrm {e}}^{n\,x}\right )-n\,x}{b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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