restart;
T := K+F(xi)*F(xi);
2
K + F(xi)
U := alpha[0]+alpha[1]*(m+F(xi))+beta[1]/(m+F(xi))+alpha[2]*(m+F(xi))*(m+F(xi))+beta[2]/(m+F(xi))^2;
beta[1]
alpha[0] + alpha[1] (m + F(xi)) + ---------
m + F(xi)
2 beta[2]
+ alpha[2] (m + F(xi)) + ------------
2
(m + F(xi))
diff(U, xi);
/ d \
beta[1] |---- F(xi)|
/ d \ \ dxi /
alpha[1] |---- F(xi)| - --------------------
\ dxi / 2
(m + F(xi))
/ d \
2 beta[2] |---- F(xi)|
/ d \ \ dxi /
+ 2 alpha[2] (m + F(xi)) |---- F(xi)| - ----------------------
\ dxi / 3
(m + F(xi))
d := alpha[1]*T-beta[1]*T/(m+F(xi))^2+2*alpha[2]*(m+F(xi))*T-2*beta[2]*T/(m+F(xi))^3;
/ 2\
/ 2\ beta[1] \K + F(xi) /
alpha[1] \K + F(xi) / - --------------------
2
(m + F(xi))
/ 2\
/ 2\ 2 beta[2] \K + F(xi) /
+ 2 alpha[2] (m + F(xi)) \K + F(xi) / - ----------------------
3
(m + F(xi))
diff(d, xi);
/ d \
2 beta[1] F(xi) |---- F(xi)|
/ d \ \ dxi /
2 alpha[1] F(xi) |---- F(xi)| - ----------------------------
\ dxi / 2
(m + F(xi))
/ 2\ / d \
2 beta[1] \K + F(xi) / |---- F(xi)|
\ dxi /
+ -----------------------------------
3
(m + F(xi))
/ d \ / 2\
+ 2 alpha[2] |---- F(xi)| \K + F(xi) /
\ dxi /
/ d \
+ 4 alpha[2] (m + F(xi)) F(xi) |---- F(xi)|
\ dxi /
/ d \
4 beta[2] F(xi) |---- F(xi)|
\ dxi /
- ----------------------------
3
(m + F(xi))
/ 2\ / d \
6 beta[2] \K + F(xi) / |---- F(xi)|
\ dxi /
+ -----------------------------------
4
(m + F(xi))
collect(%, diff);
/ / 2\
| 2 beta[1] F(xi) 2 beta[1] \K + F(xi) /
|2 alpha[1] F(xi) - --------------- + ----------------------
| 2 3
\ (m + F(xi)) (m + F(xi))
/ 2\
+ 2 alpha[2] \K + F(xi) / + 4 alpha[2] (m + F(xi)) F(xi)
/ 2\\
4 beta[2] F(xi) 6 beta[2] \K + F(xi) /| / d \
- --------------- + ----------------------| |---- F(xi)|
3 4 | \ dxi /
(m + F(xi)) (m + F(xi)) /
S := (2*alpha[1]*F(xi)-2*beta[1]*F(xi)/(m+F(xi))^2+2*beta[1]*(K+F(xi)^2)/(m+F(xi))^3+2*alpha[2]*(K+F(xi)^2)+4*alpha[2]*(m+F(xi))*F(xi)-4*beta[2]*F(xi)/(m+F(xi))^3+6*beta[2]*(K+F(xi)^2)/(m+F(xi))^4)*T;
/ / 2\
| 2 beta[1] F(xi) 2 beta[1] \K + F(xi) /
|2 alpha[1] F(xi) - --------------- + ----------------------
| 2 3
\ (m + F(xi)) (m + F(xi))
/ 2\
+ 2 alpha[2] \K + F(xi) / + 4 alpha[2] (m + F(xi)) F(xi)
/ 2\\
4 beta[2] F(xi) 6 beta[2] \K + F(xi) /| / 2\
- --------------- + ----------------------| \K + F(xi) /
3 4 |
(m + F(xi)) (m + F(xi)) /
expand((2*w*k*k)*beta*S-(2*A*k*k)*d-2*w*U+k*U*U);
2 2 2
-2 A k alpha[1] K - 2 A k alpha[1] F(xi)
2 3
- 4 A k alpha[2] F(xi) - 4 w alpha[2] F(xi) m
+ 2 k alpha[0] alpha[1] m + 2 k alpha[0] alpha[1] F(xi)
2 k alpha[0] beta[1] 2
+ -------------------- + 2 k alpha[0] alpha[2] m
m + F(xi)
2 2 k alpha[0] beta[2]
+ 2 k alpha[0] alpha[2] F(xi) + --------------------
2
(m + F(xi))
2 3
+ 2 k alpha[1] m F(xi) + 2 k alpha[1] m alpha[2]
3 2 k beta[1] beta[2]
+ 2 k alpha[1] F(xi) alpha[2] + -------------------
3
(m + F(xi))
