2.3 \(\int -\sin ^{-1}\left (\sqrt {x}-\sqrt {x+1}\right ) \,dx\)
2.3.1 Mathematica
ClearAll[x]
integrand = ArcSin[Sqrt[x + 1] - Sqrt[x]];
res = Integrate[integrand, x];
TeXForm[res]
\[
-\frac {(x+1) \left (2 x-2 \sqrt {x+1} \sqrt {x}+1\right )^2 \left (2 \sqrt {\sqrt {x} \sqrt {x+1}-x} \left (-2 x+2 \sqrt {x+1} \sqrt {x}-3\right )+3 \sqrt {-4 x+4 \sqrt {x+1} \sqrt {x}-2} \log \left (2 \sqrt {\sqrt {x} \sqrt {x+1}-x}+\sqrt {-4 x+4 \sqrt {x+1} \sqrt {x}-2}\right )\right )}{8 \sqrt {2} \left (\sqrt {x+1}-\sqrt {x}\right )^3 \left (x-\sqrt {x+1} \sqrt {x}+1\right )^2}-x \sin ^{-1}\left (\sqrt {x}-\sqrt {x+1}\right )
\]
2.3.2 Rubi
<< Rubi`
ClearAll[x]
integrand = ArcSin[Sqrt[x + 1] - Sqrt[x]];
res = Int[integrand, x];
TeXForm[res]
\[
\frac {\text {Subst}\left (\text {Int}\left (\sqrt {-x^2+\sqrt {x^2-1} x+1},x\right ),x,\sqrt {x+1}\right )}{\sqrt {2}}-x \sin ^{-1}\left (\sqrt {x}-\sqrt {x+1}\right )
\]
2.3.3 Maple
restart;
integrand:=arcsin(sqrt(x+1)-sqrt(x));
res:=int(integrand,x);
latex(res)
\[
-{\frac {1}{16} \left ( \arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \left ( \tan \left ( {\frac {1}{2}\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) } \right ) \right ) ^{8}+2\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \left ( \tan \left ( 1/2\,\arcsin \left ( -\sqrt {x+1}+ \sqrt {x} \right ) \right ) \right ) ^{6}-2\, \left ( \tan \left ( 1/2\, \arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \right ) \right ) ^{7}+ 18\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \left ( \tan \left ( 1/2\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \right ) \right ) ^{4}-6\, \left ( \tan \left ( 1/2\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \right ) \right ) ^{5}+2\,\arcsin \left ( -\sqrt {x+1}+\sqrt { x} \right ) \left ( \tan \left ( 1/2\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \right ) \right ) ^{2}+6\, \left ( \tan \left ( 1/2\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \right ) \right ) ^{3}+\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) +2\,\tan \left ( 1/2\,\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) \right ) \right ) \left ( 1+ \left ( \tan \left ( {\frac {1}{2}\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) } \right ) \right ) ^{2} \right ) ^{-2} \left ( \tan \left ( { \frac {1}{2}\arcsin \left ( -\sqrt {x+1}+\sqrt {x} \right ) } \right ) \right ) ^{-2}}
\]
2.3.4 Fricas
set output tex off
setSimplifyDenomsFlag(true)
ii:=integrate(asin(sqrt(x+1)-sqrt(x)),x)
latex(ii)
\[ {{{\left ( {3 \ {\sqrt {{x+1}}}}+{\sqrt {x}} \right )} \ {\sqrt {{{2 \ {\sqrt {x}} \ {\sqrt {{x+1}}}} -{2 \ x}}}}}+{{\left ( {8 \ x}+3 \right )} \ {\arcsin \left ( {{{\sqrt {{x+1}}} -{\sqrt {x}}}} \right )}}} \over 8 \]
2.3.5 Maxima
ii : integrate(asin(sqrt(x+1)-sqrt(x)),x);
tex(ii);
This reslult is wrong. bug.
\[
{{\pi \,x}\over {2}}
\]
2.3.6 XCAS
ii := integrate(asin(sqrt(x+1)-sqrt(x)),x);
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [92.1017843988]
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [53.1277311612]
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [5.38357630698]
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [6.79369851155]
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [84.3561567818]
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [77.6493344628]
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [72.519035968]
Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.
The choice was done assuming [t_nostep]=[0]
Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.
The choice was done assuming [t_nostep]=[0]
Algebraic extensions not allowed in a rootof
Algebraic extensions not allowed in a rootof
Warning, choosing root of [1,0,%%%{-4,[1]%%%}+%%%{-2,[0]%%%},0,1] at parameters values [69.9232513234]
latex(ii)
\[
\text {did not solve}
\]
2.3.7 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')
ii=integrate(asin(sqrt(x+1)-sqrt(x)),x);
latex(ii)
\[
\text {did not solve}
\]
2.3.8 MuPad
evalin(symengine,'int(asin(sqrt(x+1)-sqrt(x)),x)')
\[
\text {did not solve}
\]