Optimal. Leaf size=21 \[ \tanh ^{-1}\left (\sqrt {\coth (x)+1}\right )+\tanh (x) \sqrt {\coth (x)+1} \]
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Rubi [A] time = 0.05, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3516, 47, 63, 207} \[ \tanh ^{-1}\left (\sqrt {\coth (x)+1}\right )+\tanh (x) \sqrt {\coth (x)+1} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 207
Rule 3516
Rubi steps
\begin {align*} \int \sqrt {1+\coth (x)} \text {sech}^2(x) \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^2} \, dx,x,\coth (x)\right )\\ &=\sqrt {1+\coth (x)} \tanh (x)-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\coth (x)\right )\\ &=\sqrt {1+\coth (x)} \tanh (x)-\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\coth (x)}\right )\\ &=\tanh ^{-1}\left (\sqrt {1+\coth (x)}\right )+\sqrt {1+\coth (x)} \tanh (x)\\ \end {align*}
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Mathematica [C] time = 5.27, size = 160, normalized size = 7.62 \[ \frac {1}{2} \sqrt {\coth (x)+1} \left (2 \tanh (x)+\frac {(1-i) \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (\coth (x)+1)}\right )}{\sqrt {i (\coth (x)+1)}}+\frac {\sinh \left (\frac {x}{2}\right ) \left (4 \tanh ^{-1}\left (\sqrt {\tanh \left (\frac {x}{2}\right )}\right )+\sqrt {2} \left (\log \left (\tanh \left (\frac {x}{2}\right )-\sqrt {2} \sqrt {\tanh \left (\frac {x}{2}\right )}+1\right )-\log \left (\tanh \left (\frac {x}{2}\right )+\sqrt {2} \sqrt {\tanh \left (\frac {x}{2}\right )}+1\right )\right )\right ) \left (\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )}{\sqrt {\tanh \left (\frac {x}{2}\right )}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 231, normalized size = 11.00 \[ \frac {4 \, \sqrt {2} {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} + {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (\frac {2 \, \sqrt {2} {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} + 3 \, \cosh \relax (x)^{2} + 6 \, \cosh \relax (x) \sinh \relax (x) + 3 \, \sinh \relax (x)^{2} - 1}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right ) - {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (-\frac {2 \, \sqrt {2} {\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - 3 \, \cosh \relax (x)^{2} - 6 \, \cosh \relax (x) \sinh \relax (x) - 3 \, \sinh \relax (x)^{2} + 1}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}\right )}{4 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 149, normalized size = 7.10 \[ -\frac {1}{4} \, \sqrt {2} {\left (\sqrt {2} {\left (2 \, \sqrt {2} - \log \left (-\frac {\sqrt {2} - 1}{\sqrt {2} + 1}\right )\right )} + \sqrt {2} \log \left (\frac {{\left (\sqrt {e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{2} - 2 \, \sqrt {2} + 3}{{\left (\sqrt {e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{2} + 2 \, \sqrt {2} + 3}\right ) - \frac {8 \, {\left (3 \, {\left (\sqrt {e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{2} + 1\right )}}{{\left (\sqrt {e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{4} + 6 \, {\left (\sqrt {e^{\left (2 \, x\right )} - 1} - e^{x}\right )}^{2} + 1}\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \mathrm {sech}\relax (x )^{2} \sqrt {1+\coth \relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\coth \relax (x) + 1} \operatorname {sech}\relax (x)^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\sqrt {\mathrm {coth}\relax (x)+1}}{{\mathrm {cosh}\relax (x)}^2} \,d x \]
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 \[ \int \sqrt {\coth {\relax (x )} + 1} \operatorname {sech}^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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