Optimal. Leaf size=31 \[ \frac{x}{a}-\frac{\tan ^{-1}(\sinh (x))}{2 a}-\frac{\tanh (x) (2-\text{sech}(x))}{2 a} \]
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Rubi [A] time = 0.0703507, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3888, 3881, 3770} \[ \frac{x}{a}-\frac{\tan ^{-1}(\sinh (x))}{2 a}-\frac{\tanh (x) (2-\text{sech}(x))}{2 a} \]
Antiderivative was successfully verified.
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Rule 3888
Rule 3881
Rule 3770
Rubi steps
\begin{align*} \int \frac{\tanh ^4(x)}{a+a \text{sech}(x)} \, dx &=-\frac{\int (-a+a \text{sech}(x)) \tanh ^2(x) \, dx}{a^2}\\ &=-\frac{(2-\text{sech}(x)) \tanh (x)}{2 a}-\frac{\int (-2 a+a \text{sech}(x)) \, dx}{2 a^2}\\ &=\frac{x}{a}-\frac{(2-\text{sech}(x)) \tanh (x)}{2 a}-\frac{\int \text{sech}(x) \, dx}{2 a}\\ &=\frac{x}{a}-\frac{\tan ^{-1}(\sinh (x))}{2 a}-\frac{(2-\text{sech}(x)) \tanh (x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0578742, size = 41, normalized size = 1.32 \[ \frac{\cosh ^2\left (\frac{x}{2}\right ) \text{sech}(x) \left (2 \left (x-\tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )\right )+\tanh (x) (\text{sech}(x)-2)\right )}{a (\text{sech}(x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 75, normalized size = 2.4 \begin{align*}{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \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}}-{\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 = 1.66518, size = 69, normalized size = 2.23 \begin{align*} \frac{x}{a} + \frac{e^{\left (-x\right )} - 2 \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )} - 2}{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 = 2.51102, size = 711, normalized size = 22.94 \begin{align*} \frac{x \cosh \left (x\right )^{4} + x \sinh \left (x\right )^{4} +{\left (4 \, x \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + 2 \,{\left (x + 1\right )} \cosh \left (x\right )^{2} + \cosh \left (x\right )^{3} +{\left (6 \, x \cosh \left (x\right )^{2} + 2 \, x + 3 \, \cosh \left (x\right ) + 2\right )} \sinh \left (x\right )^{2} -{\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 (4 \, x \cosh \left (x\right )^{3} + 4 \,{\left (x + 1\right )} \cosh \left (x\right ) + 3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) + x - \cosh \left (x\right ) + 2}{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{\tanh ^{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 = 1.14294, size = 57, normalized size = 1.84 \begin{align*} \frac{x}{a} - \frac{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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