Optimal. Leaf size=22 \[ \frac{\tan ^{-1}(\sinh (x))}{2 a}+\frac{\tanh (x) \text{sech}(x)}{2 a} \]
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Rubi [A] time = 0.0454984, antiderivative size = 22, 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 = {3175, 3768, 3770} \[ \frac{\tan ^{-1}(\sinh (x))}{2 a}+\frac{\tanh (x) \text{sech}(x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\text{sech}(x)}{a+a \sinh ^2(x)} \, dx &=\frac{\int \text{sech}^3(x) \, dx}{a}\\ &=\frac{\text{sech}(x) \tanh (x)}{2 a}+\frac{\int \text{sech}(x) \, dx}{2 a}\\ &=\frac{\tan ^{-1}(\sinh (x))}{2 a}+\frac{\text{sech}(x) \tanh (x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0037607, size = 20, normalized size = 0.91 \[ \frac{\tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+\frac{1}{2} \tanh (x) \text{sech}(x)}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 50, normalized size = 2.3 \begin{align*} -{\frac{1}{a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3} \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+{\frac{1}{a}\arctan \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.52255, size = 54, normalized size = 2.45 \begin{align*} \frac{e^{\left (-x\right )} - e^{\left (-3 \, x\right )}}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} - \frac{\arctan \left (e^{\left (-x\right )}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.48705, size = 522, normalized size = 23.73 \begin{align*} \frac{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) +{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )}{a \cosh \left (x\right )^{4} + 4 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + a \sinh \left (x\right )^{4} + 2 \, a \cosh \left (x\right )^{2} + 2 \,{\left (3 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 4 \,{\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a} \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*} \frac{\int \frac{\operatorname{sech}{\left (x \right )}}{\sinh ^{2}{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13744, size = 70, normalized size = 3.18 \begin{align*} \frac{\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )}{4 \, a} - \frac{e^{\left (-x\right )} - e^{x}}{{\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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