Optimal. Leaf size=38 \[ \frac{x}{8}-\frac{1}{8 (1-\coth (x))}-\frac{1}{4 (\coth (x)+1)}+\frac{1}{8 (\coth (x)+1)^2} \]
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Rubi [A] time = 0.0592763, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3516, 848, 88, 207} \[ \frac{x}{8}-\frac{1}{8 (1-\coth (x))}-\frac{1}{4 (\coth (x)+1)}+\frac{1}{8 (\coth (x)+1)^2} \]
Antiderivative was successfully verified.
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Rule 3516
Rule 848
Rule 88
Rule 207
Rubi steps
\begin{align*} \int \frac{\cosh ^2(x)}{1+\coth (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2}{(1+x) \left (-1+x^2\right )^2} \, dx,x,\coth (x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{x^2}{(-1+x)^2 (1+x)^3} \, dx,x,\coth (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{8 (-1+x)^2}+\frac{1}{4 (1+x)^3}-\frac{1}{4 (1+x)^2}+\frac{1}{8 \left (-1+x^2\right )}\right ) \, dx,x,\coth (x)\right )\\ &=-\frac{1}{8 (1-\coth (x))}+\frac{1}{8 (1+\coth (x))^2}-\frac{1}{4 (1+\coth (x))}-\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\coth (x)\right )\\ &=\frac{x}{8}-\frac{1}{8 (1-\coth (x))}+\frac{1}{8 (1+\coth (x))^2}-\frac{1}{4 (1+\coth (x))}\\ \end{align*}
Mathematica [A] time = 0.047073, size = 24, normalized size = 0.63 \[ \frac{1}{32} (4 x-\sinh (4 x)+4 \cosh (2 x)+\cosh (4 x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 78, normalized size = 2.1 \begin{align*}{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-4}}- \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}+ \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{1}{8}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{1}{4} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{1}{4} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{1}{8}\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 [A] time = 1.02968, size = 30, normalized size = 0.79 \begin{align*} \frac{1}{8} \, x + \frac{1}{16} \, e^{\left (2 \, x\right )} + \frac{1}{16} \, e^{\left (-2 \, x\right )} + \frac{1}{32} \, e^{\left (-4 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53566, size = 176, normalized size = 4.63 \begin{align*} \frac{3 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + 2 \,{\left (2 \, x + 1\right )} \cosh \left (x\right ) +{\left (3 \, \cosh \left (x\right )^{2} + 4 \, x - 2\right )} \sinh \left (x\right )}{32 \,{\left (\cosh \left (x\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*} \int \frac{\cosh ^{2}{\left (x \right )}}{\coth{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11309, size = 41, normalized size = 1.08 \begin{align*} -\frac{1}{32} \,{\left (3 \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-4 \, x\right )} + \frac{1}{8} \, x + \frac{1}{16} \, e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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