Optimal. Leaf size=26 \[ \frac {x}{4}-\frac {1}{4 (1+\coth (x))^2}-\frac {1}{4 (1+\coth (x))} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 3, 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}{4}-\frac {1}{4 (\coth (x)+1)}-\frac {1}{4 (\coth (x)+1)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3560
Rubi steps
\begin {align*} \int \frac {1}{(1+\coth (x))^2} \, dx &=-\frac {1}{4 (1+\coth (x))^2}+\frac {1}{2} \int \frac {1}{1+\coth (x)} \, dx\\ &=-\frac {1}{4 (1+\coth (x))^2}-\frac {1}{4 (1+\coth (x))}+\frac {\int 1 \, dx}{4}\\ &=\frac {x}{4}-\frac {1}{4 (1+\coth (x))^2}-\frac {1}{4 (1+\coth (x))}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 30, normalized size = 1.15 \begin {gather*} \frac {1}{16} (4 x+4 \cosh (2 x)-\cosh (4 x)-4 \sinh (2 x)+\sinh (4 x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 32, normalized size = 1.23
method | result | size |
risch | \(\frac {x}{4}+\frac {{\mathrm e}^{-2 x}}{4}-\frac {{\mathrm e}^{-4 x}}{16}\) | \(17\) |
derivativedivides | \(-\frac {\ln \left (\coth \left (x \right )-1\right )}{8}-\frac {1}{4 \left (1+\coth \left (x \right )\right )^{2}}-\frac {1}{4 \left (1+\coth \left (x \right )\right )}+\frac {\ln \left (1+\coth \left (x \right )\right )}{8}\) | \(32\) |
default | \(-\frac {\ln \left (\coth \left (x \right )-1\right )}{8}-\frac {1}{4 \left (1+\coth \left (x \right )\right )^{2}}-\frac {1}{4 \left (1+\coth \left (x \right )\right )}+\frac {\ln \left (1+\coth \left (x \right )\right )}{8}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 16, normalized size = 0.62 \begin {gather*} \frac {1}{4} \, x + \frac {1}{4} \, e^{\left (-2 \, x\right )} - \frac {1}{16} \, e^{\left (-4 \, 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. 52 vs.
\(2 (20) = 40\).
time = 0.35, size = 52, normalized size = 2.00 \begin {gather*} \frac {{\left (4 \, x - 1\right )} \cosh \left (x\right )^{2} + 2 \, {\left (4 \, x + 1\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (4 \, x - 1\right )} \sinh \left (x\right )^{2} + 4}{16 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\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. 88 vs.
\(2 (20) = 40\).
time = 0.50, size = 88, normalized size = 3.38 \begin {gather*} \frac {x \tanh ^{2}{\left (x \right )}}{4 \tanh ^{2}{\left (x \right )} + 8 \tanh {\left (x \right )} + 4} + \frac {2 x \tanh {\left (x \right )}}{4 \tanh ^{2}{\left (x \right )} + 8 \tanh {\left (x \right )} + 4} + \frac {x}{4 \tanh ^{2}{\left (x \right )} + 8 \tanh {\left (x \right )} + 4} + \frac {3 \tanh {\left (x \right )}}{4 \tanh ^{2}{\left (x \right )} + 8 \tanh {\left (x \right )} + 4} + \frac {2}{4 \tanh ^{2}{\left (x \right )} + 8 \tanh {\left (x \right )} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 18, normalized size = 0.69 \begin {gather*} \frac {1}{16} \, {\left (4 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-4 \, x\right )} + \frac {1}{4} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 16, normalized size = 0.62 \begin {gather*} \frac {x}{4}+\frac {{\mathrm {e}}^{-2\,x}}{4}-\frac {{\mathrm {e}}^{-4\,x}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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