Oh, those complex values!!!
aaro...@gmail.com
the correct answer here? How to systematically avoid this pitfall?
f=x Log[a+b x+Sqrt[c+2d x+x^2]];
t = Assuming[c>d^2&&a>0&&b>0,Integrate[f, x]];
u=t/.a\[Rule]2/.b\[Rule]2/.c\[Rule]10/.d\[Rule]1;
N[u/.x\[Rule]1-u/.x\[Rule]0]
9.50705\[InvisibleSpace]-29.7626 \[ImaginaryI]
NIntegrate[f/.a\[Rule]2/.b\[Rule]2/.c\[Rule]10/.d\[Rule]1, {x, 0, 1}]
0.954442
Many thanks in advance!
Aaron Fude
David Park
I would much prefer to write your f definition as follows...
f[a_, b_, c_, d_][x_] = x Log[a + b x + Sqrt[c + 2d x + x^2]];
If you do an indefinite integration you must always remember that the answer
is the returned value plus a constant. That constant could be a complex
number that undoes the imaginary number you get in your result. Also the
returned result contains multivalued functions and this complicates the
picture more.
A general definite integral seems to take too long, but if you integrate
with specific values of a, b, c, d you can obtain exact real results in
reasonable time.
Integrate[f[2, 2, 10, 1][x], {x, 0, 1}]
% // N
NIntegrate[f[2, 2, 10, 1][x], {x, 0, 1}]
David Park
dj...@earthlink.net
http://home.earthlink.net/~djmp/
Jean-Marc Gulliet
> Integrating a real function, getting complex values again.
You seem to believe that because you start with a function supposedly
defined on the reals only you will get an primitive function that
returns only reals values.
> How to get
> the correct answer here?
The answer returned by Mathematica *IS* correct and is due to your
choice of parameter values.
> How to systematically avoid this pitfall?
Carefully choose the parameter ranges, say by visual inspection of the
integral and the white the help of functions such as Reduce[]. (See
examples at the bottom of the post.)
> f=x Log[a+b x+Sqrt[c+2d x+x^2]];
> t = Assuming[c>d^2&&a>0&&b>0,Integrate[f, x]];
> u=t/.a\[Rule]2/.b\[Rule]2/.c\[Rule]10/.d\[Rule]1;
>
> N[u/.x\[Rule]1-u/.x\[Rule]0]
> 9.50705\[InvisibleSpace]-29.7626 \[ImaginaryI]
>
> NIntegrate[f/.a\[Rule]2/.b\[Rule]2/.c\[Rule]10/.d\[Rule]1, {x, 0, 1}]
> 0.954442
For instance, the primitive function is full of expressions with
radicals. That means that whatever values you chose for a, b, c, and d,
the all of these expression must be positive or null; otherwise you will
get complex values. For instance one of these expression under radical,
with the values you assigned to a, b, c, and d, yields -27. There is no
way that Sqrt[-27] becomes real.
