Optimal. Leaf size=21 \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0114613, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3480, 206} \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \sqrt{1+\tanh (x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\tanh (x)}\right )\\ &=\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1+\tanh (x)}}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0450765, size = 21, normalized size = 1. \[ \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 17, normalized size = 0.8 \begin{align*}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{1+\tanh \left ( x \right ) }} \right ) \sqrt{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.60106, size = 58, normalized size = 2.76 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{\sqrt{2}}{\sqrt{e^{\left (-2 \, x\right )} + 1}}}{\sqrt{2} + \frac{\sqrt{2}}{\sqrt{e^{\left (-2 \, x\right )} + 1}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.21971, size = 182, normalized size = 8.67 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-2 \, \sqrt{2} \sqrt{\frac{\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - 2 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, \sinh \left (x\right )^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\tanh{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20234, size = 36, normalized size = 1.71 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (-2 \, \sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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