Optimal. Leaf size=45 \[ -\frac{2 \tanh (x)}{a}+\frac{3 \tan ^{-1}(\sinh (x))}{2 a}-\frac{\tanh (x) \text{sech}^2(x)}{a \text{sech}(x)+a}+\frac{3 \tanh (x) \text{sech}(x)}{2 a} \]
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Rubi [A] time = 0.0849706, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {3818, 3787, 3767, 8, 3768, 3770} \[ -\frac{2 \tanh (x)}{a}+\frac{3 \tan ^{-1}(\sinh (x))}{2 a}-\frac{\tanh (x) \text{sech}^2(x)}{a \text{sech}(x)+a}+\frac{3 \tanh (x) \text{sech}(x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3818
Rule 3787
Rule 3767
Rule 8
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\text{sech}^4(x)}{a+a \text{sech}(x)} \, dx &=-\frac{\text{sech}^2(x) \tanh (x)}{a+a \text{sech}(x)}-\frac{\int \text{sech}^2(x) (2 a-3 a \text{sech}(x)) \, dx}{a^2}\\ &=-\frac{\text{sech}^2(x) \tanh (x)}{a+a \text{sech}(x)}-\frac{2 \int \text{sech}^2(x) \, dx}{a}+\frac{3 \int \text{sech}^3(x) \, dx}{a}\\ &=\frac{3 \text{sech}(x) \tanh (x)}{2 a}-\frac{\text{sech}^2(x) \tanh (x)}{a+a \text{sech}(x)}-\frac{(2 i) \operatorname{Subst}(\int 1 \, dx,x,-i \tanh (x))}{a}+\frac{3 \int \text{sech}(x) \, dx}{2 a}\\ &=\frac{3 \tan ^{-1}(\sinh (x))}{2 a}-\frac{2 \tanh (x)}{a}+\frac{3 \text{sech}(x) \tanh (x)}{2 a}-\frac{\text{sech}^2(x) \tanh (x)}{a+a \text{sech}(x)}\\ \end{align*}
Mathematica [A] time = 0.0839871, size = 51, normalized size = 1.13 \[ \frac{\cosh \left (\frac{x}{2}\right ) \text{sech}(x) \left (\cosh \left (\frac{x}{2}\right ) \left (6 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+\tanh (x) (\text{sech}(x)-2)\right )-2 \sinh \left (\frac{x}{2}\right )\right )}{a (\text{sech}(x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 61, normalized size = 1.4 \begin{align*} -{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) }-3\,{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{3}}{a \left ( \left ( \tanh \left ( 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}}+3\,{\frac{\arctan \left ( \tanh \left ( x/2 \right ) \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.69542, size = 99, normalized size = 2.2 \begin{align*} -\frac{e^{\left (-x\right )} + 5 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + 4}{a e^{\left (-x\right )} + 2 \, a e^{\left (-2 \, x\right )} + 2 \, a e^{\left (-3 \, x\right )} + a e^{\left (-4 \, x\right )} + a e^{\left (-5 \, x\right )} + a} - \frac{3 \, \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 = 2.43579, size = 1099, normalized size = 24.42 \begin{align*} \frac{3 \, \cosh \left (x\right )^{4} + 3 \,{\left (4 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + 3 \, \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{3} +{\left (18 \, \cosh \left (x\right )^{2} + 9 \, \cosh \left (x\right ) + 5\right )} \sinh \left (x\right )^{2} + 3 \,{\left (\cosh \left (x\right )^{5} +{\left (5 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + \cosh \left (x\right )^{4} + 2 \,{\left (5 \, \cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + 2 \, \cosh \left (x\right )^{3} + 2 \,{\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} + 6 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 5 \, \cosh \left (x\right )^{2} +{\left (12 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )^{2} + 10 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right ) + 4}{a \cosh \left (x\right )^{5} + a \sinh \left (x\right )^{5} + a \cosh \left (x\right )^{4} +{\left (5 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{4} + 2 \, a \cosh \left (x\right )^{3} + 2 \,{\left (5 \, a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{3} + 2 \, a \cosh \left (x\right )^{2} + 2 \,{\left (5 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} + 3 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{2} + a \cosh \left (x\right ) +{\left (5 \, a \cosh \left (x\right )^{4} + 4 \, a \cosh \left (x\right )^{3} + 6 \, a \cosh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + a\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}^{4}{\left (x \right )}}{\operatorname{sech}{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15175, size = 65, normalized size = 1.44 \begin{align*} \frac{3 \, \arctan \left (e^{x}\right )}{a} + \frac{e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, x\right )} - e^{x} + 2}{a{\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} + \frac{2}{a{\left (e^{x} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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