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 = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 88
Rule 207
Rule 848
Rule 3516
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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 0.40, size = 51, normalized size = 1.34 \[ \frac {3 \, \cosh \relax (x)^{3} + 9 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + 2 \, {\left (2 \, x + 1\right )} \cosh \relax (x) + {\left (3 \, \cosh \relax (x)^{2} + 4 \, x - 2\right )} \sinh \relax (x)}{32 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 30, normalized size = 0.79 \[ -\frac {1}{32} \, {\left (3 \, e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-4 \, x\right )} + \frac {1}{8} \, x + \frac {1}{16} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 78, normalized size = 2.05 \[ \frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{4 \tanh \left (\frac {x}{2}\right )-4}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 22, normalized size = 0.58 \[ \frac {1}{8} \, x + \frac {1}{16} \, e^{\left (2 \, x\right )} + \frac {1}{16} \, e^{\left (-2 \, x\right )} + \frac {1}{32} \, e^{\left (-4 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 22, normalized size = 0.58 \[ \frac {x}{8}+\frac {{\mathrm {e}}^{-2\,x}}{16}+\frac {{\mathrm {e}}^{2\,x}}{16}+\frac {{\mathrm {e}}^{-4\,x}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{2}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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