Optimal. Leaf size=13 \[ \frac {5}{5+e^{3+e^x}} \]
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Rubi [A]
time = 0.13, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 2320, 2278,
32} \begin {gather*} \frac {5}{e^{e^x+3}+5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 2278
Rule 2320
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-\left (5 \int \frac {e^{3+e^x+x}}{25+10 e^{3+e^x}+e^{6+2 e^x}} \, dx\right )\\ &=-\left (5 \text {Subst}\left (\int \frac {e^{3+x}}{\left (5+e^{3+x}\right )^2} \, dx,x,e^x\right )\right )\\ &=-\left (5 \text {Subst}\left (\int \frac {1}{(5+x)^2} \, dx,x,e^{3+e^x}\right )\right )\\ &=\frac {5}{5+e^{3+e^x}}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 13, normalized size = 1.00 \begin {gather*} \frac {5}{5+e^{3+e^x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.18, size = 12, normalized size = 0.92
method | result | size |
risch | \(\frac {5}{5+{\mathrm e}^{3+{\mathrm e}^{x}}}\) | \(12\) |
norman | \(-\frac {{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}^{x}}}{5+{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}^{x}}}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 11, normalized size = 0.85 \begin {gather*} \frac {5}{e^{\left (e^{x} + 3\right )} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 17, normalized size = 1.31 \begin {gather*} \frac {5 \, e^{x}}{e^{\left (x + e^{x} + 3\right )} + 5 \, e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 10, normalized size = 0.77 \begin {gather*} \frac {5}{e^{3} e^{e^{x}} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 11, normalized size = 0.85 \begin {gather*} \frac {5}{e^{\left (e^{x} + 3\right )} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 11, normalized size = 0.85 \begin {gather*} \frac {5}{{\mathrm {e}}^{{\mathrm {e}}^x+3}+5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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