Optimal. Leaf size=46 \[ \frac {x}{16}-\frac {1}{8 (1+\coth (x))^4}-\frac {1}{12 (1+\coth (x))^3}-\frac {1}{16 (1+\coth (x))^2}-\frac {1}{16 (1+\coth (x))} \]
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Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3560, 8}
\begin {gather*} \frac {x}{16}-\frac {1}{16 (\coth (x)+1)}-\frac {1}{16 (\coth (x)+1)^2}-\frac {1}{12 (\coth (x)+1)^3}-\frac {1}{8 (\coth (x)+1)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3560
Rubi steps
\begin {align*} \int \frac {1}{(1+\coth (x))^4} \, dx &=-\frac {1}{8 (1+\coth (x))^4}+\frac {1}{2} \int \frac {1}{(1+\coth (x))^3} \, dx\\ &=-\frac {1}{8 (1+\coth (x))^4}-\frac {1}{12 (1+\coth (x))^3}+\frac {1}{4} \int \frac {1}{(1+\coth (x))^2} \, dx\\ &=-\frac {1}{8 (1+\coth (x))^4}-\frac {1}{12 (1+\coth (x))^3}-\frac {1}{16 (1+\coth (x))^2}+\frac {1}{8} \int \frac {1}{1+\coth (x)} \, dx\\ &=-\frac {1}{8 (1+\coth (x))^4}-\frac {1}{12 (1+\coth (x))^3}-\frac {1}{16 (1+\coth (x))^2}-\frac {1}{16 (1+\coth (x))}+\frac {\int 1 \, dx}{16}\\ &=\frac {x}{16}-\frac {1}{8 (1+\coth (x))^4}-\frac {1}{12 (1+\coth (x))^3}-\frac {1}{16 (1+\coth (x))^2}-\frac {1}{16 (1+\coth (x))}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 53, normalized size = 1.15 \begin {gather*} \frac {1}{384} (\cosh (4 x)-\sinh (4 x)) (-36+64 \cosh (2 x)+3 (-1+8 x) \cosh (4 x)+32 \sinh (2 x)+3 \sinh (4 x)+24 x \sinh (4 x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 48, normalized size = 1.04
method | result | size |
risch | \(\frac {x}{16}+\frac {{\mathrm e}^{-2 x}}{8}-\frac {3 \,{\mathrm e}^{-4 x}}{32}+\frac {{\mathrm e}^{-6 x}}{24}-\frac {{\mathrm e}^{-8 x}}{128}\) | \(29\) |
derivativedivides | \(-\frac {1}{8 \left (1+\coth \left (x \right )\right )^{4}}-\frac {1}{12 \left (1+\coth \left (x \right )\right )^{3}}-\frac {1}{16 \left (1+\coth \left (x \right )\right )^{2}}-\frac {1}{16 \left (1+\coth \left (x \right )\right )}+\frac {\ln \left (1+\coth \left (x \right )\right )}{32}-\frac {\ln \left (\coth \left (x \right )-1\right )}{32}\) | \(48\) |
default | \(-\frac {1}{8 \left (1+\coth \left (x \right )\right )^{4}}-\frac {1}{12 \left (1+\coth \left (x \right )\right )^{3}}-\frac {1}{16 \left (1+\coth \left (x \right )\right )^{2}}-\frac {1}{16 \left (1+\coth \left (x \right )\right )}+\frac {\ln \left (1+\coth \left (x \right )\right )}{32}-\frac {\ln \left (\coth \left (x \right )-1\right )}{32}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 28, normalized size = 0.61 \begin {gather*} \frac {1}{16} \, x + \frac {1}{8} \, e^{\left (-2 \, x\right )} - \frac {3}{32} \, e^{\left (-4 \, x\right )} + \frac {1}{24} \, e^{\left (-6 \, x\right )} - \frac {1}{128} \, e^{\left (-8 \, 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. 121 vs.
\(2 (36) = 72\).
time = 0.36, size = 121, normalized size = 2.63 \begin {gather*} \frac {3 \, {\left (8 \, x - 1\right )} \cosh \left (x\right )^{4} + 12 \, {\left (8 \, x + 1\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + 3 \, {\left (8 \, x - 1\right )} \sinh \left (x\right )^{4} + 2 \, {\left (9 \, {\left (8 \, x - 1\right )} \cosh \left (x\right )^{2} + 32\right )} \sinh \left (x\right )^{2} + 64 \, \cosh \left (x\right )^{2} + 4 \, {\left (3 \, {\left (8 \, x + 1\right )} \cosh \left (x\right )^{3} + 16 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 36}{384 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 299 vs.
\(2 (41) = 82\).
time = 0.89, size = 299, normalized size = 6.50 \begin {gather*} \frac {3 x \tanh ^{4}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {12 x \tanh ^{3}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {18 x \tanh ^{2}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {12 x \tanh {\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {3 x}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {45 \tanh ^{3}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {84 \tanh ^{2}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {61 \tanh {\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {16}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 30, normalized size = 0.65 \begin {gather*} \frac {1}{384} \, {\left (48 \, e^{\left (6 \, x\right )} - 36 \, e^{\left (4 \, x\right )} + 16 \, e^{\left (2 \, x\right )} - 3\right )} e^{\left (-8 \, x\right )} + \frac {1}{16} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.14, size = 28, normalized size = 0.61 \begin {gather*} \frac {x}{16}+\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {3\,{\mathrm {e}}^{-4\,x}}{32}+\frac {{\mathrm {e}}^{-6\,x}}{24}-\frac {{\mathrm {e}}^{-8\,x}}{128} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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