Optimal. Leaf size=113 \[ e^x+\frac {e^x}{1+e^{4 x}}+\frac {\text {ArcTan}\left (1-\sqrt {2} e^x\right )}{2 \sqrt {2}}-\frac {\text {ArcTan}\left (1+\sqrt {2} e^x\right )}{2 \sqrt {2}}+\frac {\log \left (1-\sqrt {2} e^x+e^{2 x}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} e^x+e^{2 x}\right )}{4 \sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {2320, 398,
294, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {\text {ArcTan}\left (1-\sqrt {2} e^x\right )}{2 \sqrt {2}}-\frac {\text {ArcTan}\left (\sqrt {2} e^x+1\right )}{2 \sqrt {2}}+e^x+\frac {e^x}{e^{4 x}+1}+\frac {\log \left (-\sqrt {2} e^x+e^{2 x}+1\right )}{4 \sqrt {2}}-\frac {\log \left (\sqrt {2} e^x+e^{2 x}+1\right )}{4 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 217
Rule 294
Rule 398
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2320
Rubi steps
\begin {align*} \int e^x \tanh ^2(2 x) \, dx &=\text {Subst}\left (\int \frac {\left (1-x^4\right )^2}{\left (1+x^4\right )^2} \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \left (1-\frac {4 x^4}{\left (1+x^4\right )^2}\right ) \, dx,x,e^x\right )\\ &=e^x-4 \text {Subst}\left (\int \frac {x^4}{\left (1+x^4\right )^2} \, dx,x,e^x\right )\\ &=e^x+\frac {e^x}{1+e^{4 x}}-\text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,e^x\right )\\ &=e^x+\frac {e^x}{1+e^{4 x}}-\frac {1}{2} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,e^x\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,e^x\right )\\ &=e^x+\frac {e^x}{1+e^{4 x}}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,e^x\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,e^x\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,e^x\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,e^x\right )}{4 \sqrt {2}}\\ &=e^x+\frac {e^x}{1+e^{4 x}}+\frac {\log \left (1-\sqrt {2} e^x+e^{2 x}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} e^x+e^{2 x}\right )}{4 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} e^x\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} e^x\right )}{2 \sqrt {2}}\\ &=e^x+\frac {e^x}{1+e^{4 x}}+\frac {\tan ^{-1}\left (1-\sqrt {2} e^x\right )}{2 \sqrt {2}}-\frac {\tan ^{-1}\left (1+\sqrt {2} e^x\right )}{2 \sqrt {2}}+\frac {\log \left (1-\sqrt {2} e^x+e^{2 x}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} e^x+e^{2 x}\right )}{4 \sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.04, size = 48, normalized size = 0.42 \begin {gather*} e^x+\frac {e^x}{1+e^{4 x}}+\frac {1}{4} \text {RootSum}\left [1+\text {$\#$1}^4\&,\frac {x-\log \left (e^x-\text {$\#$1}\right )}{\text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.81, size = 35, normalized size = 0.31
method | result | size |
risch | \({\mathrm e}^{x}+\frac {{\mathrm e}^{x}}{1+{\mathrm e}^{4 x}}+\left (\munderset {\textit {\_R} =\RootOf \left (256 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}-4 \textit {\_R} \right )\right )\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.47, size = 89, normalized size = 0.79 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {e^{x}}{e^{\left (4 \, x\right )} + 1} + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 168 vs.
\(2 (79) = 158\).
time = 0.36, size = 168, normalized size = 1.49 \begin {gather*} \frac {4 \, {\left (\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 ) + 4 \, {\left (\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 ) - {\left (\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 ) + {\left (\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 ) + 8 \, e^{\left (5 \, x\right )} + 16 \, e^{x}}{8 \, {\left (e^{\left (4 \, x\right )} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{x} \tanh ^{2}{\left (2 x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 89, normalized size = 0.79 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {e^{x}}{e^{\left (4 \, x\right )} + 1} + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.25, size = 86, normalized size = 0.76 \begin {gather*} {\mathrm {e}}^x+\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{4\,x}+1}-\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\left ({\mathrm {e}}^x-\frac {\sqrt {2}}{2}\right )\right )}{4}-\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\left ({\mathrm {e}}^x+\frac {\sqrt {2}}{2}\right )\right )}{4}+\frac {\sqrt {2}\,\ln \left ({\left ({\mathrm {e}}^x-\frac {\sqrt {2}}{2}\right )}^2+\frac {1}{2}\right )}{8}-\frac {\sqrt {2}\,\ln \left ({\left ({\mathrm {e}}^x+\frac {\sqrt {2}}{2}\right )}^2+\frac {1}{2}\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________