Optimal. Leaf size=13 \[ e^x+\frac {2}{e^x+1} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2282, 683} \[ e^x+\frac {2}{e^x+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 683
Rule 2282
Rubi steps
\begin {align*} \int \frac {e^x (1+\sinh (x))}{1+\cosh (x)} \, dx &=\operatorname {Subst}\left (\int \frac {-1+2 x+x^2}{(1+x)^2} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (1-\frac {2}{(1+x)^2}\right ) \, dx,x,e^x\right )\\ &=e^x+\frac {2}{1+e^x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 18, normalized size = 1.38 \[ \frac {e^x+e^{2 x}+2}{e^x+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^x (1+\sinh (x))}{1+\cosh (x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.69, size = 21, normalized size = 1.62 \[ \frac {3 \, \cosh \relax (x) - \sinh \relax (x) + 1}{\cosh \relax (x) - \sinh \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.60, size = 11, normalized size = 0.85 \[ \frac {2}{e^{x} + 1} + e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 12, normalized size = 0.92
method | result | size |
risch | \({\mathrm e}^{x}+\frac {2}{1+{\mathrm e}^{x}}\) | \(12\) |
default | \(-\tanh \left (\frac {x}{2}\right )-\frac {2}{\tanh \left (\frac {x}{2}\right )-1}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 11, normalized size = 0.85 \[ \frac {2}{e^{x} + 1} + e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.30, size = 11, normalized size = 0.85 \[ {\mathrm {e}}^x+\frac {2}{{\mathrm {e}}^x+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sinh {\relax (x )} + 1\right ) e^{x}}{\cosh {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________