Optimal. Leaf size=27 \[ \frac {a}{b^2 \left (a+b e^x\right )}+\frac {\log \left (a+b e^x\right )}{b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 27, 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} \[ \frac {a}{b^2 \left (a+b e^x\right )}+\frac {\log \left (a+b e^x\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{2 x}}{\left (a+b e^x\right )^2} \, dx &=\operatorname {Subst}\left (\int \frac {x}{(a+b x)^2} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx,x,e^x\right )\\ &=\frac {a}{b^2 \left (a+b e^x\right )}+\frac {\log \left (a+b e^x\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 24, normalized size = 0.89 \[ \frac {\frac {a}{a+b e^x}+\log \left (a+b e^x\right )}{b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 31, normalized size = 1.15 \[ \frac {{\left (b e^{x} + a\right )} \log \left (b e^{x} + a\right ) + a}{b^{3} e^{x} + a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 26, normalized size = 0.96 \[ \frac {\log \left ({\left | b e^{x} + a \right |}\right )}{b^{2}} + \frac {a}{{\left (b e^{x} + a\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 26, normalized size = 0.96 \[ \frac {a}{\left (b \,{\mathrm e}^{x}+a \right ) b^{2}}+\frac {\ln \left (b \,{\mathrm e}^{x}+a \right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 28, normalized size = 1.04 \[ \frac {a}{b^{3} e^{x} + a b^{2}} + \frac {\log \left (b e^{x} + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.61, size = 27, normalized size = 1.00 \[ \frac {\ln \left (a+b\,{\mathrm {e}}^x\right )}{b^2}-\frac {{\mathrm {e}}^x}{b\,\left (a+b\,{\mathrm {e}}^x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 24, normalized size = 0.89 \[ \frac {a}{a b^{2} + b^{3} e^{x}} + \frac {\log {\left (\frac {a}{b} + e^{x} \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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