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]
 

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

2.6.2 Rubi

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

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

2.6.3 Maple

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

\[ {\frac {1}{2}\sqrt { \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^ {2} \right ) ^{2}-2\, \left ( \tan \left ( x \right ) \right ) ^{2}}}-{ \frac {{\rm arcsinh} \left ( \left ( \tan \left ( x \right ) \right ) ^{2} \right )}{2}}-{\frac {\sqrt {2}}{2}{\rm arctanh} \left ({\frac { \left ( -2\, \left ( \tan \left ( x \right ) \right ) ^{2}+2 \right ) \sqrt {2}}{4 }{\frac {1}{\sqrt { \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{ 2} \right ) ^{2}-2\, \left ( \tan \left ( x \right ) \right ) ^{2}}}}} \right )} \]

2.6.4 Fricas

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

\[ {{2 \ {\log \left ( {{{\sqrt {{{{{\tan \left ( {x} \right )}} \sp {4}}+1}}} -{ {{\tan \left ( {x} \right )}} \sp {2}}}} \right )}}+{{\sqrt {2}} \ {\log \left ( {{{{{\left ( {2 \ {\sqrt {2}} \ {{{\tan \left ( {x} \right )}} \sp {2}}} -{2 \ {\sqrt {2}}} \right )} \ {\sqrt {{{{{\tan \left ( {x} \right )}} \sp {4}}+1} }}}+{3 \ {{{\tan \left ( {x} \right )}} \sp {4}}} -{2 \ {{{\tan \left ( {x} \right )}} \sp {2}}}+3} \over {{{{\tan \left ( {x} \right )}} \sp {4}}+{2 \{{{\tan \left ( {x} \right )}} \sp {2}}}+1}}} \right )}}+{2 \ {\sqrt {{{{{\tan \left ( {x} \right )}} \sp {4}}+1}}}}} \over 4 \]

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)
 

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

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)')
 

\[ \text {did not solve} \]