Optimal. Leaf size=24 \[ \frac {x}{2}-\log \left (e^x+1\right )+\frac {1}{2} \log \left (e^x+2\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2282, 705, 29, 632, 31} \[ \frac {x}{2}-\log \left (e^x+1\right )+\frac {1}{2} \log \left (e^x+2\right ) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 632
Rule 705
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{2+3 e^x+e^{2 x}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \left (2+3 x+x^2\right )} \, dx,x,e^x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {-3-x}{2+3 x+x^2} \, dx,x,e^x\right )\\ &=\frac {x}{2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{2+x} \, dx,x,e^x\right )-\operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right )\\ &=\frac {x}{2}-\log \left (1+e^x\right )+\frac {1}{2} \log \left (2+e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ \frac {x}{2}-\log \left (e^x+1\right )+\frac {1}{2} \log \left (e^x+2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 18, normalized size = 0.75 \[ \frac {1}{2} \, x + \frac {1}{2} \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 18, normalized size = 0.75 \[ \frac {1}{2} \, x + \frac {1}{2} \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 21, normalized size = 0.88 \[ -\ln \left ({\mathrm e}^{x}+1\right )+\frac {\ln \left ({\mathrm e}^{x}+2\right )}{2}+\frac {\ln \left ({\mathrm e}^{x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 18, normalized size = 0.75 \[ \frac {1}{2} \, x + \frac {1}{2} \, \log \left (e^{x} + 2\right ) - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 18, normalized size = 0.75 \[ \frac {x}{2}-\ln \left ({\mathrm {e}}^x+1\right )+\frac {\ln \left ({\mathrm {e}}^x+2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 17, normalized size = 0.71 \[ \frac {x}{2} - \log {\left (e^{x} + 1 \right )} + \frac {\log {\left (e^{x} + 2 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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