Optimal. Leaf size=3 \[ \tan ^{-1}(\tanh (x)) \]
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Rubi [A] time = 0.0322492, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3675, 203} \[ \tan ^{-1}(\tanh (x)) \]
Antiderivative was successfully verified.
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Rule 3675
Rule 203
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{1+\tanh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tanh (x)\right )\\ &=\tan ^{-1}(\tanh (x))\\ \end{align*}
Mathematica [A] time = 0.0033085, size = 3, normalized size = 1. \[ \tan ^{-1}(\tanh (x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 116, normalized size = 38.7 \begin{align*} -2\,{\frac{\sqrt{2}}{2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{2+2\,\sqrt{2}}} \right ) }-2\,{\frac{1}{2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{2+2\,\sqrt{2}}} \right ) }+2\,{\frac{\sqrt{2}}{-2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{-2+2\,\sqrt{2}}} \right ) }-2\,{\frac{1}{-2+2\,\sqrt{2}}\arctan \left ( 2\,{\frac{\tanh \left ( x/2 \right ) }{-2+2\,\sqrt{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.51455, size = 47, normalized size = 15.67 \begin{align*} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{\left (-x\right )}\right )}\right ) - \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{\left (-x\right )}\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.9952, size = 69, normalized size = 23. \begin{align*} -\arctan \left (-\frac{\cosh \left (x\right ) + \sinh \left (x\right )}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{2}{\left (x \right )}}{\tanh ^{2}{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13488, size = 7, normalized size = 2.33 \begin{align*} \arctan \left (e^{\left (2 \, x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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