Optimal. Leaf size=223 \[ -\frac {2 \text {ArcTan}\left (\frac {\tanh \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}}}\right )}{5 \sqrt {-1+(-1)^{2/5}}}-\frac {2 \sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \text {ArcTan}\left (\sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{4/5}}{1+(-1)^{4/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+(-1)^{3/5}}}+\frac {\sinh (x)}{5 (1+\cosh (x))} \]
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Rubi [A]
time = 0.38, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3292, 2727,
2738, 211, 214} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {\tanh \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}}}\right )}{5 \sqrt {(-1)^{2/5}-1}}-\frac {2 \sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \text {ArcTan}\left (\sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{4/5}}{1+(-1)^{4/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+(-1)^{3/5}}}+\frac {\sinh (x)}{5 (\cosh (x)+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 2727
Rule 2738
Rule 3292
Rubi steps
\begin {align*} \int \frac {1}{1+\cosh ^5(x)} \, dx &=\int \left (-\frac {1}{5 (-1-\cosh (x))}-\frac {1}{5 \left (-1+\sqrt [5]{-1} \cosh (x)\right )}-\frac {1}{5 \left (-1-(-1)^{2/5} \cosh (x)\right )}-\frac {1}{5 \left (-1+(-1)^{3/5} \cosh (x)\right )}-\frac {1}{5 \left (-1-(-1)^{4/5} \cosh (x)\right )}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {1}{-1-\cosh (x)} \, dx\right )-\frac {1}{5} \int \frac {1}{-1+\sqrt [5]{-1} \cosh (x)} \, dx-\frac {1}{5} \int \frac {1}{-1-(-1)^{2/5} \cosh (x)} \, dx-\frac {1}{5} \int \frac {1}{-1+(-1)^{3/5} \cosh (x)} \, dx-\frac {1}{5} \int \frac {1}{-1-(-1)^{4/5} \cosh (x)} \, dx\\ &=\frac {\sinh (x)}{5 (1+\cosh (x))}-\frac {2}{5} \text {Subst}\left (\int \frac {1}{-1+\sqrt [5]{-1}-\left (-1-\sqrt [5]{-1}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )-\frac {2}{5} \text {Subst}\left (\int \frac {1}{-1-(-1)^{2/5}-\left (-1+(-1)^{2/5}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )-\frac {2}{5} \text {Subst}\left (\int \frac {1}{-1+(-1)^{3/5}-\left (-1-(-1)^{3/5}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )-\frac {2}{5} \text {Subst}\left (\int \frac {1}{-1-(-1)^{4/5}-\left (-1+(-1)^{4/5}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\\ &=-\frac {2 \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right )}{\sqrt {-\frac {1-\sqrt [5]{-1}}{1+\sqrt [5]{-1}}}}\right )}{5 \sqrt {-1+(-1)^{2/5}}}-\frac {2 \sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tan ^{-1}\left (\sqrt {-\frac {1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \left (1+(-1)^{3/5}\right )}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1-(-1)^{4/5}}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-(-1)^{4/5}}{1+(-1)^{4/5}}} \tanh \left (\frac {x}{2}\right )\right )}{5 \sqrt {1+(-1)^{3/5}}}+\frac {\sinh (x)}{5 (1+\cosh (x))}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.07, size = 445, normalized size = 2.00 \begin {gather*} -\frac {1}{10} \text {RootSum}\left [1-2 \text {$\#$1}+8 \text {$\#$1}^2-14 \text {$\#$1}^3+30 \text {$\#$1}^4-14 \text {$\#$1}^5+8 \text {$\#$1}^6-2 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {x+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right )-4 x \text {$\#$1}-8 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}+15 x \text {$\#$1}^2+30 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^2-40 x \text {$\#$1}^3-80 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^3+15 x \text {$\#$1}^4+30 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^4-4 x \text {$\#$1}^5-8 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^5+x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )+\cosh \left (\frac {x}{2}\right ) \text {$\#$1}-\sinh \left (\frac {x}{2}\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-1+8 \text {$\#$1}-21 \text {$\#$1}^2+60 \text {$\#$1}^3-35 \text {$\#$1}^4+24 \text {$\#$1}^5-7 \text {$\#$1}^6+4 \text {$\#$1}^7}\&\right ]+\frac {1}{5} \tanh \left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.60, size = 62, normalized size = 0.28
method | result | size |
default | \(\frac {\tanh \left (\frac {x}{2}\right )}{5}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (5 \textit {\_Z}^{8}+10 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-5 \textit {\_R}^{6}+5 \textit {\_R}^{4}-5 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7}+\textit {\_R}^{3}}\right )}{50}\) | \(62\) |
risch | \(-\frac {2}{5 \left ({\mathrm e}^{x}+1\right )}+\left (\munderset {\textit {\_R} =\RootOf \left (1953125 \textit {\_Z}^{8}-156250 \textit {\_Z}^{6}+6250 \textit {\_Z}^{4}-125 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (2343750 \textit {\_R}^{7}-234375 \textit {\_R}^{6}-140625 \textit {\_R}^{5}+15625 \textit {\_R}^{4}+4375 \textit {\_R}^{3}-500 \textit {\_R}^{2}-50 \textit {\_R} +{\mathrm e}^{x}+6\right )\right )\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 3228 vs.
\(2 (150) = 300\).
time = 0.51, size = 3228, normalized size = 14.48 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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