In[1]:=
f = x*Log[a + b*x + Sqrt[c + 2*d*x + x^2]]
t = Assuming[c > d^2 && a > 0 && b > 0,
Integrate[f, x]]
u = t /. a -> 2 /. b -> 2 /. c -> 10 /. d -> 1
N[u /. x -> 1 - u /. x -> 0]
Out[1]=
2
x Log[a + b x + Sqrt[c + 2 d x + x ]]
Out[2]=
2 2
(a b - d) x x (-a + b d) Sqrt[c + 2 d x + x ]
----------- - -- + ------------------------------- +
2 4 2
2 (-1 + b ) 2 (-1 + b )
3 3 2 2 2
((a b - a b c + a b c - a d - 2 a b d + c d -
2 2 3
b c d + 3 a b d - d )
2
a b - d - x + b x
ArcTan[-----------------------------------]) /
2 2 2
Sqrt[-a + c - b c + 2 a b d - d ]
2 2 2 2 2
((-1 + b ) Sqrt[-a + c - b c + 2 a b d - d ]) +
1 2 2 2 2
------------ ((-a - a b + c - b c + 4 a b d -
2 2
4 (-1 + b )
2 2 2
2 d ) Log[a - c + 2 a b x - 2 d x - x +
2 2 1
b x ]) - ------------
2 2
2 (-1 + b )
2 3 2
((-2 a b + b c - b c + 2 a d + 2 a b d -
2 3 2
3 b d + b d )
2
Log[d + x + Sqrt[c + 2 d x + x ]]) +
1 2 2
- x Log[a + b x + Sqrt[c + 2 d x + x ]] +
2
4 4 3 2 2 3 2 5
((3 a b + a b - 4 a b c + 3 a b c + a b c +
2 3 2 5 2 3 3 2
b c - 2 b c + b c - 2 a d - 12 a b d -
3 4 2
2 a b d + 2 a c d + 6 a b c d -
4 2 2 2 3 2
8 a b c d + 11 a b d + 13 a b d -
2 3 2 3 2 3
5 b c d + 5 b c d - 2 a d - 14 a b d +
4 3 2 2
4 b d + a Sqrt[a - c + b c - 2 a b d +
2 3 2
d ] + 3 a b
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2
a c Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2 2
a b c Sqrt[a - c + b c - 2 a b d + d ] +
4 2 2 2
2 a b c Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2 2
7 a b d Sqrt[a - c + b c - 2 a b d + d ] -
2 3 2 2 2
5 a b d Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2
3 b c d Sqrt[a - c + b c - 2 a b d + d ] -
3 2 2 2
3 b c d Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2 2
2 a d Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2 2
10 a b d Sqrt[a - c + b c - 2 a b d +
2 3
d ] - 4 b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ])
2
Log[-((4 (-1 + b )
2 2 2 2 4
(-a d + 2 a b d - a b d + c d -
2 4 6
3 b c d + 3 b c d - b c d +
2 3 2 5 2 3
2 a b d - 4 a b d + 2 a b d - d +
2 3 4 3
2 b d - b d -
2 2 2
c Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2
3 b c Sqrt[a - c + b c - 2 a b d +
2
d ] -
4 2 2
3 b c Sqrt[a - c + b c - 2 a b d +
2
d ] +
6 2 2
b c Sqrt[a - c + b c - 2 a b d +
2
d ] -
2 2
a b d Sqrt[a - c + b c - 2 a b d +
2
d ] +
3
2 a b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
5 2 2
a b d Sqrt[a - c + b c - 2 a b d +
2
d ] +
2 2 2 2
d Sqrt[a - c + b c - 2 a b d + d ] -
2 2
2 b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
4 2 2 2
b d Sqrt[a - c + b c - 2 a b d +
2 2 2 2 2 4
d ] - a x + 2 a b x - a b x +
2 4 6
c x - 3 b c x + 3 b c x - b c x +
3 5
2 a b d x - 4 a b d x + 2 a b d x -
2 2 2 4 2
d x + 2 b d x - b d x -
2 2 2
a b Sqrt[a - c + b c - 2 a b d + d ]
3
x + 2 a b
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] x -
5 2 2 2
a b Sqrt[a - c + b c - 2 a b d + d ]
2
x + b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] x -
