Optimal. Leaf size=62 \[ -\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-8 \cosh ^2(x)-\sqrt{5}+5\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-8 \cosh ^2(x)+\sqrt{5}+5\right )+\frac{1}{5} \log (\cosh (x)) \]
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Rubi [A] time = 0.0833591, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857, Rules used = {4357, 1114, 705, 29, 632, 31} \[ -\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (-8 \cosh ^2(x)-\sqrt{5}+5\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (-8 \cosh ^2(x)+\sqrt{5}+5\right )+\frac{1}{5} \log (\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 4357
Rule 1114
Rule 705
Rule 29
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \text{sech}(5 x) \sinh (x) \, dx &=\operatorname{Subst}\left (\int \frac{1}{x \left (5-20 x^2+16 x^4\right )} \, dx,x,\cosh (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \left (5-20 x+16 x^2\right )} \, dx,x,\cosh ^2(x)\right )\\ &=\frac{1}{10} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\cosh ^2(x)\right )+\frac{1}{10} \operatorname{Subst}\left (\int \frac{20-16 x}{5-20 x+16 x^2} \, dx,x,\cosh ^2(x)\right )\\ &=\frac{1}{5} \log (\cosh (x))-\frac{1}{5} \left (4 \left (1-\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-10-2 \sqrt{5}+16 x} \, dx,x,\cosh ^2(x)\right )-\frac{1}{5} \left (4 \left (1+\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-10+2 \sqrt{5}+16 x} \, dx,x,\cosh ^2(x)\right )\\ &=\frac{1}{5} \log (\cosh (x))-\frac{1}{20} \left (1+\sqrt{5}\right ) \log \left (5-\sqrt{5}-8 \cosh ^2(x)\right )-\frac{1}{20} \left (1-\sqrt{5}\right ) \log \left (5+\sqrt{5}-8 \cosh ^2(x)\right )\\ \end{align*}
Mathematica [A] time = 0.083173, size = 57, normalized size = 0.92 \[ \frac{1}{20} \left (\left (\sqrt{5}-1\right ) \log \left (8 \sinh ^2(x)-\sqrt{5}+3\right )-\left (1+\sqrt{5}\right ) \log \left (8 \sinh ^2(x)+\sqrt{5}+3\right )+4 \log (\cosh (x))\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.058, size = 101, normalized size = 1.6 \begin{align*}{\frac{\ln \left ({{\rm e}^{2\,x}}+1 \right ) }{5}}-{\frac{\ln \left ({{\rm e}^{4\,x}}+ \left ( -{\frac{1}{2}}-{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{2\,x}}+1 \right ) }{20}}+{\frac{\ln \left ({{\rm e}^{4\,x}}+ \left ( -{\frac{1}{2}}-{\frac{\sqrt{5}}{2}} \right ){{\rm e}^{2\,x}}+1 \right ) \sqrt{5}}{20}}-{\frac{\ln \left ({{\rm e}^{4\,x}}+ \left ({\frac{\sqrt{5}}{2}}-{\frac{1}{2}} \right ){{\rm e}^{2\,x}}+1 \right ) }{20}}-{\frac{\ln \left ({{\rm e}^{4\,x}}+ \left ({\frac{\sqrt{5}}{2}}-{\frac{1}{2}} \right ){{\rm e}^{2\,x}}+1 \right ) \sqrt{5}}{20}} \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{2}{5} \, \int \frac{{\left (e^{\left (6 \, x\right )} - e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )} - 1\right )} e^{\left (2 \, x\right )}}{e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} + \frac{2}{5} \, \int \frac{e^{\left (6 \, x\right )}}{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}{5} \, \int \frac{e^{\left (4 \, x\right )}}{e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} - \frac{4}{5} \, \int \frac{e^{\left (2 \, x\right )}}{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}{5} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.19283, size = 583, normalized size = 9.4 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (\frac{4 \, \cosh \left (x\right )^{4} + 4 \, \sinh \left (x\right )^{4} - 4 \,{\left (\sqrt{5} + 1\right )} \cosh \left (x\right )^{2} + 4 \,{\left (6 \, \cosh \left (x\right )^{2} - \sqrt{5} - 1\right )} \sinh \left (x\right )^{2} + \sqrt{5} + 7}{2 \, \cosh \left (x\right )^{4} + 2 \, \sinh \left (x\right )^{4} + 2 \,{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 1}\right ) - \frac{1}{20} \, \log \left (\frac{2 \, \cosh \left (x\right )^{4} + 2 \, \sinh \left (x\right )^{4} + 2 \,{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 1}{\cosh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}}\right ) + \frac{1}{5} \, \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \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 \sinh{\left (x \right )} \operatorname{sech}{\left (5 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20215, size = 159, normalized size = 2.56 \begin{align*} \frac{1}{20} \,{\left (\sqrt{5} - 1\right )} \log \left (\frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{20} \,{\left (\sqrt{5} - 1\right )} \log \left (-\frac{1}{2} \, \sqrt{2 \, \sqrt{5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{20} \,{\left (\sqrt{5} + 1\right )} \log \left (\frac{1}{2} \, \sqrt{-2 \, \sqrt{5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{20} \,{\left (\sqrt{5} + 1\right )} \log \left (-\frac{1}{2} \, \sqrt{-2 \, \sqrt{5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{5} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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