Optimal. Leaf size=43 \[ i \text {Li}_2\left (-i e^x\right )-i \text {Li}_2\left (i e^x\right )-2 x \tan ^{-1}\left (e^x\right )+2 x \sinh (x)-2 \cosh (x) \]
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Rubi [A] time = 0.07, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5473, 3296, 2638, 5449, 4180, 2279, 2391} \[ i \text {PolyLog}\left (2,-i e^x\right )-i \text {PolyLog}\left (2,i e^x\right )-2 x \tan ^{-1}\left (e^x\right )+2 x \sinh (x)-2 \cosh (x) \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 2638
Rule 3296
Rule 4180
Rule 5449
Rule 5473
Rubi steps
\begin {align*} \int x \cosh (2 x) \text {sech}(x) \, dx &=\int (x \cosh (x)+x \sinh (x) \tanh (x)) \, dx\\ &=\int x \cosh (x) \, dx+\int x \sinh (x) \tanh (x) \, dx\\ &=x \sinh (x)+\int x \cosh (x) \, dx-\int x \text {sech}(x) \, dx-\int \sinh (x) \, dx\\ &=-2 x \tan ^{-1}\left (e^x\right )-\cosh (x)+2 x \sinh (x)+i \int \log \left (1-i e^x\right ) \, dx-i \int \log \left (1+i e^x\right ) \, dx-\int \sinh (x) \, dx\\ &=-2 x \tan ^{-1}\left (e^x\right )-2 \cosh (x)+2 x \sinh (x)+i \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^x\right )-i \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^x\right )\\ &=-2 x \tan ^{-1}\left (e^x\right )-2 \cosh (x)+i \text {Li}_2\left (-i e^x\right )-i \text {Li}_2\left (i e^x\right )+2 x \sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 71, normalized size = 1.65 \[ i \left (\text {Li}_2\left (-i e^{-x}\right )-\text {Li}_2\left (i e^{-x}\right )\right )+i x \left (\log \left (1-i e^{-x}\right )-\log \left (1+i e^{-x}\right )\right )+2 x \sinh (x)-2 \cosh (x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 124, normalized size = 2.88 \[ \frac {{\left (x - 1\right )} \cosh \relax (x)^{2} + 2 \, {\left (x - 1\right )} \cosh \relax (x) \sinh \relax (x) + {\left (x - 1\right )} \sinh \relax (x)^{2} + {\left (-i \, \cosh \relax (x) - i \, \sinh \relax (x)\right )} {\rm Li}_2\left (i \, \cosh \relax (x) + i \, \sinh \relax (x)\right ) + {\left (i \, \cosh \relax (x) + i \, \sinh \relax (x)\right )} {\rm Li}_2\left (-i \, \cosh \relax (x) - i \, \sinh \relax (x)\right ) + {\left (i \, x \cosh \relax (x) + i \, x \sinh \relax (x)\right )} \log \left (i \, \cosh \relax (x) + i \, \sinh \relax (x) + 1\right ) + {\left (-i \, x \cosh \relax (x) - i \, x \sinh \relax (x)\right )} \log \left (-i \, \cosh \relax (x) - i \, \sinh \relax (x) + 1\right ) - x - 1}{\cosh \relax (x) + \sinh \relax (x)} \]
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 \[ \int x \cosh \left (2 \, x\right ) \operatorname {sech}\relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 68, normalized size = 1.58 \[ 2 \left (-\frac {1}{2}+\frac {x}{2}\right ) {\mathrm e}^{x}+2 \left (-\frac {1}{2}-\frac {x}{2}\right ) {\mathrm e}^{-x}+i x \ln \left (1+i {\mathrm e}^{x}\right )-i x \ln \left (1-i {\mathrm e}^{x}\right )+i \dilog \left (1+i {\mathrm e}^{x}\right )-i \dilog \left (1-i {\mathrm e}^{x}\right ) \]
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 \[ -{\left (x + 1\right )} e^{\left (-x\right )} + {\left (x - 1\right )} e^{x} - 2 \, \int \frac {x e^{x}}{e^{\left (2 \, x\right )} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\mathrm {cosh}\left (2\,x\right )}{\mathrm {cosh}\relax (x)} \,d x \]
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 \[ \int x \cosh {\left (2 x \right )} \operatorname {sech}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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