Optimal. Leaf size=75 \[ \frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {5+2 \sqrt {5}} \tanh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {5-2 \sqrt {5}} \tanh (x)\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1166, 203} \[ \frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {5+2 \sqrt {5}} \tanh (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {5-2 \sqrt {5}} \tanh (x)\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 1166
Rubi steps
\begin {align*} \int \cosh (x) \text {sech}(5 x) \, dx &=\operatorname {Subst}\left (\int \frac {1-x^2}{1+10 x^2+5 x^4} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{2} \left (-1-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{5+2 \sqrt {5}+5 x^2} \, dx,x,\tanh (x)\right )+\frac {1}{2} \left (-1+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{5-2 \sqrt {5}+5 x^2} \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {5-2 \sqrt {5}} \tanh (x)\right )+\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {5+2 \sqrt {5}} \tanh (x)\right )\\ \end {align*}
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Mathematica [A] time = 0.11, size = 84, normalized size = 1.12 \[ \frac {\sqrt {5+\sqrt {5}} \tan ^{-1}\left (\frac {\left (5+\sqrt {5}\right ) \tanh (x)}{\sqrt {10-2 \sqrt {5}}}\right )+\sqrt {5-\sqrt {5}} \tan ^{-1}\left (\frac {\left (\sqrt {5}-5\right ) \tanh (x)}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )}{5 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 173, normalized size = 2.31 \[ -\frac {1}{5} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \arctan \left (\frac {1}{40} \, \sqrt {5} \sqrt {2} \sqrt {-32 \, {\left (\sqrt {5} + 1\right )} e^{\left (2 \, x\right )} + 64 \, e^{\left (4 \, x\right )} + 64} \sqrt {\sqrt {5} + 5} - \frac {1}{20} \, {\left (4 \, \sqrt {5} \sqrt {2} e^{\left (2 \, x\right )} - \sqrt {5} \sqrt {2} - 5 \, \sqrt {2}\right )} \sqrt {\sqrt {5} + 5}\right ) + \frac {1}{5} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \arctan \left (-\frac {1}{20} \, {\left (4 \, \sqrt {5} \sqrt {2} e^{\left (2 \, x\right )} - \sqrt {5} \sqrt {2} + 5 \, \sqrt {2}\right )} \sqrt {-\sqrt {5} + 5} + \frac {1}{5} \, \sqrt {5} \sqrt {{\left (\sqrt {5} - 1\right )} e^{\left (2 \, x\right )} + 2 \, e^{\left (4 \, x\right )} + 2} \sqrt {-\sqrt {5} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 68, normalized size = 0.91 \[ -\frac {1}{10} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {\sqrt {5} + 4 \, e^{\left (2 \, x\right )} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {1}{10} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (-\frac {\sqrt {5} - 4 \, e^{\left (2 \, x\right )} + 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 41, normalized size = 0.55 \[ 2 \left (\munderset {\textit {\_R} =\RootOf \left (32000 \textit {\_Z}^{4}+400 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-4000 \textit {\_R}^{3}+200 \textit {\_R}^{2}+{\mathrm e}^{2 x}-30 \textit {\_R} +1\right )\right ) \]
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 \[ \frac {\sqrt {5} \arctan \left (\frac {\sqrt {5} + 4 \, e^{\left (-2 \, x\right )} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {2 \, \sqrt {5} + 10}} - \frac {\sqrt {5} \arctan \left (-\frac {\sqrt {5} - 4 \, e^{\left (-2 \, x\right )} + 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right )}{5 \, \sqrt {-2 \, \sqrt {5} + 10}} - \frac {\log \left (-{\left (\sqrt {5} + 1\right )} e^{\left (-2 \, x\right )} + 2 \, e^{\left (-4 \, x\right )} + 2\right )}{10 \, {\left (\sqrt {5} + 1\right )}} + \frac {\log \left ({\left (\sqrt {5} - 1\right )} e^{\left (-2 \, x\right )} + 2 \, e^{\left (-4 \, x\right )} + 2\right )}{10 \, {\left (\sqrt {5} - 1\right )}} - \frac {1}{5} \, \int \frac {{\left (e^{\left (7 \, x\right )} - 2 \, e^{\left (5 \, x\right )} - 2 \, e^{\left (3 \, x\right )} + e^{x}\right )} e^{x}}{e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} + \frac {1}{10} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{10} \, \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.15, size = 297, normalized size = 3.96 \[ \ln \left (1-\sqrt {-\frac {\sqrt {5}}{200}-\frac {1}{40}}\,\left (4\,{\mathrm {e}}^{2\,x}+\sqrt {-\frac {\sqrt {5}}{200}-\frac {1}{40}}\,\left (48\,{\mathrm {e}}^{2\,x}+\sqrt {-\frac {\sqrt {5}}{200}-\frac {1}{40}}\,\left (360\,{\mathrm {e}}^{2\,x}-360\right )-72\right )-8\right )-{\mathrm {e}}^{2\,x}\right )\,\sqrt {-\frac {\sqrt {5}}{200}-\frac {1}{40}}-\ln \left (\sqrt {\frac {\sqrt {5}}{200}-\frac {1}{40}}\,\left (4\,{\mathrm {e}}^{2\,x}+\sqrt {\frac {\sqrt {5}}{200}-\frac {1}{40}}\,\left (\sqrt {\frac {\sqrt {5}}{200}-\frac {1}{40}}\,\left (360\,{\mathrm {e}}^{2\,x}-360\right )-48\,{\mathrm {e}}^{2\,x}+72\right )-8\right )-{\mathrm {e}}^{2\,x}+1\right )\,\sqrt {\frac {\sqrt {5}}{200}-\frac {1}{40}}-\ln \left (\sqrt {-\frac {\sqrt {5}}{200}-\frac {1}{40}}\,\left (4\,{\mathrm {e}}^{2\,x}+\sqrt {-\frac {\sqrt {5}}{200}-\frac {1}{40}}\,\left (\sqrt {-\frac {\sqrt {5}}{200}-\frac {1}{40}}\,\left (360\,{\mathrm {e}}^{2\,x}-360\right )-48\,{\mathrm {e}}^{2\,x}+72\right )-8\right )-{\mathrm {e}}^{2\,x}+1\right )\,\sqrt {-\frac {\sqrt {5}}{200}-\frac {1}{40}}+\ln \left (1-\sqrt {\frac {\sqrt {5}}{200}-\frac {1}{40}}\,\left (4\,{\mathrm {e}}^{2\,x}+\sqrt {\frac {\sqrt {5}}{200}-\frac {1}{40}}\,\left (48\,{\mathrm {e}}^{2\,x}+\sqrt {\frac {\sqrt {5}}{200}-\frac {1}{40}}\,\left (360\,{\mathrm {e}}^{2\,x}-360\right )-72\right )-8\right )-{\mathrm {e}}^{2\,x}\right )\,\sqrt {\frac {\sqrt {5}}{200}-\frac {1}{40}} \]
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 \[ \int \cosh {\relax (x )} \operatorname {sech}{\left (5 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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