Optimal. Leaf size=35 \[ e^x+\frac {e^x}{1-e^{4 x}}-\frac {1}{2} \tan ^{-1}\left (e^x\right )-\frac {1}{2} \tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2282, 390, 288, 212, 206, 203} \[ e^x+\frac {e^x}{1-e^{4 x}}-\frac {1}{2} \tan ^{-1}\left (e^x\right )-\frac {1}{2} \tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 288
Rule 390
Rule 2282
Rubi steps
\begin {align*} \int e^x \coth ^2(2 x) \, dx &=\operatorname {Subst}\left (\int \frac {\left (1+x^4\right )^2}{\left (1-x^4\right )^2} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (1+\frac {4 x^4}{\left (1-x^4\right )^2}\right ) \, dx,x,e^x\right )\\ &=e^x+4 \operatorname {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}}-\operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^x\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )\\ &=e^x+\frac {e^x}{1-e^{4 x}}-\frac {1}{2} \tan ^{-1}\left (e^x\right )-\frac {1}{2} \tanh ^{-1}\left (e^x\right )\\ \end {align*}
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Mathematica [C] time = 1.62, size = 113, normalized size = 3.23 \[ \frac {16}{585} e^{5 x} \left (e^{4 x}+1\right )^2 \, _4F_3\left (\frac {5}{4},2,2,2;1,1,\frac {17}{4};e^{4 x}\right )+\frac {1}{640} e^{-7 x} \left (5 \left (1208 e^{4 x}+102 e^{8 x}-248 e^{12 x}+e^{16 x}+729\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{4 x}\right )-6769 e^{4 x}-1483 e^{8 x}+681 e^{12 x}-3645\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.53, size = 230, normalized size = 6.57 \[ \frac {4 \, \cosh \relax (x)^{5} + 40 \, \cosh \relax (x)^{3} \sinh \relax (x)^{2} + 40 \, \cosh \relax (x)^{2} \sinh \relax (x)^{3} + 20 \, \cosh \relax (x) \sinh \relax (x)^{4} + 4 \, \sinh \relax (x)^{5} - 2 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) - {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 4 \, {\left (5 \, \cosh \relax (x)^{4} - 2\right )} \sinh \relax (x) - 8 \, \cosh \relax (x)}{4 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x)^{3} \sinh \relax (x) + 6 \, \cosh \relax (x)^{2} \sinh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 35, normalized size = 1.00 \[ -\frac {e^{x}}{e^{\left (4 \, x\right )} - 1} - \frac {1}{2} \, \arctan \left (e^{x}\right ) + e^{x} - \frac {1}{4} \, \log \left (e^{x} + 1\right ) + \frac {1}{4} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 48, normalized size = 1.37 \[ {\mathrm e}^{x}-\frac {{\mathrm e}^{x}}{{\mathrm e}^{4 x}-1}+\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{4}-\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{4}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{4}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 34, normalized size = 0.97 \[ -\frac {e^{x}}{e^{\left (4 \, x\right )} - 1} - \frac {1}{2} \, \arctan \left (e^{x}\right ) + e^{x} - \frac {1}{4} \, \log \left (e^{x} + 1\right ) + \frac {1}{4} \, \log \left (e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 38, normalized size = 1.09 \[ \frac {\ln \left (1-{\mathrm {e}}^x\right )}{4}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{4}-\frac {\mathrm {atan}\left ({\mathrm {e}}^x\right )}{2}+{\mathrm {e}}^x-\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{4\,x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{x} \coth ^{2}{\left (2 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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