Integrand size = 10, antiderivative size = 35 \[ \int e^x \coth ^2(2 x) \, dx=e^x+\frac {e^x}{1-e^{4 x}}-\frac {\arctan \left (e^x\right )}{2}-\frac {\text {arctanh}\left (e^x\right )}{2} \]
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Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2320, 398, 294, 218, 212, 209} \[ \int e^x \coth ^2(2 x) \, dx=-\frac {1}{2} \arctan \left (e^x\right )-\frac {\text {arctanh}\left (e^x\right )}{2}+e^x+\frac {e^x}{1-e^{4 x}} \]
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Rule 209
Rule 212
Rule 218
Rule 294
Rule 398
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}{1-x^2} \, dx,x,e^x\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right ) \\ & = e^x+\frac {e^x}{1-e^{4 x}}-\frac {\arctan \left (e^x\right )}{2}-\frac {\text {arctanh}\left (e^x\right )}{2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.72 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.23 \[ \int e^x \coth ^2(2 x) \, dx=\frac {1}{640} e^{-7 x} \left (-3645-6769 e^{4 x}-1483 e^{8 x}+681 e^{12 x}+5 \left (729+1208 e^{4 x}+102 e^{8 x}-248 e^{12 x}+e^{16 x}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},e^{4 x}\right )\right )+\frac {16}{585} e^{5 x} \left (1+e^{4 x}\right )^2 \, _4F_3\left (\frac {5}{4},2,2,2;1,1,\frac {17}{4};e^{4 x}\right ) \]
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Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37
method | result | size |
risch | \({\mathrm e}^{x}-\frac {{\mathrm e}^{x}}{{\mathrm e}^{4 x}-1}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{4}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{4}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{4}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{4}\) | \(48\) |
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Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 230, normalized size of antiderivative = 6.57 \[ \int e^x \coth ^2(2 x) \, dx=\frac {4 \, \cosh \left (x\right )^{5} + 40 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 40 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 20 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + 4 \, \sinh \left (x\right )^{5} - 2 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 4 \, {\left (5 \, \cosh \left (x\right )^{4} - 2\right )} \sinh \left (x\right ) - 8 \, \cosh \left (x\right )}{4 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )}} \]
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\[ \int e^x \coth ^2(2 x) \, dx=\int e^{x} \coth ^{2}{\left (2 x \right )}\, dx \]
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none
Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int e^x \coth ^2(2 x) \, dx=-\frac {e^{x}}{e^{\left (4 \, x\right )} - 1} - \frac {1}{2} \, \arctan \left (e^{x}\right ) + e^{x} - \frac {1}{4} \, \log \left (e^{x} + 1\right ) + \frac {1}{4} \, \log \left (e^{x} - 1\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int e^x \coth ^2(2 x) \, dx=-\frac {e^{x}}{e^{\left (4 \, x\right )} - 1} - \frac {1}{2} \, \arctan \left (e^{x}\right ) + e^{x} - \frac {1}{4} \, \log \left (e^{x} + 1\right ) + \frac {1}{4} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \[ \int e^x \coth ^2(2 x) \, dx=\frac {\ln \left (1-{\mathrm {e}}^x\right )}{4}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{4}-\frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{2}+{\mathrm {e}}^x-\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{4\,x}-1} \]
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