Optimal. Leaf size=24 \[ \frac{1}{2} \tanh (x) \sqrt{\tanh ^2(x)+1}+\frac{1}{2} \sinh ^{-1}(\tanh (x)) \]
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Rubi [A] time = 0.0433874, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3675, 195, 215} \[ \frac{1}{2} \tanh (x) \sqrt{\tanh ^2(x)+1}+\frac{1}{2} \sinh ^{-1}(\tanh (x)) \]
Antiderivative was successfully verified.
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Rule 3675
Rule 195
Rule 215
Rubi steps
\begin{align*} \int \text{sech}^2(x) \sqrt{1+\tanh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \sqrt{1+x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{2} \tanh (x) \sqrt{1+\tanh ^2(x)}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{2} \sinh ^{-1}(\tanh (x))+\frac{1}{2} \tanh (x) \sqrt{1+\tanh ^2(x)}\\ \end{align*}
Mathematica [B] time = 0.0917873, size = 55, normalized size = 2.29 \[ \frac{1}{4} \sqrt{\tanh ^2(x)+1} \text{sech}(x) \text{sech}(2 x) \left (-\sinh (x)+\sinh (3 x)+2 \sqrt{\cosh (2 x)} \cosh ^2(x) \tanh ^{-1}\left (\frac{\sinh (x)}{\sqrt{\cosh (2 x)}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.142, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm sech} \left (x\right ) \right ) ^{2}\sqrt{1+ \left ( \tanh \left ( x \right ) \right ) ^{2}}\, dx \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 \sqrt{\tanh \left (x\right )^{2} + 1} \operatorname{sech}\left (x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.08352, size = 1181, normalized size = 49.21 \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 \sqrt{\tanh ^{2}{\left (x \right )} + 1} \operatorname{sech}^{2}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18325, size = 196, normalized size = 8.17 \begin{align*} \frac{1}{4} \, \sqrt{2}{\left (\sqrt{2} \log \left (\frac{\sqrt{2} - \sqrt{e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1}{\sqrt{2} + \sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} - 1}\right ) - \frac{4 \,{\left (3 \,{\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{3} -{\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - \sqrt{e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} - 1\right )}}{{\left ({\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - 2 \, \sqrt{e^{\left (4 \, x\right )} + 1} + 2 \, e^{\left (2 \, x\right )} - 1\right )}^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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