Optimal. Leaf size=22 \[ \frac {e^x}{b}-\frac {a \log \left (a+b e^x\right )}{b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2248, 43} \[ \frac {e^x}{b}-\frac {a \log \left (a+b e^x\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{2 x}}{a+b e^x} \, dx &=\operatorname {Subst}\left (\int \frac {x}{a+b x} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,e^x\right )\\ &=\frac {e^x}{b}-\frac {a \log \left (a+b e^x\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 1.00 \[ \frac {e^x}{b}-\frac {a \log \left (a+b e^x\right )}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 19, normalized size = 0.86 \[ \frac {b e^{x} - a \log \left (b e^{x} + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 21, normalized size = 0.95 \[ \frac {e^{x}}{b} - \frac {a \log \left ({\left | b e^{x} + a \right |}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 0.95 \[ -\frac {a \ln \left (b \,{\mathrm e}^{x}+a \right )}{b^{2}}+\frac {{\mathrm e}^{x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 20, normalized size = 0.91 \[ \frac {e^{x}}{b} - \frac {a \log \left (b e^{x} + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.57, size = 20, normalized size = 0.91 \[ -\frac {a\,\ln \left (a+b\,{\mathrm {e}}^x\right )-b\,{\mathrm {e}}^x}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 20, normalized size = 0.91 \[ - \frac {a \log {\left (\frac {a}{b} + e^{x} \right )}}{b^{2}} + \begin {cases} \frac {e^{x}}{b} & \text {for}\: b \neq 0 \\\frac {x}{b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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