Optimal. Leaf size=25 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \tanh (x)}{\sqrt{\tanh ^2(x)+1}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0185679, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3661, 377, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \tanh (x)}{\sqrt{\tanh ^2(x)+1}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 3661
Rule 377
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+\tanh ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{1+x^2}} \, dx,x,\tanh (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\frac{\tanh (x)}{\sqrt{1+\tanh ^2(x)}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \tanh (x)}{\sqrt{1+\tanh ^2(x)}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0249142, size = 35, normalized size = 1.4 \[ \frac{\sinh ^{-1}\left (\sqrt{2} \sinh (x)\right ) \sqrt{\cosh (2 x)} \text{sech}(x)}{\sqrt{2} \sqrt{\tanh ^2(x)+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 62, normalized size = 2.5 \begin{align*} -{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\frac{ \left ( 2-2\,\tanh \left ( x \right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{ \left ( 1+\tanh \left ( x \right ) \right ) ^{2}-2\,\tanh \left ( x \right ) }}}} \right ) }+{\frac{\sqrt{2}}{4}{\it Artanh} \left ({\frac{ \left ( 2\,\tanh \left ( x \right ) +2 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{ \left ( \tanh \left ( x \right ) -1 \right ) ^{2}+2\,\tanh \left ( x \right ) }}}} \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*} \int \frac{1}{\sqrt{\tanh \left (x\right )^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.54609, size = 1831, normalized size = 73.24 \begin{align*} \text{result too large to display} \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 \frac{1}{\sqrt{\tanh ^{2}{\left (x \right )} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30805, size = 78, normalized size = 3.12 \begin{align*} -\frac{1}{4} \, \sqrt{2}{\left (\log \left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + \log \left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - \log \left (-\sqrt{e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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