Optimal. Leaf size=60 \[ \frac {x}{16}+\frac {1}{32 (1-\coth (x))^2}-\frac {1}{8 (1-\coth (x))}-\frac {1}{24 (1+\coth (x))^3}+\frac {5}{32 (1+\coth (x))^2}-\frac {3}{16 (1+\coth (x))} \]
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Rubi [A]
time = 0.05, antiderivative size = 60, 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 = {3597, 862, 90,
213} \begin {gather*} \frac {x}{16}-\frac {1}{8 (1-\coth (x))}-\frac {3}{16 (\coth (x)+1)}+\frac {1}{32 (1-\coth (x))^2}+\frac {5}{32 (\coth (x)+1)^2}-\frac {1}{24 (\coth (x)+1)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 213
Rule 862
Rule 3597
Rubi steps
\begin {align*} \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx &=-\text {Subst}\left (\int \frac {x^4}{(1+x) \left (-1+x^2\right )^3} \, dx,x,\coth (x)\right )\\ &=-\text {Subst}\left (\int \frac {x^4}{(-1+x)^3 (1+x)^4} \, dx,x,\coth (x)\right )\\ &=-\text {Subst}\left (\int \left (\frac {1}{16 (-1+x)^3}+\frac {1}{8 (-1+x)^2}-\frac {1}{8 (1+x)^4}+\frac {5}{16 (1+x)^3}-\frac {3}{16 (1+x)^2}+\frac {1}{16 \left (-1+x^2\right )}\right ) \, dx,x,\coth (x)\right )\\ &=\frac {1}{32 (1-\coth (x))^2}-\frac {1}{8 (1-\coth (x))}-\frac {1}{24 (1+\coth (x))^3}+\frac {5}{32 (1+\coth (x))^2}-\frac {3}{16 (1+\coth (x))}-\frac {1}{16} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\coth (x)\right )\\ &=\frac {x}{16}+\frac {1}{32 (1-\coth (x))^2}-\frac {1}{8 (1-\coth (x))}-\frac {1}{24 (1+\coth (x))^3}+\frac {5}{32 (1+\coth (x))^2}-\frac {3}{16 (1+\coth (x))}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 42, normalized size = 0.70 \begin {gather*} \frac {1}{192} (12 x+15 \cosh (2 x)+6 \cosh (4 x)+\cosh (6 x)+3 \sinh (2 x)-3 \sinh (4 x)-\sinh (6 x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(117\) vs.
\(2(48)=96\).
time = 0.42, size = 118, normalized size = 1.97
method | result | size |
risch | \(\frac {x}{16}+\frac {{\mathrm e}^{4 x}}{128}+\frac {3 \,{\mathrm e}^{2 x}}{64}+\frac {{\mathrm e}^{-2 x}}{32}+\frac {3 \,{\mathrm e}^{-4 x}}{128}+\frac {{\mathrm e}^{-6 x}}{192}\) | \(35\) |
default | \(\frac {1}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{4 \tanh \left (\frac {x}{2}\right )-4}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{16}+\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{6}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {13}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {19}{12 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{16}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 36, normalized size = 0.60 \begin {gather*} \frac {1}{128} \, {\left (6 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (4 \, x\right )} + \frac {1}{16} \, x + \frac {1}{32} \, e^{\left (-2 \, x\right )} + \frac {3}{128} \, e^{\left (-4 \, x\right )} + \frac {1}{192} \, e^{\left (-6 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs.
\(2 (44) = 88\).
time = 0.39, size = 92, normalized size = 1.53 \begin {gather*} \frac {5 \, \cosh \left (x\right )^{5} + 25 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + {\left (10 \, \cosh \left (x\right )^{2} + 9\right )} \sinh \left (x\right )^{3} + 27 \, \cosh \left (x\right )^{3} + {\left (50 \, \cosh \left (x\right )^{3} + 81 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 12 \, {\left (2 \, x + 1\right )} \cosh \left (x\right ) + {\left (5 \, \cosh \left (x\right )^{4} + 27 \, \cosh \left (x\right )^{2} + 24 \, x - 12\right )} \sinh \left (x\right )}{384 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\cosh ^{4}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 42, normalized size = 0.70 \begin {gather*} -\frac {1}{384} \, {\left (22 \, e^{\left (6 \, x\right )} - 12 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-6 \, x\right )} + \frac {1}{16} \, x + \frac {1}{128} \, e^{\left (4 \, x\right )} + \frac {3}{64} \, e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.42, size = 34, normalized size = 0.57 \begin {gather*} \frac {x}{16}+\frac {{\mathrm {e}}^{-2\,x}}{32}+\frac {3\,{\mathrm {e}}^{2\,x}}{64}+\frac {3\,{\mathrm {e}}^{-4\,x}}{128}+\frac {{\mathrm {e}}^{4\,x}}{128}+\frac {{\mathrm {e}}^{-6\,x}}{192} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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