Optimal. Leaf size=13 \[ e^x+\frac {2}{1+e^x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, 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 = {2320, 697}
\begin {gather*} e^x+\frac {2}{e^x+1} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 697
Rule 2320
Rubi steps
\begin {align*} \int \frac {e^x (1+\sinh (x))}{1+\cosh (x)} \, dx &=\text {Subst}\left (\int \frac {-1+2 x+x^2}{(1+x)^2} \, dx,x,e^x\right )\\ &=\text {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 = 0.02, size = 18, normalized size = 1.38 \begin {gather*} \frac {2+e^x+e^{2 x}}{1+e^x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.04, size = 18, normalized size = 1.38
method | result | size |
risch | \({\mathrm e}^{x}+\frac {2}{1+{\mathrm e}^{x}}\) | \(12\) |
default | \(-\tanh \left (\frac {x}{2}\right )-\frac {2}{-1+\tanh \left (\frac {x}{2}\right )}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.24, size = 11, normalized size = 0.85 \begin {gather*} \frac {2}{e^{x} + 1} + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.30, size = 21, normalized size = 1.62 \begin {gather*} \frac {3 \, \cosh \left (x\right ) - \sinh \left (x\right ) + 1}{\cosh \left (x\right ) - \sinh \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\sinh {\left (x \right )} + 1\right ) e^{x}}{\cosh {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 10, normalized size = 0.77 \begin {gather*} \mathrm {e}^{x}+\frac {2}{\mathrm {e}^{x}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.30, size = 11, normalized size = 0.85 \begin {gather*} {\mathrm {e}}^x+\frac {2}{{\mathrm {e}}^x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________