Optimal. Leaf size=18 \[ -e^{-x} \sqrt {e^{2 x}+1} \]
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Rubi [A] time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2249, 191} \[ -e^{-x} \sqrt {e^{2 x}+1} \]
Antiderivative was successfully verified.
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Rule 191
Rule 2249
Rubi steps
\begin {align*} \int \frac {e^{-x}}{\sqrt {1+e^{2 x}}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {1}{x^2}}} \, dx,x,e^{-x}\right )\\ &=-e^{-x} \sqrt {1+e^{2 x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ -e^{-x} \sqrt {e^{2 x}+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 10, normalized size = 0.56 \[ -\sqrt {e^{\left (-2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 21, normalized size = 1.17 \[ \frac {2}{{\left (\sqrt {e^{\left (2 \, x\right )} + 1} - e^{x}\right )}^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 15, normalized size = 0.83 \[ -\sqrt {{\mathrm e}^{2 x}+1}\, {\mathrm e}^{-x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 14, normalized size = 0.78 \[ -\sqrt {e^{\left (2 \, x\right )} + 1} e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {{\mathrm {e}}^{-x}}{\sqrt {{\mathrm {e}}^{2\,x}+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{- x}}{\sqrt {e^{2 x} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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