Optimal. Leaf size=55 \[ \frac{x}{a}-\frac{\coth ^5(x) (1-\text{sech}(x))}{5 a}-\frac{\coth ^3(x) (5-4 \text{sech}(x))}{15 a}-\frac{\coth (x) (15-8 \text{sech}(x))}{15 a} \]
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Rubi [A] time = 0.120491, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3888, 3882, 8} \[ \frac{x}{a}-\frac{\coth ^5(x) (1-\text{sech}(x))}{5 a}-\frac{\coth ^3(x) (5-4 \text{sech}(x))}{15 a}-\frac{\coth (x) (15-8 \text{sech}(x))}{15 a} \]
Antiderivative was successfully verified.
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Rule 3888
Rule 3882
Rule 8
Rubi steps
\begin{align*} \int \frac{\coth ^4(x)}{a+a \text{sech}(x)} \, dx &=-\frac{\int \coth ^6(x) (-a+a \text{sech}(x)) \, dx}{a^2}\\ &=-\frac{\coth ^5(x) (1-\text{sech}(x))}{5 a}+\frac{\int \coth ^4(x) (5 a-4 a \text{sech}(x)) \, dx}{5 a^2}\\ &=-\frac{\coth ^3(x) (5-4 \text{sech}(x))}{15 a}-\frac{\coth ^5(x) (1-\text{sech}(x))}{5 a}-\frac{\int \coth ^2(x) (-15 a+8 a \text{sech}(x)) \, dx}{15 a^2}\\ &=-\frac{\coth (x) (15-8 \text{sech}(x))}{15 a}-\frac{\coth ^3(x) (5-4 \text{sech}(x))}{15 a}-\frac{\coth ^5(x) (1-\text{sech}(x))}{5 a}+\frac{\int 15 a \, dx}{15 a^2}\\ &=\frac{x}{a}-\frac{\coth (x) (15-8 \text{sech}(x))}{15 a}-\frac{\coth ^3(x) (5-4 \text{sech}(x))}{15 a}-\frac{\coth ^5(x) (1-\text{sech}(x))}{5 a}\\ \end{align*}
Mathematica [A] time = 0.102212, size = 69, normalized size = 1.25 \[ \frac{\text{csch}^3(x) \text{sech}(x) (-90 x \sinh (x)-30 x \sinh (2 x)+30 x \sinh (3 x)+15 x \sinh (4 x)+8 \cosh (x)+16 \cosh (2 x)-16 \cosh (3 x)-23 \cosh (4 x)-25)}{120 a (\text{sech}(x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 78, normalized size = 1.4 \begin{align*} -{\frac{1}{80\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{1}{8\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{48\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}-{\frac{3}{8\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{1}{a}\ln \left ( \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 [B] time = 1.19771, size = 142, normalized size = 2.58 \begin{align*} \frac{x}{a} - \frac{2 \,{\left (31 \, e^{\left (-x\right )} - 31 \, e^{\left (-2 \, x\right )} - 73 \, e^{\left (-3 \, x\right )} + 25 \, e^{\left (-4 \, x\right )} + 65 \, e^{\left (-5 \, x\right )} + 15 \, e^{\left (-6 \, x\right )} - 15 \, e^{\left (-7 \, x\right )} + 23\right )}}{15 \,{\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.44348, size = 502, normalized size = 9.13 \begin{align*} -\frac{23 \, \cosh \left (x\right )^{4} - 2 \,{\left (2 \,{\left (15 \, x + 23\right )} \cosh \left (x\right ) + 15 \, x + 23\right )} \sinh \left (x\right )^{3} + 23 \, \sinh \left (x\right )^{4} + 16 \, \cosh \left (x\right )^{3} + 2 \,{\left (69 \, \cosh \left (x\right )^{2} + 24 \, \cosh \left (x\right ) - 8\right )} \sinh \left (x\right )^{2} - 16 \, \cosh \left (x\right )^{2} - 2 \,{\left (2 \,{\left (15 \, x + 23\right )} \cosh \left (x\right )^{3} + 3 \,{\left (15 \, x + 23\right )} \cosh \left (x\right )^{2} - 2 \,{\left (15 \, x + 23\right )} \cosh \left (x\right ) - 45 \, x - 69\right )} \sinh \left (x\right ) - 8 \, \cosh \left (x\right ) + 25}{30 \,{\left ({\left (2 \, a \cosh \left (x\right ) + a\right )} \sinh \left (x\right )^{3} +{\left (2 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} - 2 \, a \cosh \left (x\right ) - 3 \, a\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*} \frac{\int \frac{\coth ^{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 = 1.11255, size = 86, normalized size = 1.56 \begin{align*} \frac{x}{a} - \frac{21 \, e^{\left (2 \, x\right )} - 36 \, e^{x} + 19}{24 \, a{\left (e^{x} - 1\right )}^{3}} + \frac{115 \, e^{\left (4 \, x\right )} + 380 \, e^{\left (3 \, x\right )} + 530 \, e^{\left (2 \, x\right )} + 340 \, e^{x} + 91}{40 \, a{\left (e^{x} + 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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