Optimal. Leaf size=26 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\tanh (4 x)+1}}{\sqrt{2}}\right )}{2 \sqrt{2}} \]
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Rubi [A] time = 0.0134693, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3480, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\tanh (4 x)+1}}{\sqrt{2}}\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \sqrt{1+\tanh (4 x)} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\tanh (4 x)}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{1+\tanh (4 x)}}{\sqrt{2}}\right )}{2 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0167125, size = 26, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{\tanh (4 x)+1}}{\sqrt{2}}\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 20, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{1+\tanh \left ( 4\,x \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.82197, size = 58, normalized size = 2.23 \begin{align*} -\frac{1}{8} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{\sqrt{2}}{\sqrt{e^{\left (-8 \, x\right )} + 1}}}{\sqrt{2} + \frac{\sqrt{2}}{\sqrt{e^{\left (-8 \, x\right )} + 1}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.20844, size = 207, normalized size = 7.96 \begin{align*} \frac{1}{8} \, \sqrt{2} \log \left (-2 \, \sqrt{2} \sqrt{\frac{\cosh \left (4 \, x\right )}{\cosh \left (4 \, x\right ) - \sinh \left (4 \, x\right )}}{\left (\cosh \left (4 \, x\right ) + \sinh \left (4 \, x\right )\right )} - 2 \, \cosh \left (4 \, x\right )^{2} - 4 \, \cosh \left (4 \, x\right ) \sinh \left (4 \, x\right ) - 2 \, \sinh \left (4 \, 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 (4 x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08865, size = 41, normalized size = 1.58 \begin{align*} \frac{1}{8} \, \sqrt{2}{\left (\log \left (\sqrt{e^{\left (-8 \, x\right )} + 1} + 1\right ) - \log \left (\sqrt{e^{\left (-8 \, x\right )} + 1} - 1\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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