4 2 2
2 b d Sqrt[a - c + b c - 2 a b d +
2
d ] x +
6 2 2 2
b d Sqrt[a - c + b c - 2 a b d + d ]
x)) /
2 2 2 2 4
(Sqrt[a + a b - b c + b c - 2 a b d -
3 2 2
2 a b d + 2 b d +
2 2
2 a b Sqrt[a - c + b c - 2 a b d +
2
d ] -
2 2 2
2 b d Sqrt[a - c + b c - 2 a b d +
2
d ]]
4 4 3 2 2 3
(3 a b + a b - 4 a b c + 3 a b c +
2 5 2 3 2 5 2
a b c + b c - 2 b c + b c -
3 3 2 3 4
2 a d - 12 a b d - 2 a b d +
2 4
2 a c d + 6 a b c d - 8 a b c d +
2 2 2 3 2 2
11 a b d + 13 a b d - 5 b c d +
3 2 3 2 3
5 b c d - 2 a d - 14 a b d +
4
4 b d +
3 2 2 2
a Sqrt[a - c + b c - 2 a b d + d ] +
3 2
3 a b
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
2 2
a c Sqrt[a - c + b c - 2 a b d +
2
d ] -
2 2 2
a b c Sqrt[a - c + b c - 2 a b d +
2
d ] +
4
2 a b c
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
2
7 a b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
2 3
5 a b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
3 b c d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
3
3 b c d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2
2 a d Sqrt[a - c + b c - 2 a b d +
2
d ] +
2 2
10 a b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
3 2 2
4 b d Sqrt[a - c + b c - 2 a b d +
2
d ])
(a b - d +
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
2
x + b x))) -
2 3 2 2
(8 (-1 + b ) Sqrt[a - c + b c - 2 a b d +
2 2
d ] Sqrt[c + 2 d x + x ]) /
4 4 3 2 2 3
((3 a b + a b - 4 a b c + 3 a b c +
2 5 2 3 2 5 2
a b c + b c - 2 b c + b c -
3 3 2 3 4
2 a d - 12 a b d - 2 a b d +
2 4
2 a c d + 6 a b c d - 8 a b c d +
2 2 2 3 2 2
11 a b d + 13 a b d - 5 b c d +
3 2 3 2 3
5 b c d - 2 a d - 14 a b d +
4 3
4 b d + a
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
3 2 2 2
3 a b Sqrt[a - c + b c - 2 a b d +
2
d ] - a c
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2
a b c Sqrt[a - c + b c - 2 a b d +
2 4
d ] + 2 a b c
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2
7 a b d Sqrt[a - c + b c - 2 a b d +
2 2 3
d ] - 5 a b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
2 2
3 b c d Sqrt[a - c + b c - 2 a b d +
2 3
d ] - 3 b c d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2
2 a d Sqrt[a - c + b c - 2 a b d +
2 2 2
d ] + 10 a b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
3 2 2 2
4 b d Sqrt[a - c + b c - 2 a b d + d ])
(2 (a b - d +
2 2 2
Sqrt[a - c + b c - 2 a b d + d ]) +
2
2 (-1 + b ) x))]) /
2 2 2 2 2
(4 (-1 + b ) Sqrt[a - c + b c - 2 a b d + d ]
2 2 2 2 4
Sqrt[a + a b - b c + b c - 2 a b d -
3 2 2
2 a b d + 2 b d +
2 2 2
2 a b Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2 2
2 b d Sqrt[a - c + b c - 2 a b d + d ]]) +
4 4 3 2 2 3
((-3 a b - a b + 4 a b c - 3 a b c -
2 5 2 3 2 5 2 3
a b c - b c + 2 b c - b c + 2 a d +
3 2 3 4
12 a b d + 2 a b d - 2 a c d -
2 4 2 2
6 a b c d + 8 a b c d - 11 a b d -
2 3 2 2 3 2 3
13 a b d + 5 b c d - 5 b c d + 2 a d +
2 3 4
14 a b d - 4 b d +
3 2 2 2
a Sqrt[a - c + b c - 2 a b d + d ] +
