2.9 \(\int \sin (x) \tan ^{-1}\left (\sqrt {\sec (x)-1}\right ) \,dx\)

2.9.1 Mathematica

ClearAll[x]; 
integrand = Sin[x] ArcTan[Sqrt[Sec[x] - 1]]; 
res = Integrate[integrand, x] 
TeXForm[res]
 

\[ -\frac {1}{2} \left (-3-2 \sqrt {2}\right ) \left (\left (\sqrt {2}-2\right ) \cos \left (\frac {x}{2}\right )-\sqrt {2}+1\right ) \cos ^2\left (\frac {x}{4}\right ) \sqrt {-\tan ^2\left (\frac {x}{4}\right )-2 \sqrt {2}+3} \sqrt {\left (2 \sqrt {2}-3\right ) \tan ^2\left (\frac {x}{4}\right )+1} \cot \left (\frac {x}{4}\right ) \sqrt {\sec (x)-1} \sec (x) \sqrt {\left (\left (10-7 \sqrt {2}\right ) \cos \left (\frac {x}{2}\right )-5 \sqrt {2}+7\right ) \sec ^2\left (\frac {x}{4}\right )} \sqrt {\left (\left (2+\sqrt {2}\right ) \cos \left (\frac {x}{2}\right )-\sqrt {2}-1\right ) \sec ^2\left (\frac {x}{4}\right )} \left (\text {EllipticF}\left (\sin ^{-1}\left (\frac {\tan \left (\frac {x}{4}\right )}{\sqrt {3-2 \sqrt {2}}}\right ),17-12 \sqrt {2}\right )+2 \text {EllipticPi}\left (2 \sqrt {2}-3,-\sin ^{-1}\left (\frac {\tan \left (\frac {x}{4}\right )}{\sqrt {3-2 \sqrt {2}}}\right ),17-12 \sqrt {2}\right )\right )+\frac {1}{2} \cos (x) \sqrt {\sec (x)-1}-\cos (x) \tan ^{-1}\left (\sqrt {\sec (x)-1}\right ) \]

2.9.2 Rubi

<< Rubi` 
ClearAll[x] 
integrand = Sin[x] ArcTan[Sqrt[Sec[x] - 1]]; 
res = Int[integrand, x]; 
TeXForm[res]
 

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

2.9.3 Maple

restart; 
integrand := sin(x)*arctan(sqrt(sec(x) - 1)); 
res:=int(integrand,x); 
latex(res)
 

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

2.9.4 Fricas

set output tex off 
setSimplifyDenomsFlag(true) 
integrand := sin(x)*atan(sqrt(sec(x) - 1)); 
res:=integrate(integrand,x); 
latex(res)
 

\[ {{{\left ( -{2 \ {\cos \left ( {x} \right )}}+1 \right )} \ {\arctan \left ( {{ \sqrt {{{\sec \left ( {x} \right )} -1}}}} \right )}}+{{\cos \left ( {x} \right )} \ {\sqrt {{{-{\cos \left ( {x} \right )}+1} \over {\cos \left ( {x} \right )}}} }}} \over 2 \]

2.9.5 Maxima

integrand : sin(x)*atan(sqrt(sec(x) - 1)); 
res : integrate(integrand,x); 
tex(res);
 

\[ -\cos x\,\arctan \left ({{\sqrt {1-\cos x}}\over {\sqrt {\cos x}}} \right )+{{\arctan \left ({{\sqrt {1-\cos x}}\over {\sqrt {\cos x}}} \right )}\over {2}}+{{\sqrt {1-\cos x}}\over {\left (2-{{2\,\left (\cos x- 1\right )}\over {\cos x}}\right )\,\sqrt {\cos x}}} \]

2.9.6 XCAS

integrand := sin(x)*atan(sqrt(sec(x) - 1)); 
res := integrate(integrand,x); 
 
Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real): 
Check [abs(cos(x))] 
Discontinuities at zeroes of cos(x) were not checked 
 
latex(res)
 

Unable to conver to latex. Here is the raw output

expr:=(1/2*asin(2*cos(x)-1)/(sign(cos(x))^2-1)+sign(cos(x))^2* 
   atan(1/2*(sign(cos(x))^2+(-2*sqrt(-cos(x)^2+cos(x))+1)/(-2*cos(x)+1) 
   +(-2*sqrt(-cos(x)^2+cos(x))+1)*sign(cos(x))^2/(-2*cos(x)+1)-1)/sign(cos(x)))/ 
   ((sign(cos(x))^2-1)*sign(cos(x))))*sign(cos(x)) 
   -cos(x)*atan(sqrt(-cos(x)^2+cos(x))*sign(cos(x))/cos(x));
 

2.9.7 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 = sin(x)*atan(sqrt(sec(x) - 1)); 
res = integrate(integrand,x); 
latex(res)
 

\[ \text {did not solve} \]

2.9.8 MuPad

evalin(symengine,'int(sin(x)*arctan(sqrt(sec(x) - 1)),x)')
 

\[ -\arctan \left ( \sqrt { \left ( \cos \left ( x \right ) \right ) ^{-1}-1} \right ) \cos \left ( x \right ) -{\frac {\cos \left ( x \right ) }{3} \left ( {\frac {3}{2}\arcsin \left ( \sqrt {\cos \left ( x \right ) } \right ) \left ( \cos \left ( x \right ) \right ) ^{-{\frac {3}{2}}}}-{ \frac {3}{2\,\cos \left ( x \right ) }\sqrt {1-\cos \left ( x \right ) }} \right ) \sqrt {1-\cos \left ( x \right ) }{\frac {1}{\sqrt { \left ( \cos \left ( x \right ) \right ) ^{-1}-1}}}} \]