2.6 $\int \tan (x)\sqrt{{\tan }^4(x)+1}dx$

2.6.1 Mathematica

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

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

2.6.2 Rubi

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

$$\frac{1}{2}\sqrt{{\tan }^4(x)+1}-\frac{{\tanh }^{-1}\left(\frac{1-{\tan }^2(x)}{\sqrt{2}\sqrt{{\tan }^4(x)+1}}\right)}{\sqrt{2}}-\frac{1}{2}{\sinh }^{-1}({\tan }^2(x))$$

2.6.3 Maple

restart; 
integrand := tan(x)*sqrt(1 + tan(x)^4); 
res:=int(integrand,x); 
latex(res)
 

$$\frac{1}{2}\sqrt{{(1+{(\tan \left(x\right))}^2)}^2-2{(\tan \left(x\right))}^2}-\frac{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left({(\tan \left(x\right))}^2\right)}{2}-\frac{\sqrt{2}}{2}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\frac{(-2{(\tan \left(x\right))}^2+2)\sqrt{2}}{4}\frac{1}{\sqrt{{(1+{(\tan \left(x\right))}^2)}^2-2{(\tan \left(x\right))}^2}}\right)$$

2.6.4 Fricas

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

$$\frac{2\log \left(\sqrt{{\tan \left(x\right)}^4+1}-{\tan \left(x\right)}^2\right)+\sqrt{2}\log \left(\frac{(2\sqrt{2}{\tan \left(x\right)}^2-2\sqrt{2})\sqrt{{\tan \left(x\right)}^4+1}+3{\tan \left(x\right)}^4-2{\tan \left(x\right)}^2+3}{{\tan \left(x\right)}^4+2{\tan \left(x\right)}^2+1}\right)+2\sqrt{{\tan \left(x\right)}^4+1}}{4}$$

2.6.5 Maxima

integrand : tan(x)*sqrt(1 + tan(x)^4); 
res : integrate(integrand,x); 
tex(res);
 

$$\text{did not solve}$$

2.6.6 XCAS

integrand := tan(x)*sqrt(1 + tan(x)^4); 
res := integrate(integrand,x); 
latex(res)
 

$$\frac{\sqrt{{\tan }^{4}x+1}+2(\frac{\ln (\sqrt{{\tan }^{4}x+1}-{\tan }^{2}x)}{2}+\frac{\ln \left(\frac{-2(\sqrt{{\tan }^{4}x+1}-{\tan }^{2}x)+2+2\sqrt{2}}{2(\sqrt{{\tan }^{4}x+1}-{\tan }^{2}x)-2+2\sqrt{2}}\right)}{\sqrt{2}})}{2}$$

2.6.7 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 = tan(x)*sqrt(1 + tan(x)**4); 
res = integrate(integrand,x); 
latex(res)
 

$$\text{did not solve}$$

2.6.8 MuPad

evalin(symengine,'int(tan(x)*sqrt(1 + tan(x)^4),x)')
 

$$\text{did not solve}$$