3 2 2 2 2
3 a b Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2
a c Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2 2
a b c Sqrt[a - c + b c - 2 a b d + d ] +
4 2 2 2
2 a b c Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2 2
7 a b d Sqrt[a - c + b c - 2 a b d + d ] -
2 3 2 2 2
5 a b d Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2
3 b c d Sqrt[a - c + b c - 2 a b d + d ] -
3 2 2 2
3 b c d Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2 2
2 a d Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2 2
10 a b d Sqrt[a - c + b c - 2 a b d +
2 3
d ] - 4 b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ])
2
Log[(4 (-1 + b )
2 2 2 2 4
(a d - 2 a b d + a b d - c d +
2 4 6 2
3 b c d - 3 b c d + b c d - 2 a b d +
3 2 5 2 3 2 3
4 a b d - 2 a b d + d - 2 b d +
4 3
b d - c
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2
3 b c Sqrt[a - c + b c - 2 a b d +
2 4
d ] - 3 b c
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
6 2 2 2
b c Sqrt[a - c + b c - 2 a b d + d ] -
2 2
a b d Sqrt[a - c + b c - 2 a b d +
2 3
d ] + 2 a b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
5 2 2
a b d Sqrt[a - c + b c - 2 a b d +
2 2
d ] + d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2 2
2 b d Sqrt[a - c + b c - 2 a b d +
2 4 2
d ] + b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2 2 4
a x - 2 a b x + a b x - c x +
2 4 6
3 b c x - 3 b c x + b c x -
3 5
2 a b d x + 4 a b d x - 2 a b d x +
2 2 2 4 2
d x - 2 b d x + b d x -
2 2 2
a b Sqrt[a - c + b c - 2 a b d + d ]
3
x + 2 a b
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] x -
5 2 2 2
a b Sqrt[a - c + b c - 2 a b d + d ]
2
x + b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] x -
4 2 2 2
2 b d Sqrt[a - c + b c - 2 a b d + d ]
6
x + b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] x)) /
2 2 2 2 4
(Sqrt[a + a b - b c + b c - 2 a b d -
3 2 2
2 a b d + 2 b d -
2 2
2 a b Sqrt[a - c + b c - 2 a b d +
2 2
d ] + 2 b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ]]
4 4 3 2 2 3
(3 a b + a b - 4 a b c + 3 a b c +
2 5 2 3 2 5 2
a b c + b c - 2 b c + b c -
3 3 2 3 4
2 a d - 12 a b d - 2 a b d +
2 4
2 a c d + 6 a b c d - 8 a b c d +
2 2 2 3 2 2
11 a b d + 13 a b d - 5 b c d +
3 2 3 2 3
5 b c d - 2 a d - 14 a b d +
4 3
4 b d - a
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
3 2 2 2
3 a b Sqrt[a - c + b c - 2 a b d +
2
d ] + a c
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2
a b c Sqrt[a - c + b c - 2 a b d +
2 4
d ] - 2 a b c
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2
7 a b d Sqrt[a - c + b c - 2 a b d +
2 2 3
d ] + 5 a b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
2 2
3 b c d Sqrt[a - c + b c - 2 a b d +
2 3
d ] + 3 b c d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2
2 a d Sqrt[a - c + b c - 2 a b d +
2 2 2
d ] - 10 a b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
3 2 2 2
4 b d Sqrt[a - c + b c - 2 a b d + d ])
