Optimal. Leaf size=13 \[ -\tanh ^{-1}\left (\sqrt{\cosh ^2(x)+1}\right ) \]
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Rubi [A] time = 0.0408253, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3194, 63, 207} \[ -\tanh ^{-1}\left (\sqrt{\cosh ^2(x)+1}\right ) \]
Antiderivative was successfully verified.
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Rule 3194
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{\tanh (x)}{\sqrt{1+\cosh ^2(x)}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,\cosh ^2(x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+\cosh ^2(x)}\right )\\ &=-\tanh ^{-1}\left (\sqrt{1+\cosh ^2(x)}\right )\\ \end{align*}
Mathematica [A] time = 0.0104219, size = 13, normalized size = 1. \[ -\tanh ^{-1}\left (\sqrt{\cosh ^2(x)+1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 12, normalized size = 0.9 \begin{align*} -{\it Artanh} \left ({\frac{1}{\sqrt{1+ \left ( \cosh \left ( x \right ) \right ) ^{2}}}} \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{\tanh \left (x\right )}{\sqrt{\cosh \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 = 2.33288, size = 224, normalized size = 17.23 \begin{align*} \log \left (\frac{\sqrt{2} \sqrt{\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 2 \, \cosh \left (x\right ) - 2 \, \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \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 \frac{\tanh{\left (x \right )}}{\sqrt{\cosh ^{2}{\left (x \right )} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (x\right )}{\sqrt{\cosh \left (x\right )^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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