Optimal. Leaf size=71 \[ \frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}} \]
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Rubi [A] time = 0.07, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4357, 1093, 207} \[ \frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}} \]
Antiderivative was successfully verified.
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Rule 207
Rule 1093
Rule 4357
Rubi steps
\begin {align*} \int \text {sech}(4 x) \sinh (x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-8 x^2+8 x^4} \, dx,x,\cosh (x)\right )\\ &=\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-4-2 \sqrt {2}+8 x^2} \, dx,x,\cosh (x)\right )-\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-4+2 \sqrt {2}+8 x^2} \, dx,x,\cosh (x)\right )\\ &=\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2-\sqrt {2}}}\right )}{2 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\tanh ^{-1}\left (\frac {2 \cosh (x)}{\sqrt {2+\sqrt {2}}}\right )}{2 \sqrt {2 \left (2+\sqrt {2}\right )}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 110, normalized size = 1.55 \[ \frac {1}{16} \text {RootSum}\left [\text {$\#$1}^8+1\& ,\frac {\text {$\#$1}^2 x+2 \text {$\#$1}^2 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-2 \log \left (-\text {$\#$1} \sinh \left (\frac {x}{2}\right )+\text {$\#$1} \cosh \left (\frac {x}{2}\right )-\sinh \left (\frac {x}{2}\right )-\cosh \left (\frac {x}{2}\right )\right )-x}{\text {$\#$1}^5}\& \right ] \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 215, normalized size = 3.03 \[ \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + {\left ({\left (\sqrt {2} - 1\right )} \cosh \relax (x) + {\left (\sqrt {2} - 1\right )} \sinh \relax (x)\right )} \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - {\left ({\left (\sqrt {2} - 1\right )} \cosh \relax (x) + {\left (\sqrt {2} - 1\right )} \sinh \relax (x)\right )} \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + {\left ({\left (\sqrt {2} + 1\right )} \cosh \relax (x) + {\left (\sqrt {2} + 1\right )} \sinh \relax (x)\right )} \sqrt {-\sqrt {2} + 2} + 1\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} - {\left ({\left (\sqrt {2} + 1\right )} \cosh \relax (x) + {\left (\sqrt {2} + 1\right )} \sinh \relax (x)\right )} \sqrt {-\sqrt {2} + 2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 115, normalized size = 1.62 \[ -\frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 40, normalized size = 0.56 \[ 2 \left (\munderset {\textit {\_R} =\RootOf \left (32768 \textit {\_Z}^{4}-512 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (4096 \textit {\_R}^{3}-48 \textit {\_R} \right ) {\mathrm e}^{x}+1\right )\right ) \]
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 \[ \int \operatorname {sech}\left (4 \, x\right ) \sinh \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 251, normalized size = 3.54 \[ \ln \left (3\,{\mathrm {e}}^{2\,x}-2\,\sqrt {2}+8\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-8\,\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}+3\right )\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-\ln \left (3\,{\mathrm {e}}^{2\,x}-2\,\sqrt {2}-8\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}+8\,\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}+3\right )\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{64}}-\ln \left (3\,{\mathrm {e}}^{2\,x}+2\,\sqrt {2}-8\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}-8\,\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+3\right )\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+\ln \left (3\,{\mathrm {e}}^{2\,x}+2\,\sqrt {2}+8\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+2\,\sqrt {2}\,{\mathrm {e}}^{2\,x}+8\,\sqrt {2}\,{\mathrm {e}}^x\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}}+3\right )\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{32}} \]
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 \[ \int \sinh {\relax (x )} \operatorname {sech}{\left (4 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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