(a b - d -
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] - x +
2 2 3
b x)) - (8 (-1 + b )
2 2 2
Sqrt[a - c + b c - 2 a b d + d ]
2
Sqrt[c + 2 d x + x ]) /
4 4 3 2 2 3
((-3 a b - a b + 4 a b c - 3 a b c -
2 5 2 3 2 5 2
a b c - b c + 2 b c - b c +
3 3 2 3 4
2 a d + 12 a b d + 2 a b d -
2 4
2 a c d - 6 a b c d + 8 a b c d -
2 2 2 3 2 2
11 a b d - 13 a b d + 5 b c d -
3 2 3 2 3
5 b c d + 2 a d + 14 a b d -
4 3
4 b d + a
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
3 2 2 2
3 a b Sqrt[a - c + b c - 2 a b d +
2
d ] - a c
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2
a b c Sqrt[a - c + b c - 2 a b d +
2 4
d ] + 2 a b c
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
2 2 2
7 a b d Sqrt[a - c + b c - 2 a b d +
2 2 3
d ] - 5 a b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
2 2
3 b c d Sqrt[a - c + b c - 2 a b d +
2 3
d ] - 3 b c d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2
2 a d Sqrt[a - c + b c - 2 a b d +
2 2 2
d ] + 10 a b d
2 2 2
Sqrt[a - c + b c - 2 a b d + d ] -
3 2 2 2
4 b d Sqrt[a - c + b c - 2 a b d + d ])
(-2 (-a b + d +
2 2 2
Sqrt[a - c + b c - 2 a b d + d ]) +
2
2 (-1 + b ) x))]) /
2 2 2 2 2
(4 (-1 + b ) Sqrt[a - c + b c - 2 a b d + d ]
2 2 2 2 4
Sqrt[a + a b - b c + b c - 2 a b d -
3 2 2
2 a b d + 2 b d -
2 2 2
2 a b Sqrt[a - c + b c - 2 a b d + d ] +
2 2 2 2
2 b d Sqrt[a - c + b c - 2 a b d + d ]])
Out[3]=
2
x x 3 + 3 x
- - -- - Sqrt[3] ArcTanh[---------] -
2 4 3 Sqrt[3]
2
Log[-6 + 6 x + 3 x ] +
2
3 Log[1 + x + Sqrt[10 + 2 x + x ]] +
1 2 2
- x Log[2 + 2 x + Sqrt[10 + 2 x + x ]] +
2
1
---- ((-1944 + 972 Sqrt[3])
1944
Log[(2 (243 + 729 Sqrt[3] + 243 x)) /
(Sqrt[3] (1944 - 972 Sqrt[3])
(3 - 3 Sqrt[3] + 3 x)) -
2
(648 Sqrt[3] Sqrt[10 + 2 x + x ]) /
((-1944 + 972 Sqrt[3])
(-2 (-3 + 3 Sqrt[3]) + 6 x))]) +
1
---- ((1944 + 972 Sqrt[3])
1944
Log[-((2 (-243 + 729 Sqrt[3] - 243 x)) /
(Sqrt[3] (1944 + 972 Sqrt[3])
(3 + 3 Sqrt[3] + 3 x))) -
2
(648 Sqrt[3] Sqrt[10 + 2 x + x ]) /
((1944 + 972 Sqrt[3])
(2 (3 + 3 Sqrt[3]) + 6 x))])
Out[4]=
9.50705 - 29.7626 I
In[5]:=
-a^2 + c - b^2*c + 2*a*b*d - d^2 /. a -> 2 /.
b -> 2 /. c -> 10 /. d -> 1
Out[5]=
-27
In[6]:=
Reduce[c > d^2 && a > 0 && b > 0 &&
-a^2 + c - b^2*c + 2*a*b*d - d^2 >= 0]
Out[6]=
(d <= 0 && a > 0 && 0 < b < 1 &&
c >= (-a^2 + 2*a*b*d - d^2)/(-1 + b^2)) ||
(d > 0 && ((0 < a < d && ((0 < b < a/d &&
c >= (-a^2 + 2*a*b*d - d^2)/(-1 + b^2)) ||
(b == a/d && c > (-a^2 + 2*a*b*d - d^2)/
(-1 + b^2)) || (a/d < b < 1 &&
c >= (-a^2 + 2*a*b*d - d^2)/(-1 + b^2)))) ||
(a == d && ((0 < b < 1 &&
c >= (-a^2 + 2*a*b*d - d^2)/(-1 + b^2)) ||
(b == 1 && c > d^2))) || (a > d && 0 < b < 1 &&
c >= (-a^2 + 2*a*b*d - d^2)/(-1 + b^2))))
Regards,
Jean-Marc
David W. Cantrell
> Integrating a real function, getting complex values again. How to get
> the correct answer here?
Sorry I don't have time for a detailed answer, but maybe this will help.
> How to systematically avoid this pitfall?
