Integrand size = 10, antiderivative size = 113 \[ \int e^x \tanh ^2(2 x) \, dx=e^x+\frac {e^x}{1+e^{4 x}}+\frac {\arctan \left (1-\sqrt {2} e^x\right )}{2 \sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} e^x\right )}{2 \sqrt {2}}+\frac {\log \left (1-\sqrt {2} e^x+e^{2 x}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} e^x+e^{2 x}\right )}{4 \sqrt {2}} \]
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Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {2320, 398, 294, 217, 1179, 642, 1176, 631, 210} \[ \int e^x \tanh ^2(2 x) \, dx=\frac {\arctan \left (1-\sqrt {2} e^x\right )}{2 \sqrt {2}}-\frac {\arctan \left (\sqrt {2} e^x+1\right )}{2 \sqrt {2}}+e^x+\frac {e^x}{e^{4 x}+1}+\frac {\log \left (-\sqrt {2} e^x+e^{2 x}+1\right )}{4 \sqrt {2}}-\frac {\log \left (\sqrt {2} e^x+e^{2 x}+1\right )}{4 \sqrt {2}} \]
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Rule 210
Rule 217
Rule 294
Rule 398
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \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 {\arctan \left (1-\sqrt {2} e^x\right )}{2 \sqrt {2}}-\frac {\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}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.42 \[ \int e^x \tanh ^2(2 x) \, dx=e^x+\frac {e^x}{1+e^{4 x}}+\frac {1}{4} \text {RootSum}\left [1+\text {$\#$1}^4\&,\frac {x-\log \left (e^x-\text {$\#$1}\right )}{\text {$\#$1}^3}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.31
method | result | size |
risch | \({\mathrm e}^{x}+\frac {{\mathrm e}^{x}}{1+{\mathrm e}^{4 x}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (256 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{x}-4 \textit {\_R} \right )\right )\) | \(35\) |
default | \(-\frac {2}{\tanh \left (\frac {x}{2}\right )-1}-\frac {2 \left (-\frac {\tanh \left (\frac {x}{2}\right )^{3}}{2}-\frac {3 \tanh \left (\frac {x}{2}\right )^{2}}{2}+\frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {1}{2}\right )}{\tanh \left (\frac {x}{2}\right )^{4}+6 \tanh \left (\frac {x}{2}\right )^{2}+1}+\frac {\sqrt {2}\, \ln \left (\tanh \left (\frac {x}{2}\right )^{2}+3-2 \sqrt {2}\right )}{8}+\frac {\left (\sqrt {2}-2\right ) \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{2 \sqrt {2}-2}\right )}{4 \sqrt {2}-4}-\frac {\sqrt {2}\, \ln \left (\tanh \left (\frac {x}{2}\right )^{2}+3+2 \sqrt {2}\right )}{8}-\frac {\left (2+\sqrt {2}\right ) \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right )}{2+2 \sqrt {2}}\right )}{2 \left (2+2 \sqrt {2}\right )}\) | \(158\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 397, normalized size of antiderivative = 3.51 \[ \int e^x \tanh ^2(2 x) \, dx=\frac {8 \, \cosh \left (x\right )^{5} + 80 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 80 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 40 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + 8 \, \sinh \left (x\right )^{5} + {\left (-\left (i + 1\right ) \, \sqrt {2} \cosh \left (x\right )^{4} - \left (4 i + 4\right ) \, \sqrt {2} \cosh \left (x\right )^{3} \sinh \left (x\right ) - \left (6 i + 6\right ) \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} - \left (4 i + 4\right ) \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} - \left (i + 1\right ) \, \sqrt {2} \sinh \left (x\right )^{4} - \left (i + 1\right ) \, \sqrt {2}\right )} \log \left (\left (i + 1\right ) \, \sqrt {2} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right ) + {\left (\left (i - 1\right ) \, \sqrt {2} \cosh \left (x\right )^{4} + \left (4 i - 4\right ) \, \sqrt {2} \cosh \left (x\right )^{3} \sinh \left (x\right ) + \left (6 i - 6\right ) \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + \left (4 i - 4\right ) \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \left (i - 1\right ) \, \sqrt {2} \sinh \left (x\right )^{4} + \left (i - 1\right ) \, \sqrt {2}\right )} \log \left (-\left (i - 1\right ) \, \sqrt {2} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right ) + {\left (-\left (i - 1\right ) \, \sqrt {2} \cosh \left (x\right )^{4} - \left (4 i - 4\right ) \, \sqrt {2} \cosh \left (x\right )^{3} \sinh \left (x\right ) - \left (6 i - 6\right ) \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} - \left (4 i - 4\right ) \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} - \left (i - 1\right ) \, \sqrt {2} \sinh \left (x\right )^{4} - \left (i - 1\right ) \, \sqrt {2}\right )} \log \left (\left (i - 1\right ) \, \sqrt {2} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \cosh \left (x\right )^{4} + \left (4 i + 4\right ) \, \sqrt {2} \cosh \left (x\right )^{3} \sinh \left (x\right ) + \left (6 i + 6\right ) \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + \left (4 i + 4\right ) \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \left (i + 1\right ) \, \sqrt {2} \sinh \left (x\right )^{4} + \left (i + 1\right ) \, \sqrt {2}\right )} \log \left (-\left (i + 1\right ) \, \sqrt {2} + 2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right )\right ) + 8 \, {\left (5 \, \cosh \left (x\right )^{4} + 2\right )} \sinh \left (x\right ) + 16 \, \cosh \left (x\right )}{8 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 1\right )}} \]
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\[ \int e^x \tanh ^2(2 x) \, dx=\int e^{x} \tanh ^{2}{\left (2 x \right )}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int e^x \tanh ^2(2 x) \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {e^{x}}{e^{\left (4 \, x\right )} + 1} + e^{x} \]
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Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int e^x \tanh ^2(2 x) \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, e^{x}\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, e^{x}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} e^{x} + e^{\left (2 \, x\right )} + 1\right ) + \frac {e^{x}}{e^{\left (4 \, x\right )} + 1} + e^{x} \]
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Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.76 \[ \int e^x \tanh ^2(2 x) \, dx={\mathrm {e}}^x+\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{4\,x}+1}-\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\left ({\mathrm {e}}^x-\frac {\sqrt {2}}{2}\right )\right )}{4}-\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\left ({\mathrm {e}}^x+\frac {\sqrt {2}}{2}\right )\right )}{4}+\frac {\sqrt {2}\,\ln \left ({\left ({\mathrm {e}}^x-\frac {\sqrt {2}}{2}\right )}^2+\frac {1}{2}\right )}{8}-\frac {\sqrt {2}\,\ln \left ({\left ({\mathrm {e}}^x+\frac {\sqrt {2}}{2}\right )}^2+\frac {1}{2}\right )}{8} \]
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