Optimal. Leaf size=30 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right )}{\sqrt{2}}+\frac{1}{\sqrt{\tanh (x)+1}} \]
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Rubi [A] time = 0.0365176, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3526, 3480, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right )}{\sqrt{2}}+\frac{1}{\sqrt{\tanh (x)+1}} \]
Antiderivative was successfully verified.
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Rule 3526
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \frac{\tanh (x)}{\sqrt{1+\tanh (x)}} \, dx &=\frac{1}{\sqrt{1+\tanh (x)}}+\frac{1}{2} \int \sqrt{1+\tanh (x)} \, dx\\ &=\frac{1}{\sqrt{1+\tanh (x)}}+\operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\tanh (x)}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{1+\tanh (x)}}{\sqrt{2}}\right )}{\sqrt{2}}+\frac{1}{\sqrt{1+\tanh (x)}}\\ \end{align*}
Mathematica [A] time = 0.0539926, size = 30, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right )}{\sqrt{2}}+\frac{1}{\sqrt{\tanh (x)+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 25, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{1+\tanh \left ( x \right ) }} \right ) }+{\frac{1}{\sqrt{1+\tanh \left ( x \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, \sqrt{2} \sqrt{e^{\left (-2 \, x\right )} + 1} + \int \frac{e^{\left (-x\right )}}{\frac{\sqrt{2} e^{\left (-x\right )}}{\sqrt{e^{\left (-2 \, x\right )} + 1}} + \frac{\sqrt{2} e^{\left (-3 \, x\right )}}{\sqrt{e^{\left (-2 \, x\right )} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15684, size = 301, normalized size = 10.03 \begin{align*} \frac{{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \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 ) + 4 \, \sqrt{\frac{\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}}}{4 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.40823, size = 66, normalized size = 2.2 \begin{align*} - \begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left (\frac{\sqrt{2} \sqrt{\tanh{\left (x \right )} + 1}}{2} \right )}}{2} & \text{for}\: \tanh{\left (x \right )} + 1 > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{\tanh{\left (x \right )} + 1}}{2} \right )}}{2} & \text{for}\: \tanh{\left (x \right )} + 1 < 2 \end{cases} + \frac{1}{\sqrt{\tanh{\left (x \right )} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26666, size = 78, normalized size = 2.6 \begin{align*} -\frac{1}{4} \, \sqrt{2} \log \left (-2 \, \sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{2} \, \sqrt{2} + \frac{\sqrt{2}}{2 \,{\left (\sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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