Optimal. Leaf size=69 \[ \cosh (x)-\frac{1}{4} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{2}}}\right )-\frac{1}{4} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{2}}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0867444, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {12, 1279, 1166, 207} \[ \cosh (x)-\frac{1}{4} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{2}}}\right )-\frac{1}{4} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{2}}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 1279
Rule 1166
Rule 207
Rubi steps
\begin{align*} \int \cosh (x) \tanh (4 x) \, dx &=\operatorname{Subst}\left (\int \frac{4 x^2 \left (-1+2 x^2\right )}{1-8 x^2+8 x^4} \, dx,x,\cosh (x)\right )\\ &=4 \operatorname{Subst}\left (\int \frac{x^2 \left (-1+2 x^2\right )}{1-8 x^2+8 x^4} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{2-8 x^2}{1-8 x^2+8 x^4} \, dx,x,\cosh (x)\right )\\ &=\cosh (x)-\left (-2+\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{-4+2 \sqrt{2}+8 x^2} \, dx,x,\cosh (x)\right )+\left (2+\sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{-4-2 \sqrt{2}+8 x^2} \, dx,x,\cosh (x)\right )\\ &=-\frac{1}{4} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2-\sqrt{2}}}\right )-\frac{1}{4} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left (\frac{2 \cosh (x)}{\sqrt{2+\sqrt{2}}}\right )+\cosh (x)\\ \end{align*}
Mathematica [C] time = 0.0229881, size = 113, normalized size = 1.64 \[ \frac{1}{16} \text{RootSum}\left [\text{$\#$1}^8+1\& ,\frac{\text{$\#$1}^6 x+2 \text{$\#$1}^6 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-2 \log \left (-\text{$\#$1} \sinh \left (\frac{x}{2}\right )+\text{$\#$1} \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )-\cosh \left (\frac{x}{2}\right )\right )-x}{\text{$\#$1}^7}\& \right ]+\cosh (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.054, size = 66, normalized size = 1. \begin{align*} \cosh \left ( x \right ) -{\frac{ \left ( \sqrt{2}-1 \right ) \sqrt{2}}{4\,\sqrt{2-\sqrt{2}}}{\it Artanh} \left ( 2\,{\frac{\cosh \left ( x \right ) }{\sqrt{2-\sqrt{2}}}} \right ) }-{\frac{ \left ( 1+\sqrt{2} \right ) \sqrt{2}}{4\,\sqrt{2+\sqrt{2}}}{\it Artanh} \left ( 2\,{\frac{\cosh \left ( x \right ) }{\sqrt{2+\sqrt{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{x}\right )}}{e^{\left (8 \, x\right )} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (x \right )} \tanh{\left (4 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.23464, size = 161, normalized size = 2.33 \begin{align*} -\frac{1}{8} \, \sqrt{\sqrt{2} + 2} \log \left (\sqrt{\sqrt{2} + 2} + e^{\left (-x\right )} + e^{x}\right ) + \frac{1}{8} \, \sqrt{\sqrt{2} + 2} \log \left (-\sqrt{\sqrt{2} + 2} + e^{\left (-x\right )} + e^{x}\right ) - \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \log \left (\sqrt{-\sqrt{2} + 2} + e^{\left (-x\right )} + e^{x}\right ) + \frac{1}{8} \, \sqrt{-\sqrt{2} + 2} \log \left (-\sqrt{-\sqrt{2} + 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.
[In]
[Out]