Optimal. Leaf size=4 \[ x+\tanh (x) \]
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Rubi [A] time = 0.0523521, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {206} \[ x+\tanh (x) \]
Antiderivative was successfully verified.
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Rule 206
Rubi steps
\begin{align*} \int \text{sech}^2(x) \left (1+\frac{1}{1-\tanh ^2(x)}\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+\frac{1}{1-x^2}\right ) \, dx,x,\tanh (x)\right )\\ &=\tanh (x)+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=x+\tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0063401, size = 4, normalized size = 1. \[ x+\tanh (x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.032, size = 34, normalized size = 8.5 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +2\,{\frac{\tanh \left ( x/2 \right ) }{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02392, size = 16, normalized size = 4. \begin{align*} x + \frac{2}{e^{\left (-2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.31436, size = 50, normalized size = 12.5 \begin{align*} \frac{{\left (x - 1\right )} \cosh \left (x\right ) + \sinh \left (x\right )}{\cosh \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.48378, size = 29, normalized size = 7.25 \begin{align*} - \frac{x \operatorname{sech}^{2}{\left (x \right )}}{\tanh ^{2}{\left (x \right )} - 1} - \frac{\tanh{\left (x \right )} \operatorname{sech}^{2}{\left (x \right )}}{\tanh ^{2}{\left (x \right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15623, size = 16, normalized size = 4. \begin{align*} x - \frac{2}{e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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