Optimal. Leaf size=62 \[ -\frac {1}{20} \left (1+\sqrt {5}\right ) \log \left (-8 \cosh ^2(x)-\sqrt {5}+5\right )-\frac {1}{20} \left (1-\sqrt {5}\right ) \log \left (-8 \cosh ^2(x)+\sqrt {5}+5\right )+\frac {1}{5} \log (\cosh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 62, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 29
Rule 31
Rule 632
Rule 705
Rule 1114
Rule 4357
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 = 0.08, 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.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.44, size = 182, normalized size = 2.94 \[ \frac {1}{20} \, \sqrt {5} \log \left (\frac {4 \, \cosh \relax (x)^{4} + 4 \, \sinh \relax (x)^{4} - 4 \, {\left (\sqrt {5} + 1\right )} \cosh \relax (x)^{2} + 4 \, {\left (6 \, \cosh \relax (x)^{2} - \sqrt {5} - 1\right )} \sinh \relax (x)^{2} + \sqrt {5} + 7}{2 \, \cosh \relax (x)^{4} + 2 \, \sinh \relax (x)^{4} + 2 \, {\left (6 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 1}\right ) - \frac {1}{20} \, \log \left (\frac {2 \, \cosh \relax (x)^{4} + 2 \, \sinh \relax (x)^{4} + 2 \, {\left (6 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 1}{\cosh \relax (x)^{4} - 4 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} - 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4}}\right ) + \frac {1}{5} \, \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.16, size = 118, normalized size = 1.90 \[ \frac {1}{20} \, {\left (\sqrt {5} - 1\right )} \log \left (\frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{20} \, {\left (\sqrt {5} - 1\right )} \log \left (-\frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{20} \, {\left (\sqrt {5} + 1\right )} \log \left (\frac {1}{2} \, \sqrt {-2 \, \sqrt {5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{20} \, {\left (\sqrt {5} + 1\right )} \log \left (-\frac {1}{2} \, \sqrt {-2 \, \sqrt {5} + 10} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{5} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.24, size = 101, normalized size = 1.63 \[ \frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{5}-\frac {\ln \left ({\mathrm e}^{4 x}+\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{2 x}+1\right )}{20}+\frac {\ln \left ({\mathrm e}^{4 x}+\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) {\mathrm e}^{2 x}+1\right ) \sqrt {5}}{20}-\frac {\ln \left ({\mathrm e}^{4 x}+\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) {\mathrm e}^{2 x}+1\right )}{20}-\frac {\ln \left ({\mathrm e}^{4 x}+\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) {\mathrm e}^{2 x}+1\right ) \sqrt {5}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2}{5} \, \int \frac {{\left (e^{\left (6 \, x\right )} - e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )} - 1\right )} e^{\left (2 \, x\right )}}{e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} + \frac {2}{5} \, \int \frac {e^{\left (6 \, x\right )}}{e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} + \frac {1}{5} \, \int \frac {e^{\left (4 \, x\right )}}{e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} - \frac {4}{5} \, \int \frac {e^{\left (2 \, x\right )}}{e^{\left (8 \, x\right )} - e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )} + 1}\,{d x} + \frac {1}{5} \, \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.52, size = 100, normalized size = 1.61 \[ \frac {\ln \left (5\,{\mathrm {e}}^{2\,x}+5\right )}{5}-\ln \left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right )\,\left (20\,{\mathrm {e}}^{2\,x}+30\,{\mathrm {e}}^{4\,x}+30\right )+2\right )\,\left (\frac {\sqrt {5}}{20}+\frac {1}{20}\right )+\ln \left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}-\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right )\,\left (20\,{\mathrm {e}}^{2\,x}+30\,{\mathrm {e}}^{4\,x}+30\right )+2\right )\,\left (\frac {\sqrt {5}}{20}-\frac {1}{20}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh {\relax (x )} \operatorname {sech}{\left (5 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________