2 3 2 2 2
+ 4 k alpha[2] m F(xi) + 6 k alpha[2] m F(xi)
2 3 2
+ 4 k alpha[2] F(xi) m - 2 w alpha[0] + k alpha[0]
2 3 2 2
+ 4 w k beta alpha[1] F(xi) + 4 w k beta alpha[2] K
2
2 4 2 A k beta[1] K
+ 12 w k beta alpha[2] F(xi) + ----------------
2
(m + F(xi))
2 2
2 A k beta[1] F(xi) 2
+ --------------------- - 4 A k alpha[2] m K
2
(m + F(xi))
2 2 2
- 4 A k alpha[2] m F(xi) - 4 A k alpha[2] F(xi) K
2 2 2
4 A k beta[2] K 4 A k beta[2] F(xi)
+ ---------------- + ---------------------
3 3
(m + F(xi)) (m + F(xi))
2 k alpha[1] m beta[1]
+ 4 k alpha[0] alpha[2] F(xi) m + ----------------------
m + F(xi)
2
+ 6 k alpha[1] m alpha[2] F(xi)
2 2 k alpha[1] m beta[2]
+ 6 k alpha[1] m alpha[2] F(xi) + ----------------------
2
(m + F(xi))
2 k alpha[1] F(xi) beta[1] 2 k alpha[1] F(xi) beta[2]
+ -------------------------- + --------------------------
m + F(xi) 2
(m + F(xi))
2 2
2 k beta[1] alpha[2] m 2 k beta[1] alpha[2] F(xi)
+ ----------------------- + ---------------------------
m + F(xi) m + F(xi)
2 2
2 k alpha[2] m beta[2] 2 k alpha[2] F(xi) beta[2]
+ ----------------------- + ---------------------------
2 2
(m + F(xi)) (m + F(xi))
2
2 2 2 2 k beta[1]
+ k alpha[1] m + k alpha[1] F(xi) + ------------
2
(m + F(xi))
2
2 4 2 4 k beta[2]
+ k alpha[2] m + k alpha[2] F(xi) + ------------
4
(m + F(xi))
2 w beta[1]
- 2 w alpha[1] m - 2 w alpha[1] F(xi) - -----------
m + F(xi)
2 2 2 w beta[2]
- 2 w alpha[2] m - 2 w alpha[2] F(xi) - ------------
2
(m + F(xi))
2 2 2
4 w k beta beta[1] F(xi) K 8 w k beta beta[1] K F(xi)
- --------------------------- + ----------------------------
2 3
(m + F(xi)) (m + F(xi))
2
2 8 w k beta beta[2] F(xi) K
+ 8 w k beta alpha[2] F(xi) m K - ---------------------------
3
(m + F(xi))
2 2
24 w k beta beta[2] K F(xi) 2
+ ----------------------------- + 4 w k beta alpha[1] F(xi) K
4
(m + F(xi))
2 3 2 2
4 w k beta beta[1] F(xi) 4 w k beta beta[1] K
- -------------------------- + ----------------------
2 3
(m + F(xi)) (m + F(xi))
2 4
4 w k beta beta[1] F(xi) 2 2
+ -------------------------- + 16 w k beta alpha[2] K F(xi)
3
(m + F(xi))
2 3
2 3 8 w k beta beta[2] F(xi)
+ 8 w k beta alpha[2] F(xi) m - --------------------------
3
(m + F(xi))
2 2 2 4
12 w k beta beta[2] K 12 w k beta beta[2] F(xi)
+ ----------------------- + ---------------------------
4 4
(m + F(xi)) (m + F(xi))
4 k beta[1] alpha[2] F(xi) m 4 k alpha[2] F(xi) m beta[2]
+ ---------------------------- + ----------------------------
m + F(xi) 2
(m + F(xi))
value(%);
2 2 2
-2 A k alpha[1] K - 2 A k alpha[1] F(xi)
2 3
- 4 A k alpha[2] F(xi) - 4 w alpha[2] F(xi) m
+ 2 k alpha[0] alpha[1] m + 2 k alpha[0] alpha[1] F(xi)
2 k alpha[0] beta[1] 2
+ -------------------- + 2 k alpha[0] alpha[2] m
m + F(xi)
2 2 k alpha[0] beta[2]
+ 2 k alpha[0] alpha[2] F(xi) + --------------------
2
(m + F(xi))
2 3
