Optimal. Leaf size=130 \[ -\frac{3 e^{3 x}}{4 \left (e^{4 x}+1\right )}+\frac{e^{3 x}}{\left (e^{4 x}+1\right )^2}+\frac{5 \log \left (-\sqrt{2} e^x+e^{2 x}+1\right )}{16 \sqrt{2}}-\frac{5 \log \left (\sqrt{2} e^x+e^{2 x}+1\right )}{16 \sqrt{2}}-\frac{5 \tan ^{-1}\left (1-\sqrt{2} e^x\right )}{8 \sqrt{2}}+\frac{5 \tan ^{-1}\left (\sqrt{2} e^x+1\right )}{8 \sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.106234, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {2282, 12, 463, 457, 297, 1162, 617, 204, 1165, 628} \[ -\frac{3 e^{3 x}}{4 \left (e^{4 x}+1\right )}+\frac{e^{3 x}}{\left (e^{4 x}+1\right )^2}+\frac{5 \log \left (-\sqrt{2} e^x+e^{2 x}+1\right )}{16 \sqrt{2}}-\frac{5 \log \left (\sqrt{2} e^x+e^{2 x}+1\right )}{16 \sqrt{2}}-\frac{5 \tan ^{-1}\left (1-\sqrt{2} e^x\right )}{8 \sqrt{2}}+\frac{5 \tan ^{-1}\left (\sqrt{2} e^x+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 12
Rule 463
Rule 457
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int e^x \text{sech}(2 x) \tanh ^2(2 x) \, dx &=\operatorname{Subst}\left (\int \frac{2 x^2 \left (1-x^4\right )^2}{\left (1+x^4\right )^3} \, dx,x,e^x\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x^2 \left (1-x^4\right )^2}{\left (1+x^4\right )^3} \, dx,x,e^x\right )\\ &=\frac{e^{3 x}}{\left (1+e^{4 x}\right )^2}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2 \left (4-8 x^4\right )}{\left (1+x^4\right )^2} \, dx,x,e^x\right )\\ &=\frac{e^{3 x}}{\left (1+e^{4 x}\right )^2}-\frac{3 e^{3 x}}{4 \left (1+e^{4 x}\right )}+\frac{5}{4} \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,e^x\right )\\ &=\frac{e^{3 x}}{\left (1+e^{4 x}\right )^2}-\frac{3 e^{3 x}}{4 \left (1+e^{4 x}\right )}-\frac{5}{8} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,e^x\right )+\frac{5}{8} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,e^x\right )\\ &=\frac{e^{3 x}}{\left (1+e^{4 x}\right )^2}-\frac{3 e^{3 x}}{4 \left (1+e^{4 x}\right )}+\frac{5}{16} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,e^x\right )+\frac{5}{16} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,e^x\right )+\frac{5 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,e^x\right )}{16 \sqrt{2}}+\frac{5 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,e^x\right )}{16 \sqrt{2}}\\ &=\frac{e^{3 x}}{\left (1+e^{4 x}\right )^2}-\frac{3 e^{3 x}}{4 \left (1+e^{4 x}\right )}+\frac{5 \log \left (1-\sqrt{2} e^x+e^{2 x}\right )}{16 \sqrt{2}}-\frac{5 \log \left (1+\sqrt{2} e^x+e^{2 x}\right )}{16 \sqrt{2}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} e^x\right )}{8 \sqrt{2}}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} e^x\right )}{8 \sqrt{2}}\\ &=\frac{e^{3 x}}{\left (1+e^{4 x}\right )^2}-\frac{3 e^{3 x}}{4 \left (1+e^{4 x}\right )}-\frac{5 \tan ^{-1}\left (1-\sqrt{2} e^x\right )}{8 \sqrt{2}}+\frac{5 \tan ^{-1}\left (1+\sqrt{2} e^x\right )}{8 \sqrt{2}}+\frac{5 \log \left (1-\sqrt{2} e^x+e^{2 x}\right )}{16 \sqrt{2}}-\frac{5 \log \left (1+\sqrt{2} e^x+e^{2 x}\right )}{16 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0603184, size = 58, normalized size = 0.45 \[ \frac{e^{3 x}-3 e^{7 x}}{4 \left (e^{4 x}+1\right )^2}-\frac{5}{16} \text{RootSum}\left [\text{$\#$1}^4+1\& ,\frac{x-\log \left (e^x-\text{$\#$1}\right )}{\text{$\#$1}}\& \right ] \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.061, size = 48, normalized size = 0.4 \begin{align*} -{\frac{{{\rm e}^{3\,x}} \left ( 3\,{{\rm e}^{4\,x}}-1 \right ) }{4\, \left ( 1+{{\rm e}^{4\,x}} \right ) ^{2}}}+2\,\sum _{{\it \_R}={\it RootOf} \left ( 1048576\,{{\it \_Z}}^{4}+625 \right ) }{\it \_R}\,\ln \left ({{\rm e}^{x}}+{\frac{32768\,{{\it \_R}}^{3}}{125}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.623, size = 142, normalized size = 1.09 \begin{align*} \frac{5}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) + \frac{5}{16} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) - \frac{5}{32} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{5}{32} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{3 \, e^{\left (7 \, x\right )} - e^{\left (3 \, x\right )}}{4 \,{\left (e^{\left (8 \, x\right )} + 2 \, e^{\left (4 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.10761, size = 651, normalized size = 5.01 \begin{align*} -\frac{20 \,{\left (\sqrt{2} e^{\left (8 \, x\right )} + 2 \, \sqrt{2} e^{\left (4 \, x\right )} + \sqrt{2}\right )} \arctan \left (-\sqrt{2} e^{x} + \sqrt{2} \sqrt{\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1} - 1\right ) + 20 \,{\left (\sqrt{2} e^{\left (8 \, x\right )} + 2 \, \sqrt{2} e^{\left (4 \, x\right )} + \sqrt{2}\right )} \arctan \left (-\sqrt{2} e^{x} + \frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4} + 1\right ) + 5 \,{\left (\sqrt{2} e^{\left (8 \, x\right )} + 2 \, \sqrt{2} e^{\left (4 \, x\right )} + \sqrt{2}\right )} \log \left (4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) - 5 \,{\left (\sqrt{2} e^{\left (8 \, x\right )} + 2 \, \sqrt{2} e^{\left (4 \, x\right )} + \sqrt{2}\right )} \log \left (-4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + 24 \, e^{\left (7 \, x\right )} - 8 \, e^{\left (3 \, x\right )}}{32 \,{\left (e^{\left (8 \, x\right )} + 2 \, e^{\left (4 \, x\right )} + 1\right )}} \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 e^{x} \tanh ^{2}{\left (2 x \right )} \operatorname{sech}{\left (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15151, size = 134, normalized size = 1.03 \begin{align*} \frac{5}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) + \frac{5}{16} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) - \frac{5}{32} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{5}{32} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) - \frac{3 \, e^{\left (7 \, x\right )} - e^{\left (3 \, x\right )}}{4 \,{\left (e^{\left (4 \, x\right )} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]