Optimal. Leaf size=82 \[ \cosh (x)-\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tanh ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} \cosh (x)\right )-\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} \cosh (x)\right ) \]
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Rubi [A] time = 0.116861, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {1676, 1166, 207} \[ \cosh (x)-\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tanh ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} \cosh (x)\right )-\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} \cosh (x)\right ) \]
Antiderivative was successfully verified.
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Rule 1676
Rule 1166
Rule 207
Rubi steps
\begin{align*} \int \cosh (x) \tanh (5 x) \, dx &=\operatorname{Subst}\left (\int \frac{1-12 x^2+16 x^4}{5-20 x^2+16 x^4} \, dx,x,\cosh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (1-\frac{4 \left (1-2 x^2\right )}{5-20 x^2+16 x^4}\right ) \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-4 \operatorname{Subst}\left (\int \frac{1-2 x^2}{5-20 x^2+16 x^4} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)+\frac{1}{5} \left (4 \left (5-\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-10+2 \sqrt{5}+16 x^2} \, dx,x,\cosh (x)\right )+\frac{1}{5} \left (4 \left (5+\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-10-2 \sqrt{5}+16 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{1}{5} \sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tanh ^{-1}\left (2 \sqrt{\frac{2}{5+\sqrt{5}}} \cosh (x)\right )-\frac{1}{5} \sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{5} \left (5+\sqrt{5}\right )} \cosh (x)\right )+\cosh (x)\\ \end{align*}
Mathematica [C] time = 0.0275745, size = 249, normalized size = 3.04 \[ \frac{1}{4} \text{RootSum}\left [\text{$\#$1}^8-\text{$\#$1}^6+\text{$\#$1}^4-\text{$\#$1}^2+1\& ,\frac{\text{$\#$1}^6 x-\text{$\#$1}^4 x+\text{$\#$1}^2 x+2 \text{$\#$1}^6 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-2 \text{$\#$1}^4 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )+2 \text{$\#$1}^2 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-2 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-x}{4 \text{$\#$1}^7-3 \text{$\#$1}^5+2 \text{$\#$1}^3-\text{$\#$1}}\& \right ]+\cosh (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 70, normalized size = 0.9 \begin{align*} \cosh \left ( x \right ) -{\frac{ \left ( \sqrt{5}-1 \right ) \sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}{\it Artanh} \left ( 4\,{\frac{\cosh \left ( x \right ) }{\sqrt{10-2\,\sqrt{5}}}} \right ) }-{\frac{\sqrt{5} \left ( \sqrt{5}+1 \right ) }{5\,\sqrt{10+2\,\sqrt{5}}}{\it Artanh} \left ( 4\,{\frac{\cosh \left ( x \right ) }{\sqrt{10+2\,\sqrt{5}}}} \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{1}{2} \,{\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} + \frac{1}{2} \, \int \frac{2 \,{\left (e^{\left (7 \, x\right )} - e^{\left (5 \, x\right )} + e^{\left (3 \, x\right )} - e^{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.15884, size = 1019, normalized size = 12.43 \begin{align*} -\frac{{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{\sqrt{5} + 5} \log \left (2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} +{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{\sqrt{5} + 5} + 2\right ) -{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{\sqrt{5} + 5} \log \left (2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} -{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{\sqrt{5} + 5} + 2\right ) +{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{-\sqrt{5} + 5} \log \left (2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} +{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{-\sqrt{5} + 5} + 2\right ) -{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{-\sqrt{5} + 5} \log \left (2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} -{\left (\sqrt{2} \cosh \left (x\right ) + \sqrt{2} \sinh \left (x\right )\right )} \sqrt{-\sqrt{5} + 5} + 2\right ) - 10 \, \cosh \left (x\right )^{2} - 20 \, \cosh \left (x\right ) \sinh \left (x\right ) - 10 \, \sinh \left (x\right )^{2} - 10}{20 \,{\left (\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 \cosh{\left (x \right )} \tanh{\left (5 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23672, size = 171, normalized size = 2.09 \begin{align*} -\frac{1}{20} \, \sqrt{2 \, \sqrt{5} + 10} \log \left (\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{5}{2}} + e^{\left (-x\right )} + e^{x}\right ) + \frac{1}{20} \, \sqrt{2 \, \sqrt{5} + 10} \log \left (-\sqrt{\frac{1}{2} \, \sqrt{5} + \frac{5}{2}} + e^{\left (-x\right )} + e^{x}\right ) - \frac{1}{20} \, \sqrt{-2 \, \sqrt{5} + 10} \log \left (\sqrt{-\frac{1}{2} \, \sqrt{5} + \frac{5}{2}} + e^{\left (-x\right )} + e^{x}\right ) + \frac{1}{20} \, \sqrt{-2 \, \sqrt{5} + 10} \log \left (-\sqrt{-\frac{1}{2} \, \sqrt{5} + \frac{5}{2}} + e^{\left (-x\right )} + e^{x}\right ) + \frac{1}{2} \, e^{\left (-x\right )} + \frac{1}{2} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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