2.7 \(\int \frac {\tan (x)}{\sqrt {\sec ^3(x)+1}} \,dx\)

2.7.1 Mathematica

ClearAll[x] 
integrand = Tan[x]/Sqrt[Sec[x]^3 + 1]; 
res = Integrate[integrand, x] 
TeXForm[res]
 

\[ -\frac {2}{3} \tanh ^{-1}\left (\sqrt {\sec ^3(x)+1}\right ) \]

2.7.2 Rubi

<< Rubi` 
ClearAll[x] 
integrand = Tan[x]/Sqrt[Sec[x]^3 + 1]; 
res = Int[integrand, x] 
TeXForm[res]
 

\[ -\frac {2}{3} \tanh ^{-1}\left (\sqrt {\sec ^3(x)+1}\right ) \]

2.7.3 Maple

restart; 
integrand := tan(x)/sqrt(sec(x)^3 + 1); 
res:=int(integrand,x); 
latex(res)
 

\[ -{\frac {2}{3}{\rm arctanh} \left (\sqrt { \left ( \sec \left ( x \right ) \right ) ^{3}+1}\right )} \]

2.7.4 Fricas

set output tex off 
setSimplifyDenomsFlag(true) 
integrand := tan(x)/sqrt(sec(x)^3 + 1); 
res:=integrate(integrand,x); 
latex(res)
 

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

2.7.5 Maxima

integrand : tan(x)/sqrt(sec(x)^3 + 1); 
res : integrate(integrand,x); 
tex(res)
 

\[ {{\log \left (\sqrt {{{1}\over {\cos ^3x}}+1}-1\right )}\over {3}}-{{ \log \left (\sqrt {{{1}\over {\cos ^3x}}+1}+1\right )}\over {3}} \]

2.7.6 XCAS

integrand := tan(x)/sqrt(sec(x)^3 + 1); 
res := integrate(integrand,x); 
latex(res)
 

\[ 2 \left (-\frac {\ln \left (\sqrt {\left (\frac 1{\cos x}\right )^{3}+1}+1\right )}{6}+\frac {\ln \left |\sqrt {\left (\frac 1{\cos x}\right )^{3}+1}-1\right |}{6}\right ) \]

2.7.7 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(sec(x)**3 + 1); 
res = integrate(integrand,x); 
latex(res)
 

\[ \text {did not solve} \]

2.7.8 MuPad

evalin(symengine,'int(tan(x)/sqrt(sec(x)^3 + 1),x)')
 

\[ \text {did not solve} \]