Optimal. Leaf size=20 \[ -\text {sech}(x)-\frac {1}{2} \tan ^{-1}(\sinh (x))+\frac {1}{2} \tanh (x) \text {sech}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3518, 3108, 3107, 2606, 8, 2611, 3770} \[ -\text {sech}(x)-\frac {1}{2} \tan ^{-1}(\sinh (x))+\frac {1}{2} \tanh (x) \text {sech}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2606
Rule 2611
Rule 3107
Rule 3108
Rule 3518
Rule 3770
Rubi steps
\begin {align*} \int \frac {\text {sech}^3(x)}{1+\coth (x)} \, dx &=-\left (i \int \frac {\text {sech}^2(x) \tanh (x)}{-i \cosh (x)-i \sinh (x)} \, dx\right )\\ &=-\int \text {sech}^2(x) (-\cosh (x)+\sinh (x)) \tanh (x) \, dx\\ &=i \int \left (-i \text {sech}(x) \tanh (x)+i \text {sech}(x) \tanh ^2(x)\right ) \, dx\\ &=\int \text {sech}(x) \tanh (x) \, dx-\int \text {sech}(x) \tanh ^2(x) \, dx\\ &=\frac {1}{2} \text {sech}(x) \tanh (x)-\frac {1}{2} \int \text {sech}(x) \, dx-\operatorname {Subst}(\int 1 \, dx,x,\text {sech}(x))\\ &=-\frac {1}{2} \tan ^{-1}(\sinh (x))-\text {sech}(x)+\frac {1}{2} \text {sech}(x) \tanh (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 20, normalized size = 1.00 \[ \frac {1}{2} (\tanh (x)-2) \text {sech}(x)-\tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.38, size = 140, normalized size = 7.00 \[ -\frac {\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + 3 \, {\left (\cosh \relax (x)^{2} + 1\right )} \sinh \relax (x) + 3 \, \cosh \relax (x)}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 25, normalized size = 1.25 \[ -\frac {e^{\left (3 \, x\right )} + 3 \, e^{x}}{{\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} - \arctan \left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.11, size = 45, normalized size = 2.25 \[ \frac {-\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )-2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+\tanh \left (\frac {x}{2}\right )-2}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}-\arctan \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.42, size = 33, normalized size = 1.65 \[ -\frac {e^{\left (-x\right )} + 3 \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + \arctan \left (e^{\left (-x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.25, size = 22, normalized size = 1.10 \[ -\mathrm {atan}\left ({\mathrm {e}}^x\right )-\frac {1}{2\,\mathrm {cosh}\relax (x)}-\frac {{\mathrm {e}}^{-x}}{2\,{\mathrm {cosh}\relax (x)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{3}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________