Optimal. Leaf size=25 \[ -2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+e^{-x}}}{\sqrt {2}}\right ) \]
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Rubi [A]
time = 0.04, 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 = {2320, 12, 1460,
1483, 641, 65, 212} \begin {gather*} -2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {e^{-x}+1}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 212
Rule 641
Rule 1460
Rule 1483
Rule 2320
Rubi steps
\begin {align*} \int \sqrt {1+e^{-x}} \text {csch}(x) \, dx &=\text {Subst}\left (\int \frac {2 \sqrt {1+\frac {1}{x}}}{-1+x^2} \, dx,x,e^x\right )\\ &=2 \text {Subst}\left (\int \frac {\sqrt {1+\frac {1}{x}}}{-1+x^2} \, dx,x,e^x\right )\\ &=2 \text {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 \text {Subst}\left (\int \frac {\sqrt {1+x}}{1-x^2} \, dx,x,e^{-x}\right )\right )\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x}} \, dx,x,e^{-x}\right )\right )\\ &=-\left (4 \text {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] Leaf count is larger than twice the leaf count of optimal. \(126\) vs. \(2(25)=50\).
time = 0.09, size = 126, normalized size = 5.04 \begin {gather*} \frac {\sqrt {2} e^{x/2} \sqrt {1+e^{-x}} \left (\log \left (1-e^{-x/2}\right )+\log \left (1+e^{-x/2}\right )-\log \left (e^{-x/2} \left (-1+e^{x/2}+\sqrt {2} \sqrt {1+e^x}\right )\right )-\log \left (e^{-x/2} \left (1+e^{x/2}+\sqrt {2} \sqrt {1+e^x}\right )\right )\right )}{\sqrt {1+e^x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 33, normalized size = 1.32
method | result | size |
default | \(-2 \sqrt {2}\, \sqrt {\frac {1}{\tanh \left (\frac {x}{2}\right )+1}}\, \sqrt {\tanh \left (\frac {x}{2}\right )+1}\, \arctanh \left (\sqrt {\tanh \left (\frac {x}{2}\right )+1}\right )\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.64, size = 35, normalized size = 1.40 \begin {gather*} \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {e^{\left (-x\right )} + 1}}{\sqrt {2} + \sqrt {e^{\left (-x\right )} + 1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs.
\(2 (19) = 38\).
time = 1.06, size = 55, normalized size = 2.20 \begin {gather*} \sqrt {2} \log \left (\frac {2 \, {\left (\sqrt {2} \cosh \left (x\right ) + \sqrt {2} \sinh \left (x\right )\right )} \sqrt {\frac {\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right )}} - 3 \, \cosh \left (x\right ) - 3 \, \sinh \left (x\right ) - 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {1 + e^{- x}}}{\sinh {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (19) = 38\).
time = 0.56, size = 54, normalized size = 2.16 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {{\mathrm {e}}^{-x}+1}}{\mathrm {sinh}\left (x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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