Integrand size = 25, antiderivative size = 25 \[ \int \frac {\sqrt {1+e^{-x}}}{-e^{-x}+e^x} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+e^{-x}}}{\sqrt {2}}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2320, 1460, 1483, 641, 65, 212} \[ \int \frac {\sqrt {1+e^{-x}}}{-e^{-x}+e^x} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {e^{-x}+1}}{\sqrt {2}}\right ) \]
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Rule 65
Rule 212
Rule 641
Rule 1460
Rule 1483
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {1+\frac {1}{x}}}{-1+x^2} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \frac {\sqrt {1+\frac {1}{x}}}{\left (1-\frac {1}{x^2}\right ) x^2} \, dx,x,e^x\right ) \\ & = -\text {Subst}\left (\int \frac {\sqrt {1+x}}{1-x^2} \, dx,x,e^{-x}\right ) \\ & = -\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x}} \, dx,x,e^{-x}\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+e^{-x}}\right )\right ) \\ & = -\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+e^{-x}}}{\sqrt {2}}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(25)=50\).
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.60 \[ \int \frac {\sqrt {1+e^{-x}}}{-e^{-x}+e^x} \, dx=-\frac {\sqrt {2} e^{x/2} \sqrt {1+e^{-x}} \text {arctanh}\left (\frac {1-e^x+e^{x/2} \sqrt {1+e^x}}{\sqrt {2}}\right )}{\sqrt {1+e^x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(19)=38\).
Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96
method | result | size |
default | \(-\frac {\sqrt {\left (1+{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}\, {\mathrm e}^{x} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (1+3 \,{\mathrm e}^{x}\right ) \sqrt {2}}{4 \sqrt {{\mathrm e}^{x}+{\mathrm e}^{2 x}}}\right )}{2 \sqrt {\left (1+{\mathrm e}^{x}\right ) {\mathrm e}^{x}}}\) | \(49\) |
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none
Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {1+e^{-x}}}{-e^{-x}+e^x} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, \sqrt {2} \sqrt {e^{x} + 1} e^{\left (\frac {1}{2} \, x\right )} - 3 \, e^{x} - 1}{e^{x} - 1}\right ) \]
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\[ \int \frac {\sqrt {1+e^{-x}}}{-e^{-x}+e^x} \, dx=\int \frac {\sqrt {1 + e^{- x}} e^{x}}{\left (e^{x} - 1\right ) \left (e^{x} + 1\right )}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {1+e^{-x}}}{-e^{-x}+e^x} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {e^{\left (-x\right )} + 1}}{\sqrt {2} + \sqrt {e^{\left (-x\right )} + 1}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (19) = 38\).
Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {\sqrt {1+e^{-x}}}{-e^{-x}+e^x} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {e^{\left (2 \, x\right )} + e^{x}} - 2 \, e^{x} + 2 \right |}}{{\left | 2 \, \sqrt {2} + 2 \, \sqrt {e^{\left (2 \, x\right )} + e^{x}} - 2 \, e^{x} + 2 \right |}}\right ) \]
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Timed out. \[ \int \frac {\sqrt {1+e^{-x}}}{-e^{-x}+e^x} \, dx=-\int \frac {\sqrt {{\mathrm {e}}^{-x}+1}}{{\mathrm {e}}^{-x}-{\mathrm {e}}^x} \,d x \]
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