Optimal. Leaf size=43 \[ \frac {5 x}{2}-2 \log (\cosh (x))-\frac {5 \tanh (x)}{2}+\tanh ^2(x)-\frac {5 \tanh ^3(x)}{6}+\frac {\tanh ^3(x)}{2 (1+\coth (x))} \]
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Rubi [A]
time = 0.08, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3633, 3610,
3612, 3556} \begin {gather*} \frac {5 x}{2}-\frac {5 \tanh ^3(x)}{6}+\tanh ^2(x)-\frac {5 \tanh (x)}{2}-2 \log (\cosh (x))+\frac {\tanh ^3(x)}{2 (\coth (x)+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3610
Rule 3612
Rule 3633
Rubi steps
\begin {align*} \int \frac {\tanh ^4(x)}{1+\coth (x)} \, dx &=\frac {\tanh ^3(x)}{2 (1+\coth (x))}-\frac {1}{2} \int (-5+4 \coth (x)) \tanh ^4(x) \, dx\\ &=-\frac {5}{6} \tanh ^3(x)+\frac {\tanh ^3(x)}{2 (1+\coth (x))}-\frac {1}{2} i \int (-4 i+5 i \coth (x)) \tanh ^3(x) \, dx\\ &=\tanh ^2(x)-\frac {5 \tanh ^3(x)}{6}+\frac {\tanh ^3(x)}{2 (1+\coth (x))}+\frac {1}{2} \int (5-4 \coth (x)) \tanh ^2(x) \, dx\\ &=-\frac {5 \tanh (x)}{2}+\tanh ^2(x)-\frac {5 \tanh ^3(x)}{6}+\frac {\tanh ^3(x)}{2 (1+\coth (x))}+\frac {1}{2} i \int (4 i-5 i \coth (x)) \tanh (x) \, dx\\ &=\frac {5 x}{2}-\frac {5 \tanh (x)}{2}+\tanh ^2(x)-\frac {5 \tanh ^3(x)}{6}+\frac {\tanh ^3(x)}{2 (1+\coth (x))}-2 \int \tanh (x) \, dx\\ &=\frac {5 x}{2}-2 \log (\cosh (x))-\frac {5 \tanh (x)}{2}+\tanh ^2(x)-\frac {5 \tanh ^3(x)}{6}+\frac {\tanh ^3(x)}{2 (1+\coth (x))}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 40, normalized size = 0.93 \begin {gather*} \frac {1}{12} \left (30 x+3 \cosh (2 x)-24 \log (\cosh (x))-3 \sinh (2 x)-28 \tanh (x)+\text {sech}^2(x) (-6+4 \tanh (x))\right ) \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. \(95\) vs.
\(2(35)=70\).
time = 0.57, size = 96, normalized size = 2.23
method | result | size |
risch | \(\frac {9 x}{2}+\frac {{\mathrm e}^{-2 x}}{4}+\frac {4 \,{\mathrm e}^{4 x}+6 \,{\mathrm e}^{2 x}+\frac {14}{3}}{\left (1+{\mathrm e}^{2 x}\right )^{3}}-2 \ln \left (1+{\mathrm e}^{2 x}\right )\) | \(44\) |
default | \(\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{\tanh \left (\frac {x}{2}\right )+1}+\frac {9 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}-\frac {4 \left (\tanh ^{5}\left (\frac {x}{2}\right )-\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{2}+\frac {8 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}-\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{2}+\tanh \left (\frac {x}{2}\right )\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-2 \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 55, normalized size = 1.28 \begin {gather*} \frac {1}{2} \, x - \frac {2 \, {\left (15 \, e^{\left (-2 \, x\right )} + 12 \, e^{\left (-4 \, x\right )} + 7\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} + \frac {1}{4} \, e^{\left (-2 \, x\right )} - 2 \, \log \left (e^{\left (-2 \, x\right )} + 1\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. 571 vs.
\(2 (35) = 70\).
time = 0.36, size = 571, normalized size = 13.28 \begin {gather*} \frac {54 \, x \cosh \left (x\right )^{8} + 432 \, x \cosh \left (x\right ) \sinh \left (x\right )^{7} + 54 \, x \sinh \left (x\right )^{8} + 3 \, {\left (54 \, x + 17\right )} \cosh \left (x\right )^{6} + 3 \, {\left (504 \, x \cosh \left (x\right )^{2} + 54 \, x + 17\right )} \sinh \left (x\right )^{6} + 18 \, {\left (168 \, x \cosh \left (x\right )^{3} + {\left (54 \, x + 17\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 81 \, {\left (2 \, x + 1\right )} \cosh \left (x\right )^{4} + 9 \, {\left (420 \, x \cosh \left (x\right )^{4} + 5 \, {\left (54 \, x + 17\right )} \cosh \left (x\right )^{2} + 18 \, x + 9\right )} \sinh \left (x\right )^{4} + 12 \, {\left (252 \, x \cosh \left (x\right )^{5} + 5 \, {\left (54 \, x + 17\right )} \cosh \left (x\right )^{3} + 27 \, {\left (2 \, x + 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (54 \, x + 65\right )} \cosh \left (x\right )^{2} + {\left (1512 \, x \cosh \left (x\right )^{6} + 45 \, {\left (54 \, x + 17\right )} \cosh \left (x\right )^{4} + 486 \, {\left (2 \, x + 1\right )} \cosh \left (x\right )^{2} + 54 \, x + 65\right )} \sinh \left (x\right )^{2} - 24 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + {\left (28 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{6} + 3 \, \cosh \left (x\right )^{6} + 2 \, {\left (28 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + {\left (70 \, \cosh \left (x\right )^{4} + 45 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (14 \, \cosh \left (x\right )^{5} + 15 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (28 \, \cosh \left (x\right )^{6} + 45 \, \cosh \left (x\right )^{4} + 18 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (4 \, \cosh \left (x\right )^{7} + 9 \, \cosh \left (x\right )^{5} + 6 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (216 \, x \cosh \left (x\right )^{7} + 9 \, {\left (54 \, x + 17\right )} \cosh \left (x\right )^{5} + 162 \, {\left (2 \, x + 1\right )} \cosh \left (x\right )^{3} + {\left (54 \, x + 65\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3}{12 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + {\left (28 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{6} + 3 \, \cosh \left (x\right )^{6} + 2 \, {\left (28 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + {\left (70 \, \cosh \left (x\right )^{4} + 45 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (14 \, \cosh \left (x\right )^{5} + 15 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (28 \, \cosh \left (x\right )^{6} + 45 \, \cosh \left (x\right )^{4} + 18 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (4 \, \cosh \left (x\right )^{7} + 9 \, \cosh \left (x\right )^{5} + 6 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\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 {\tanh ^{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 = 0.41, size = 47, normalized size = 1.09 \begin {gather*} \frac {9}{2} \, x + \frac {{\left (51 \, e^{\left (6 \, x\right )} + 81 \, e^{\left (4 \, x\right )} + 65 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-2 \, x\right )}}{12 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} - 2 \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.30, size = 69, normalized size = 1.60 \begin {gather*} \frac {9\,x}{2}-2\,\ln \left ({\mathrm {e}}^{2\,x}+1\right )+\frac {{\mathrm {e}}^{-2\,x}}{4}+\frac {8}{3\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}-\frac {2}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {4}{{\mathrm {e}}^{2\,x}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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