2.5 $\int \frac{{\cos }^2(x)}{\sqrt{{\cos }^4(x)+{\cos }^2(x)+1}}dx$

2.5.1 Mathematica

ClearAll[x] 
integrand = Cos[x]^2/Sqrt[Cos[x]^4 + Cos[x]^2 + 1]; 
res = Integrate[integrand, x]; 
TeXForm[res]
 

$$-\frac{2i{\cos }^2(x)\sqrt{1-\frac{2i{\tan }^2(x)}{\sqrt{3}-3i}}\sqrt{1+\frac{2i{\tan }^2(x)}{\sqrt{3}+3i}}\text{EllipticPi}(\frac{3}{2}+\frac{i\sqrt{3}}{2},i{\sinh }^{-1}(\sqrt{-\frac{2i}{\sqrt{3}-3i}}\tan (x)),\frac{-\sqrt{3}+3i}{\sqrt{3}+3i})}{\sqrt{-\frac{i}{\sqrt{3}-3i}}\sqrt{8\cos (2x)+\cos (4x)+15}}$$

2.5.2 Rubi

<< Rubi` 
ClearAll[x] 
integrand = Cos[x]^2/Sqrt[Cos[x]^4 + Cos[x]^2 + 1]; 
res = Int[integrand, x]; 
TeXForm[res]
 

$$-\frac{(1+\sqrt{3}){\cos }^2(x)({\tan }^2(x)+\sqrt{3})\sqrt{\frac{{\tan }^4(x)+3{\tan }^2(x)+3}{{({\tan }^2(x)+\sqrt{3})}^2}}\text{EllipticF}(2{\tan }^{-1}\left(\frac{\tan (x)}{\sqrt[4]{3}}\right),\frac{1}{4}(2-\sqrt{3}))}{4\sqrt[4]{3}\sqrt{{\cos }^4(x)({\tan }^4(x)+3{\tan }^2(x)+3)}}+\frac{(2+\sqrt{3}){\cos }^2(x)({\tan }^2(x)+\sqrt{3})\sqrt{\frac{{\tan }^4(x)+3{\tan }^2(x)+3}{{({\tan }^2(x)+\sqrt{3})}^2}}\text{EllipticPi}(\frac{1}{6}(3-2\sqrt{3}),2{\tan }^{-1}\left(\frac{\tan (x)}{\sqrt[4]{3}}\right),\frac{1}{4}(2-\sqrt{3}))}{4\sqrt[4]{3}\sqrt{{\cos }^4(x)({\tan }^4(x)+3{\tan }^2(x)+3)}}+\frac{{\cos }^2(x){\tan }^{-1}\left(\frac{\tan (x)}{\sqrt{{\tan }^4(x)+3{\tan }^2(x)+3}}\right)\sqrt{{\tan }^4(x)+3{\tan }^2(x)+3}}{2\sqrt{{\cos }^4(x)({\tan }^4(x)+3{\tan }^2(x)+3)}}$$

2.5.3 Maple

restart; 
integrand:= cos(x)^2/sqrt(cos(x)^4 + cos(x)^2 + 1); 
res:=int(integrand,x); 
latex(res)
 

$$-2\frac{\sqrt{({(\cos \left(2x\right))}^2+4\cos \left(2x\right)+7){(\sin \left(2x\right))}^2}(-3+i\sqrt{3}){(\cos \left(2x\right)+1)}^2}{(-1+i\sqrt{3})\sqrt{(\cos \left(2x\right)-1)(\cos \left(2x\right)+1)(\cos \left(2x\right)+2+i\sqrt{3})(i\sqrt{3}-\cos \left(2x\right)-2)}\sin \left(2x\right)\sqrt{{(\cos \left(2x\right))}^2+4\cos \left(2x\right)+7}}\sqrt{\frac{(-1+i\sqrt{3})(\cos \left(2x\right)-1)}{(-3+i\sqrt{3})(\cos \left(2x\right)+1)}}\sqrt{\frac{\cos \left(2x\right)+2+i\sqrt{3}}{(i\sqrt{3}+3)(\cos \left(2x\right)+1)}}\sqrt{\frac{i\sqrt{3}-\cos \left(2x\right)-2}{(-3+i\sqrt{3})(\cos \left(2x\right)+1)}}\mathit{E}\mathit{l}\mathit{l}\mathit{i}\mathit{p}\mathit{t}\mathit{i}\mathit{c}\mathit{P}\mathit{i}(\sqrt{\frac{(-1+i\sqrt{3})(\cos \left(2x\right)-1)}{(-3+i\sqrt{3})(\cos \left(2x\right)+1)}},\frac{-3+i\sqrt{3}}{-1+i\sqrt{3}},\sqrt{\frac{(-3+i\sqrt{3})(1+i\sqrt{3})}{(i\sqrt{3}+3)(-1+i\sqrt{3})}})$$

2.5.4 Fricas

set output tex off 
setSimplifyDenomsFlag(true) 
integrand:= cos(x)^2/sqrt(cos(x)^4 + cos(x)^2 + 1); 
res:=integrate(integrand,x); 
latex(res)
 

$$\frac{\arctan \left(\frac{2{\cos \left(x\right)}^3\sin \left(x\right)\sqrt{{\cos \left(x\right)}^4+{\cos \left(x\right)}^2+1}}{2{\cos \left(x\right)}^6-1}\right)}{6}$$

2.5.5 Maxima

integrand : cos(x)^2/sqrt(cos(x)^4 + cos(x)^2 + 1); 
res : integrate(integrand,x); 
latex(res)
 

$$\text{did not solve}$$

2.5.6 XCAS

integrand := cos(x)^2/sqrt(cos(x)^4 + cos(x)^2 + 1); 
res := integrate(integrand,x); 
latex(res)
 

$$\text{did not solve}$$

2.5.7 Sympy

>python 
Python 3.7.3 (default, Mar 27 2019, 22:11:17) 
[GCC 7.3.0] :: Anaconda, Inc. on linux 
 
from sympy import * 
x = symbols('x') 
integrand =  integrand = cos(x)**2/sqrt(cos(x)**4 + cos(x)**2 + 1); 
res = integrate(integrand,x); 
latex(res)
 

$$\text{did not solve}$$

2.5.8 MuPad

evalin(symengine,'int(cos(x)^2/sqrt(cos(x)^4 + cos(x)^2 + 1),x)')
 

$$\text{did not solve}$$