Optimal. Leaf size=10 \[ -\sinh (x)+\cosh (x)+\tan ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.11, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3518, 3108, 3107, 2638, 2592, 321, 203} \[ -\sinh (x)+\cosh (x)+\tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 203
Rule 321
Rule 2592
Rule 2638
Rule 3107
Rule 3108
Rule 3518
Rubi steps
\begin {align*} \int \frac {\text {sech}(x)}{1+\coth (x)} \, dx &=-\left (i \int \frac {\tanh (x)}{-i \cosh (x)-i \sinh (x)} \, dx\right )\\ &=-\int (-\cosh (x)+\sinh (x)) \tanh (x) \, dx\\ &=i \int (-i \sinh (x)+i \sinh (x) \tanh (x)) \, dx\\ &=\int \sinh (x) \, dx-\int \sinh (x) \tanh (x) \, dx\\ &=\cosh (x)-\operatorname {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\sinh (x)\right )\\ &=\cosh (x)-\sinh (x)+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (x)\right )\\ &=\tan ^{-1}(\sinh (x))+\cosh (x)-\sinh (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 16, normalized size = 1.60 \[ -\sinh (x)+\cosh (x)+2 \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 23, normalized size = 2.30 \[ \frac {2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + 1}{\cosh \relax (x) + \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 10, normalized size = 1.00 \[ 2 \, \arctan \left (e^{x}\right ) + e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 19, normalized size = 1.90 \[ \frac {2}{\tanh \left (\frac {x}{2}\right )+1}+2 \arctan \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 12, normalized size = 1.20 \[ -2 \, \arctan \left (e^{\left (-x\right )}\right ) + e^{\left (-x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 10, normalized size = 1.00 \[ {\mathrm {e}}^{-x}+2\,\mathrm {atan}\left ({\mathrm {e}}^x\right ) \]
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 \[ \int \frac {\operatorname {sech}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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