>
> f=x Log[a+b x+Sqrt[c+2d x+x^2]];
> t = Assuming[c>d^2&&a>0&&b>0,Integrate[f, x]];
> u=t/.a\[Rule]2/.b\[Rule]2/.c\[Rule]10/.d\[Rule]1;
>
> N[u/.x\[Rule]1-u/.x\[Rule]0]
> 9.50705\[InvisibleSpace]-29.7626 \[ImaginaryI]
I don't know what you did above, but surely it's not what you intended.
Note the parentheses below.
In[32]:= N[(u /. x -> 1) - (u /. x -> 0)]
Out[32]= 0.954442 + 6.28319 I
The real part is correct, and the imaginary part is exactly 2 Pi:
In[33]:= FullSimplify[Im[(u /. x -> 1) - (u /. x -> 0)]]
Out[33]= 2 Pi
which I suppose is due to some branch cut. Anyway, we don't want that, so
to get what you want from (u /. x -> 1) - (u /. x -> 0), you could either
just subtract 2 Pi I from it or take its real part:
In[34]:= FullSimplify[Re[(u /. x -> 1) - (u /. x -> 0)]]
Out[34]=
(1/4)*(1 + 4*Sqrt[3]*ArcCoth[Sqrt[3]] - 4*Log[3] + 4*Log[6] - 12*Log[1 +
Sqrt[10]] + 12*Log[2 + Sqrt[13]] + 2*Log[4 + Sqrt[13]] + 2*(2 + Sqrt[3])*
Log[(18*(5 + 3*Sqrt[3]))/(9 - Sqrt[3] + 2*Sqrt[30])] +
2*(-2 + Sqrt[3])*Log[(18*(-5 + 3*Sqrt[3]))/(9 + Sqrt[3] + 2*Sqrt[30])] +
2*(2 + Sqrt[3])*Log[(1/18)*(7 - 4*Sqrt[3])*(9 - 2*Sqrt[3] + 2*Sqrt[39])]
+ 2*(-2 + Sqrt[3])*Log[(1/18)*(7 + 4*Sqrt[3])*(9 + 2*Sqrt[3] +
2*Sqrt[39])] - 4*Sqrt[3]*Re[ArcTanh[2/Sqrt[3]]])
In[35]:= N[%]
Out[35]= 0.954442
David W. Cantrell
aaro...@gmail.com
the correct answer.
Below, I integrate the function symbolically and then numecially and
get two different answers. (The code runs in about 10 min on my
laptop.) That seems like an error. Is this the multi-valued function
phenomenon?
In any case, I would really like a symbolic expression for this
integral - any suggestions?
Thank you!
F=(a + b x) Log[(c+d x+e Sqrt[f+ 2g x + h x^2])];
In[53]:=
\!\(\(st\ = \
Simplify[Integrate[F, \ x],
e > 0 && g > 0 && h > 0 && x > 0 && x < 1 &&
f + 2\ g\ x + h\ x\^2\ > \ 0\ && \
c + d\ x + e\ Sqrt[f + 2\ g\ x + h\ x\^2] > 0\ ];\)\)
In[54]:=
specific =
st /. { a->0.86715714970012 , b\[Rule]-0.15020415937719,
c->-0.27885748971898, d->-0.43809550060395,
e\[Rule]0.76700694910877,
f->0.68676104348272,g->-0.37812253188014, h->0.96322559609812};
In[55]:=
(specific /. x\[Rule]1) - (specific/.x\[Rule]0)
Out[55]=
\!\(\(-2.952838260511225`\) + 4.440892098500626`*^-15\ \[ImaginaryI]\)
In[56]:=
NIntegrate[
F/. { a->0.86715714970012 , b\[Rule]-0.15020415937719,
c->-0.27885748971898, d->-0.43809550060395, e\[Rule]
0.76700694910877,
f->0.68676104348272,g->-0.37812253188014, h->0.96322559609812},
{x, 0,
1}]
Out[56]=
-2.22949