+ 2 k alpha[1] m F(xi) + 2 k alpha[1] m alpha[2]
3 2 k beta[1] beta[2]
+ 2 k alpha[1] F(xi) alpha[2] + -------------------
3
(m + F(xi))
2 3 2 2 2
+ 4 k alpha[2] m F(xi) + 6 k alpha[2] m F(xi)
2 3 2
+ 4 k alpha[2] F(xi) m - 2 w alpha[0] + k alpha[0]
2 3 2 2
+ 4 w k beta alpha[1] F(xi) + 4 w k beta alpha[2] K
2
2 4 2 A k beta[1] K
+ 12 w k beta alpha[2] F(xi) + ----------------
2
(m + F(xi))
2 2
2 A k beta[1] F(xi) 2
+ --------------------- - 4 A k alpha[2] m K
2
(m + F(xi))
2 2 2
- 4 A k alpha[2] m F(xi) - 4 A k alpha[2] F(xi) K
2 2 2
4 A k beta[2] K 4 A k beta[2] F(xi)
+ ---------------- + ---------------------
3 3
(m + F(xi)) (m + F(xi))
2 k alpha[1] m beta[1]
+ 4 k alpha[0] alpha[2] F(xi) m + ----------------------
m + F(xi)
2
+ 6 k alpha[1] m alpha[2] F(xi)
2 2 k alpha[1] m beta[2]
+ 6 k alpha[1] m alpha[2] F(xi) + ----------------------
2
(m + F(xi))
2 k alpha[1] F(xi) beta[1] 2 k alpha[1] F(xi) beta[2]
+ -------------------------- + --------------------------
m + F(xi) 2
(m + F(xi))
2 2
2 k beta[1] alpha[2] m 2 k beta[1] alpha[2] F(xi)
+ ----------------------- + ---------------------------
m + F(xi) m + F(xi)
2 2
2 k alpha[2] m beta[2] 2 k alpha[2] F(xi) beta[2]
+ ----------------------- + ---------------------------
2 2
(m + F(xi)) (m + F(xi))
2
2 2 2 2 k beta[1]
+ k alpha[1] m + k alpha[1] F(xi) + ------------
2
(m + F(xi))
2
2 4 2 4 k beta[2]
+ k alpha[2] m + k alpha[2] F(xi) + ------------
4
(m + F(xi))
2 w beta[1]
- 2 w alpha[1] m - 2 w alpha[1] F(xi) - -----------
m + F(xi)
2 2 2 w beta[2]
- 2 w alpha[2] m - 2 w alpha[2] F(xi) - ------------
2
(m + F(xi))
2 2 2
4 w k beta beta[1] F(xi) K 8 w k beta beta[1] K F(xi)
- --------------------------- + ----------------------------
2 3
(m + F(xi)) (m + F(xi))
2
2 8 w k beta beta[2] F(xi) K
+ 8 w k beta alpha[2] F(xi) m K - ---------------------------
3
(m + F(xi))
2 2
24 w k beta beta[2] K F(xi) 2
+ ----------------------------- + 4 w k beta alpha[1] F(xi) K
4
(m + F(xi))
2 3 2 2
4 w k beta beta[1] F(xi) 4 w k beta beta[1] K
- -------------------------- + ----------------------
2 3
(m + F(xi)) (m + F(xi))
2 4
4 w k beta beta[1] F(xi) 2 2
+ -------------------------- + 16 w k beta alpha[2] K F(xi)
3
(m + F(xi))
2 3
2 3 8 w k beta beta[2] F(xi)
+ 8 w k beta alpha[2] F(xi) m - --------------------------
3
(m + F(xi))
2 2 2 4
12 w k beta beta[2] K 12 w k beta beta[2] F(xi)
+ ----------------------- + ---------------------------
4 4
(m + F(xi)) (m + F(xi))
4 k beta[1] alpha[2] F(xi) m 4 k alpha[2] F(xi) m beta[2]
+ ---------------------------- + ----------------------------
m + F(xi) 2
(m + F(xi))
simplify(%);
2 2 2
-2 A k alpha[1] K - 2 A k alpha[1] F(xi)
2 3
- 4 A k alpha[2] F(xi) - 4 w alpha[2] F(xi) m
+ 2 k alpha[0] alpha[1] m + 2 k alpha[0] alpha[1] F(xi)
2 k alpha[0] beta[1] 2
