Optimal. Leaf size=95 \[ e^x+\frac{\log \left (-\sqrt{2} e^x+e^{2 x}+1\right )}{2 \sqrt{2}}-\frac{\log \left (\sqrt{2} e^x+e^{2 x}+1\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (1-\sqrt{2} e^x\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} e^x+1\right )}{\sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0620859, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {2282, 388, 211, 1165, 628, 1162, 617, 204} \[ e^x+\frac{\log \left (-\sqrt{2} e^x+e^{2 x}+1\right )}{2 \sqrt{2}}-\frac{\log \left (\sqrt{2} e^x+e^{2 x}+1\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (1-\sqrt{2} e^x\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} e^x+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 388
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int e^x \tanh (2 x) \, dx &=\operatorname{Subst}\left (\int \frac{-1+x^4}{1+x^4} \, dx,x,e^x\right )\\ &=e^x-2 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,e^x\right )\\ &=e^x-\operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,e^x\right )-\operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,e^x\right )\\ &=e^x-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,e^x\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,e^x\right )+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,e^x\right )}{2 \sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,e^x\right )}{2 \sqrt{2}}\\ &=e^x+\frac{\log \left (1-\sqrt{2} e^x+e^{2 x}\right )}{2 \sqrt{2}}-\frac{\log \left (1+\sqrt{2} e^x+e^{2 x}\right )}{2 \sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} e^x\right )}{\sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} e^x\right )}{\sqrt{2}}\\ &=e^x+\frac{\tan ^{-1}\left (1-\sqrt{2} e^x\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (1+\sqrt{2} e^x\right )}{\sqrt{2}}+\frac{\log \left (1-\sqrt{2} e^x+e^{2 x}\right )}{2 \sqrt{2}}-\frac{\log \left (1+\sqrt{2} e^x+e^{2 x}\right )}{2 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0297731, size = 95, normalized size = 1. \[ e^x+\frac{\log \left (-\sqrt{2} e^x+e^{2 x}+1\right )}{2 \sqrt{2}}-\frac{\log \left (\sqrt{2} e^x+e^{2 x}+1\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (1-\sqrt{2} e^x\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} e^x+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.052, size = 24, normalized size = 0.3 \begin{align*}{{\rm e}^{x}}+\sum _{{\it \_R}={\it RootOf} \left ( 16\,{{\it \_Z}}^{4}+1 \right ) }{\it \_R}\,\ln \left ({{\rm e}^{x}}-2\,{\it \_R} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.5883, size = 105, normalized size = 1.11 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) - \frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.43529, size = 360, normalized size = 3.79 \begin{align*} \sqrt{2} \arctan \left (-\sqrt{2} e^{x} + \sqrt{2} \sqrt{\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1} - 1\right ) + \sqrt{2} \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 ) - \frac{1}{4} \, \sqrt{2} \log \left (4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-4 \, \sqrt{2} e^{x} + 4 \, e^{\left (2 \, x\right )} + 4\right ) + e^{x} \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{\left (2 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27455, size = 105, normalized size = 1.11 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, e^{x}\right )}\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, e^{x}\right )}\right ) - \frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]