Optimal. Leaf size=5 \[ \tan ^{-1}(\tanh (x)+1) \]
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Rubi [A] time = 0.0491681, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4342, 617, 204} \[ \tan ^{-1}(\tanh (x)+1) \]
Antiderivative was successfully verified.
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Rule 4342
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{2+2 \tanh (x)+\tanh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{2+2 x+x^2} \, dx,x,\tanh (x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\tanh (x)\right )\\ &=\tan ^{-1}(1+\tanh (x))\\ \end{align*}
Mathematica [F] time = 0.0386977, size = 0, normalized size = 0. \[ \int \frac{\text{sech}^2(x)}{2+2 \tanh (x)+\tanh ^2(x)} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 0.059, size = 42, normalized size = 8.4 \begin{align*}{\frac{i}{2}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+ \left ( 1-i \right ) \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{\frac{i}{2}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+ \left ( 1+i \right ) \tanh \left ({\frac{x}{2}} \right ) +1 \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{\operatorname{sech}\left (x\right )^{2}}{\tanh \left (x\right )^{2} + 2 \, \tanh \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 = 2.0324, size = 74, normalized size = 14.8 \begin{align*} -\arctan \left (-\frac{3 \, \cosh \left (x\right ) + 2 \, \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 )} + 2 \tanh{\left (x \right )} + 2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16817, size = 12, normalized size = 2.4 \begin{align*} \arctan \left (\frac{5}{2} \, e^{\left (2 \, x\right )} + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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