3.22 \(\int \frac{1}{(1+\cos (x)+\sin (x))^2} \, dx\)

Optimal. Leaf size=29 \[ -\log \left (\tan \left (\frac{x}{2}\right )+1\right )-\frac{\cos (x)-\sin (x)}{\sin (x)+\cos (x)+1} \]

[Out]

-Log[1 + Tan[x/2]] - (Cos[x] - Sin[x])/(1 + Cos[x] + Sin[x])

_______________________________________________________________________________________

Rubi [A]  time = 0.0334933, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\log \left (\tan \left (\frac{x}{2}\right )+1\right )-\frac{\cos (x)-\sin (x)}{\sin (x)+\cos (x)+1} \]

Antiderivative was successfully verified.

[In]  Int[(1 + Cos[x] + Sin[x])^(-2),x]

[Out]

-Log[1 + Tan[x/2]] - (Cos[x] - Sin[x])/(1 + Cos[x] + Sin[x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.06855, size = 22, normalized size = 0.76 \[ \frac{\sin{\left (x \right )} - \cos{\left (x \right )}}{\sin{\left (x \right )} + \cos{\left (x \right )} + 1} - \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+cos(x)+sin(x))**2,x)

[Out]

(sin(x) - cos(x))/(sin(x) + cos(x) + 1) - log(tan(x/2) + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.040985, size = 56, normalized size = 1.93 \[ \frac{1}{2} \tan \left (\frac{x}{2}\right )+\log \left (\cos \left (\frac{x}{2}\right )\right )+\frac{\sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}-\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + Cos[x] + Sin[x])^(-2),x]

[Out]

Log[Cos[x/2]] - Log[Cos[x/2] + Sin[x/2]] + Sin[x/2]/(Cos[x/2] + Sin[x/2]) + Tan[
x/2]/2

_______________________________________________________________________________________

Maple [A]  time = 0.062, size = 27, normalized size = 0.9 \[{\frac{1}{2}\tan \left ({\frac{x}{2}} \right ) }-\ln \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) - \left ( 1+\tan \left ({\frac{x}{2}} \right ) \right ) ^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+cos(x)+sin(x))^2,x)

[Out]

1/2*tan(1/2*x)-ln(1+tan(1/2*x))-1/(1+tan(1/2*x))

_______________________________________________________________________________________

Maxima [A]  time = 1.47513, size = 54, normalized size = 1.86 \[ \frac{\sin \left (x\right )}{2 \,{\left (\cos \left (x\right ) + 1\right )}} - \frac{1}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1} - \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + sin(x) + 1)^(-2),x, algorithm="maxima")

[Out]

1/2*sin(x)/(cos(x) + 1) - 1/(sin(x)/(cos(x) + 1) + 1) - log(sin(x)/(cos(x) + 1)
+ 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.231173, size = 62, normalized size = 2.14 \[ \frac{{\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )} \log \left (\sin \left (x\right ) + 1\right ) - 2 \, \cos \left (x\right ) + 2 \, \sin \left (x\right )}{2 \,{\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + sin(x) + 1)^(-2),x, algorithm="fricas")

[Out]

1/2*((cos(x) + sin(x) + 1)*log(1/2*cos(x) + 1/2) - (cos(x) + sin(x) + 1)*log(sin
(x) + 1) - 2*cos(x) + 2*sin(x))/(cos(x) + sin(x) + 1)

_______________________________________________________________________________________

Sympy [A]  time = 1.46813, size = 66, normalized size = 2.28 \[ - \frac{2 \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )} \tan{\left (\frac{x}{2} \right )}}{2 \tan{\left (\frac{x}{2} \right )} + 2} - \frac{2 \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )}}{2 \tan{\left (\frac{x}{2} \right )} + 2} + \frac{\tan ^{2}{\left (\frac{x}{2} \right )}}{2 \tan{\left (\frac{x}{2} \right )} + 2} - \frac{3}{2 \tan{\left (\frac{x}{2} \right )} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+cos(x)+sin(x))**2,x)

[Out]

-2*log(tan(x/2) + 1)*tan(x/2)/(2*tan(x/2) + 2) - 2*log(tan(x/2) + 1)/(2*tan(x/2)
+ 2) + tan(x/2)**2/(2*tan(x/2) + 2) - 3/(2*tan(x/2) + 2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.204712, size = 41, normalized size = 1.41 \[ \frac{\tan \left (\frac{1}{2} \, x\right )}{\tan \left (\frac{1}{2} \, x\right ) + 1} -{\rm ln}\left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 1 \right |}\right ) + \frac{1}{2} \, \tan \left (\frac{1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((cos(x) + sin(x) + 1)^(-2),x, algorithm="giac")

[Out]

tan(1/2*x)/(tan(1/2*x) + 1) - ln(abs(tan(1/2*x) + 1)) + 1/2*tan(1/2*x)