Optimal. Leaf size=53 \[ -\frac{3}{2} i \text{PolyLog}\left (2,-i e^x\right )+\frac{3}{2} i \text{PolyLog}\left (2,i e^x\right )+3 x \tan ^{-1}\left (e^x\right )-\frac{\text{sech}(x)}{2}-\frac{1}{2} x \tanh (x) \text{sech}(x) \]
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Rubi [A] time = 0.131812, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5473, 4180, 2279, 2391, 5455, 4185} \[ -\frac{3}{2} i \text{PolyLog}\left (2,-i e^x\right )+\frac{3}{2} i \text{PolyLog}\left (2,i e^x\right )+3 x \tan ^{-1}\left (e^x\right )-\frac{\text{sech}(x)}{2}-\frac{1}{2} x \tanh (x) \text{sech}(x) \]
Antiderivative was successfully verified.
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Rule 5473
Rule 4180
Rule 2279
Rule 2391
Rule 5455
Rule 4185
Rubi steps
\begin{align*} \int x \cosh (2 x) \text{sech}^3(x) \, dx &=\int \left (x \text{sech}(x)+x \text{sech}(x) \tanh ^2(x)\right ) \, dx\\ &=\int x \text{sech}(x) \, dx+\int x \text{sech}(x) \tanh ^2(x) \, dx\\ &=2 x \tan ^{-1}\left (e^x\right )-i \int \log \left (1-i e^x\right ) \, dx+i \int \log \left (1+i e^x\right ) \, dx+\int x \text{sech}(x) \, dx-\int x \text{sech}^3(x) \, dx\\ &=4 x \tan ^{-1}\left (e^x\right )-\frac{\text{sech}(x)}{2}-\frac{1}{2} x \text{sech}(x) \tanh (x)-i \int \log \left (1-i e^x\right ) \, dx+i \int \log \left (1+i e^x\right ) \, dx-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 )-\frac{1}{2} \int x \text{sech}(x) \, dx\\ &=3 x \tan ^{-1}\left (e^x\right )-i \text{Li}_2\left (-i e^x\right )+i \text{Li}_2\left (i e^x\right )-\frac{\text{sech}(x)}{2}-\frac{1}{2} x \text{sech}(x) \tanh (x)+\frac{1}{2} i \int \log \left (1-i e^x\right ) \, dx-\frac{1}{2} i \int \log \left (1+i e^x\right ) \, dx-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 )\\ &=3 x \tan ^{-1}\left (e^x\right )-2 i \text{Li}_2\left (-i e^x\right )+2 i \text{Li}_2\left (i e^x\right )-\frac{\text{sech}(x)}{2}-\frac{1}{2} x \text{sech}(x) \tanh (x)+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^x\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^x\right )\\ &=3 x \tan ^{-1}\left (e^x\right )-\frac{3}{2} i \text{Li}_2\left (-i e^x\right )+\frac{3}{2} i \text{Li}_2\left (i e^x\right )-\frac{\text{sech}(x)}{2}-\frac{1}{2} x \text{sech}(x) \tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0640587, size = 78, normalized size = 1.47 \[ -\frac{1}{2} i \left (3 \text{PolyLog}\left (2,-i e^{-x}\right )-3 \text{PolyLog}\left (2,i e^{-x}\right )+3 x \log \left (1-i e^{-x}\right )-3 x \log \left (1+i e^{-x}\right )-i \text{sech}(x)-i x \tanh (x) \text{sech}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 75, normalized size = 1.4 \begin{align*} -{\frac{{{\rm e}^{x}} \left ( x{{\rm e}^{2\,x}}+{{\rm e}^{2\,x}}-x+1 \right ) }{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}-{\frac{3\,i}{2}}x\ln \left ( 1+i{{\rm e}^{x}} \right ) +{\frac{3\,i}{2}}x\ln \left ( 1-i{{\rm e}^{x}} \right ) -{\frac{3\,i}{2}}{\it dilog} \left ( 1+i{{\rm e}^{x}} \right ) +{\frac{3\,i}{2}}{\it dilog} \left ( 1-i{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (x + 1\right )} e^{\left (3 \, x\right )} -{\left (x - 1\right )} e^{x}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} + 12 \, \int \frac{x e^{x}}{4 \,{\left (e^{\left (2 \, x\right )} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.93736, size = 1442, normalized size = 27.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh{\left (2 x \right )} \operatorname{sech}^{3}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh \left (2 \, x\right ) \operatorname{sech}\left (x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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