+ -------------------- + 2 k alpha[0] alpha[2] m
m + F(xi)
2 2 k alpha[0] beta[2]
+ 2 k alpha[0] alpha[2] F(xi) + --------------------
2
(m + F(xi))
2 3
+ 2 k alpha[1] m F(xi) + 2 k alpha[1] m alpha[2]
3 2 k beta[1] beta[2]
+ 2 k alpha[1] F(xi) alpha[2] + -------------------
3
(m + F(xi))
2 3 2 2 2
+ 4 k alpha[2] m F(xi) + 6 k alpha[2] m F(xi)
2 3 2
+ 4 k alpha[2] F(xi) m - 2 w alpha[0] + k alpha[0]
2 3 2 2
+ 4 w k beta alpha[1] F(xi) + 4 w k beta alpha[2] K
2
2 4 2 A k beta[1] K
+ 12 w k beta alpha[2] F(xi) + ----------------
2
(m + F(xi))
2 2
2 A k beta[1] F(xi) 2
+ --------------------- - 4 A k alpha[2] m K
2
(m + F(xi))
2 2 2
- 4 A k alpha[2] m F(xi) - 4 A k alpha[2] F(xi) K
2 2 2
4 A k beta[2] K 4 A k beta[2] F(xi)
+ ---------------- + ---------------------
3 3
(m + F(xi)) (m + F(xi))
2 k alpha[1] m beta[1]
+ 4 k alpha[0] alpha[2] F(xi) m + ----------------------
m + F(xi)
2
+ 6 k alpha[1] m alpha[2] F(xi)
2 2 k alpha[1] m beta[2]
+ 6 k alpha[1] m alpha[2] F(xi) + ----------------------
2
(m + F(xi))
2 k alpha[1] F(xi) beta[1] 2 k alpha[1] F(xi) beta[2]
+ -------------------------- + --------------------------
m + F(xi) 2
(m + F(xi))
2 2
2 k beta[1] alpha[2] m 2 k beta[1] alpha[2] F(xi)
+ ----------------------- + ---------------------------
m + F(xi) m + F(xi)
2 2
2 k alpha[2] m beta[2] 2 k alpha[2] F(xi) beta[2]
+ ----------------------- + ---------------------------
2 2
(m + F(xi)) (m + F(xi))
2
2 2 2 2 k beta[1]
+ k alpha[1] m + k alpha[1] F(xi) + ------------
2
(m + F(xi))
2
2 4 2 4 k beta[2]
+ k alpha[2] m + k alpha[2] F(xi) + ------------
4
(m + F(xi))
2 w beta[1]
- 2 w alpha[1] m - 2 w alpha[1] F(xi) - -----------
m + F(xi)
2 2 2 w beta[2]
- 2 w alpha[2] m - 2 w alpha[2] F(xi) - ------------
2
(m + F(xi))
2 2 2
4 w k beta beta[1] F(xi) K 8 w k beta beta[1] K F(xi)
- --------------------------- + ----------------------------
2 3
(m + F(xi)) (m + F(xi))
2
2 8 w k beta beta[2] F(xi) K
+ 8 w k beta alpha[2] F(xi) m K - ---------------------------
3
(m + F(xi))
2 2
24 w k beta beta[2] K F(xi) 2
+ ----------------------------- + 4 w k beta alpha[1] F(xi) K
4
(m + F(xi))
2 3 2 2
4 w k beta beta[1] F(xi) 4 w k beta beta[1] K
- -------------------------- + ----------------------
2 3
(m + F(xi)) (m + F(xi))
2 4
4 w k beta beta[1] F(xi) 2 2
+ -------------------------- + 16 w k beta alpha[2] K F(xi)
3
(m + F(xi))
2 3
2 3 8 w k beta beta[2] F(xi)
+ 8 w k beta alpha[2] F(xi) m - --------------------------
3
(m + F(xi))
2 2 2 4
12 w k beta beta[2] K 12 w k beta beta[2] F(xi)
+ ----------------------- + ---------------------------
4 4
(m + F(xi)) (m + F(xi))
4 k beta[1] alpha[2] F(xi) m 4 k alpha[2] F(xi) m beta[2]
+ ---------------------------- + ----------------------------
m + F(xi) 2
(m + F(xi))
collect(%, m+F(xi));
Error, (in collect) cannot collect m+F(xi)
