Optimal. Leaf size=25 \[ -2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {e^{-x}+1}}{\sqrt {2}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2282, 12, 1446, 1469, 627, 63, 206} \[ -2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {e^{-x}+1}}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 206
Rule 627
Rule 1446
Rule 1469
Rule 2282
Rubi steps
\begin {align*} \int \sqrt {1+e^{-x}} \text {csch}(x) \, dx &=\operatorname {Subst}\left (\int \frac {2 \sqrt {1+\frac {1}{x}}}{-1+x^2} \, dx,x,e^x\right )\\ &=2 \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {1}{x}}}{-1+x^2} \, dx,x,e^x\right )\\ &=2 \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {1}{x}}}{\left (1-\frac {1}{x^2}\right ) x^2} \, dx,x,e^x\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{1-x^2} \, dx,x,e^{-x}\right )\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x}} \, dx,x,e^{-x}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+e^{-x}}\right )\right )\\ &=-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+e^{-x}}}{\sqrt {2}}\right )\\ \end {align*}
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Mathematica [B] time = 0.12, size = 126, normalized size = 5.04 \[ \frac {\sqrt {2} e^{x/2} \sqrt {e^{-x}+1} \left (\log \left (1-e^{-x/2}\right )+\log \left (e^{-x/2}+1\right )-\log \left (e^{-x/2} \left (\sqrt {2} \sqrt {e^x+1}+e^{x/2}-1\right )\right )-\log \left (e^{-x/2} \left (\sqrt {2} \sqrt {e^x+1}+e^{x/2}+1\right )\right )\right )}{\sqrt {e^x+1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 55, normalized size = 2.20 \[ \sqrt {2} \log \left (\frac {2 \, {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\frac {\cosh \relax (x) + \sinh \relax (x) + 1}{\cosh \relax (x) + \sinh \relax (x)}} - 3 \, \cosh \relax (x) - 3 \, \sinh \relax (x) - 1}{\cosh \relax (x) + \sinh \relax (x) - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 74, normalized size = 2.96 \[ -\sqrt {2} \log \left (\frac {\sqrt {2} - 1}{\sqrt {2} + 1}\right ) + \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {e^{\left (2 \, x\right )} + e^{x}} - 2 \, e^{x} + 2 \right |}}{{\left | 2 \, \sqrt {2} + 2 \, \sqrt {e^{\left (2 \, x\right )} + e^{x}} - 2 \, e^{x} + 2 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 33, normalized size = 1.32 \[ -2 \sqrt {2}\, \sqrt {\frac {1}{\tanh \left (\frac {x}{2}\right )+1}}\, \sqrt {\tanh \left (\frac {x}{2}\right )+1}\, \arctanh \left (\sqrt {\tanh \left (\frac {x}{2}\right )+1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 35, normalized size = 1.40 \[ \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {e^{\left (-x\right )} + 1}}{\sqrt {2} + \sqrt {e^{\left (-x\right )} + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {{\mathrm {e}}^{-x}+1}}{\mathrm {sinh}\relax (x)} \,d x \]
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 \[ \int \frac {\sqrt {1 + e^{- x}}}{\sinh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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