Optimal. Leaf size=23 \[ 4 x-2 \coth (x)-\frac {1}{2} (1+\coth (x))^2+4 \log (\sinh (x)) \]
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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3559, 3558,
3556} \begin {gather*} 4 x-\frac {1}{2} (\coth (x)+1)^2-2 \coth (x)+4 \log (\sinh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3558
Rule 3559
Rubi steps
\begin {align*} \int (1+\coth (x))^3 \, dx &=-\frac {1}{2} (1+\coth (x))^2+2 \int (1+\coth (x))^2 \, dx\\ &=4 x-2 \coth (x)-\frac {1}{2} (1+\coth (x))^2+4 \int \coth (x) \, dx\\ &=4 x-2 \coth (x)-\frac {1}{2} (1+\coth (x))^2+4 \log (\sinh (x))\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.11, size = 61, normalized size = 2.65 \begin {gather*} \frac {1}{4} \text {csch}^2(x) \left (-1-2 x-8 \log (\cosh (x))-8 \log (\tanh (x))+\cosh (2 x) (-1+2 x+8 \log (\cosh (x))+8 \log (\tanh (x)))-6 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(x)\right ) \sinh (2 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 19, normalized size = 0.83
method | result | size |
derivativedivides | \(-\frac {\left (\coth ^{2}\left (x \right )\right )}{2}-3 \coth \left (x \right )-4 \ln \left (\coth \left (x \right )-1\right )\) | \(19\) |
default | \(-\frac {\left (\coth ^{2}\left (x \right )\right )}{2}-3 \coth \left (x \right )-4 \ln \left (\coth \left (x \right )-1\right )\) | \(19\) |
risch | \(-\frac {2 \left (4 \,{\mathrm e}^{2 x}-3\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2}}+4 \ln \left ({\mathrm e}^{2 x}-1\right )\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs.
\(2 (21) = 42\).
time = 0.27, size = 55, normalized size = 2.39 \begin {gather*} 5 \, x + \frac {2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {6}{e^{\left (-2 \, x\right )} - 1} + \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) + 3 \, \log \left (\sinh \left (x\right )\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. 142 vs.
\(2 (21) = 42\).
time = 0.34, size = 142, normalized size = 6.17 \begin {gather*} -\frac {2 \, {\left (4 \, \cosh \left (x\right )^{2} - 2 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 8 \, \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, \sinh \left (x\right )^{2} - 3\right )}}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.39, size = 31, normalized size = 1.35 \begin {gather*} 8 x - 4 \log {\left (\tanh {\left (x \right )} + 1 \right )} + 4 \log {\left (\tanh {\left (x \right )} \right )} - \frac {3}{\tanh {\left (x \right )}} - \frac {1}{2 \tanh ^{2}{\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 29, normalized size = 1.26 \begin {gather*} -\frac {2 \, {\left (4 \, e^{\left (2 \, x\right )} - 3\right )}}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + 4 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 36, normalized size = 1.57 \begin {gather*} 4\,\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\frac {2}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {8}{{\mathrm {e}}^{2\,x}-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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