3.599 \(\int \frac{x+\cosh (x)+\sinh (x)}{1+\cosh (x)} \, dx\)

Optimal. Leaf size=15 \[ x-(1-x) \tanh \left (\frac{x}{2}\right ) \]

[Out]

x - (1 - x)*Tanh[x/2]

_______________________________________________________________________________________

Rubi [A]  time = 0.20023, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ x-(1-x) \tanh \left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x + Cosh[x] + Sinh[x])/(1 + Cosh[x]),x]

[Out]

x - (1 - x)*Tanh[x/2]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x + e^{x}}{\cosh{\left (x \right )} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x+cosh(x)+sinh(x))/(1+cosh(x)),x)

[Out]

Integral((x + exp(x))/(cosh(x) + 1), x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0392386, size = 20, normalized size = 1.33 \[ \frac{\sinh (x) \left (x+x \coth \left (\frac{x}{2}\right )-1\right )}{\cosh (x)+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(x + Cosh[x] + Sinh[x])/(1 + Cosh[x]),x]

[Out]

((-1 + x + x*Coth[x/2])*Sinh[x])/(1 + Cosh[x])

_______________________________________________________________________________________

Maple [A]  time = 0.032, size = 16, normalized size = 1.1 \[ 2\,x-2\,{\frac{-1+x}{1+{{\rm e}^{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x+cosh(x)+sinh(x))/(1+cosh(x)),x)

[Out]

2*x-2*(-1+x)/(1+exp(x))

_______________________________________________________________________________________

Maxima [A]  time = 1.3595, size = 47, normalized size = 3.13 \[ x + \frac{2 \, x e^{x}}{e^{x} + 1} - \frac{2}{e^{\left (-x\right )} + 1} + \log \left (\cosh \left (x\right ) + 1\right ) - 2 \, \log \left (e^{x} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + cosh(x) + sinh(x))/(cosh(x) + 1),x, algorithm="maxima")

[Out]

x + 2*x*e^x/(e^x + 1) - 2/(e^(-x) + 1) + log(cosh(x) + 1) - 2*log(e^x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.203108, size = 27, normalized size = 1.8 \[ \frac{2 \,{\left (x \cosh \left (x\right ) + x \sinh \left (x\right ) + 1\right )}}{\cosh \left (x\right ) + \sinh \left (x\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + cosh(x) + sinh(x))/(cosh(x) + 1),x, algorithm="fricas")

[Out]

2*(x*cosh(x) + x*sinh(x) + 1)/(cosh(x) + sinh(x) + 1)

_______________________________________________________________________________________

Sympy [A]  time = 0.651869, size = 12, normalized size = 0.8 \[ x \tanh{\left (\frac{x}{2} \right )} + x - \tanh{\left (\frac{x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x+cosh(x)+sinh(x))/(1+cosh(x)),x)

[Out]

x*tanh(x/2) + x - tanh(x/2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21597, size = 19, normalized size = 1.27 \[ \frac{2 \,{\left (x e^{x} + 1\right )}}{e^{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + cosh(x) + sinh(x))/(cosh(x) + 1),x, algorithm="giac")

[Out]

2*(x*e^x + 1)/(e^x + 1)