# Simplify a complicated solution by a simpler way?

GROUPS:
 eq50 has given a complicated solution, and it should be simplified into the sum of several proper-fraction-alike parts. I can get a simpler one in eq55, but it takes 5 abstruse steps and isn't the simplest yet. Can you give a easy and straight way? In[159]:= eq12 = (cinf \[Theta]c \[Lambda])/(s + \[Theta]c) - ( E^((lh - x) /Sqrt[Dc ] Sqrt[s + \[Theta]c]) F1 Sqrt[ Dc ] \[Lambda])/ Sqrt[ (s + \[Theta]c)] /. {Sqrt[Dc] F1 \[Lambda] -> A1, cinf \[Theta]c \[Lambda] -> A2} Out[159]= A2/(s + \[Theta]c) - ( A1 E^(((lh - x) Sqrt[s + \[Theta]c])/Sqrt[Dc]))/Sqrt[s + \[Theta]c] In[160]:= eq13 = s *q[x, s] - Dp*D[q[x, s], {x, 2}] - pinf + eq12 == 0 Out[160]= -pinf + A2/(s + \[Theta]c) - ( A1 E^(((lh - x) Sqrt[s + \[Theta]c])/Sqrt[Dc]))/Sqrt[ s + \[Theta]c] + s q[x, s] - Dp \!$$\*SuperscriptBox[\(q$$, \* TagBox[ RowBox[{"(", RowBox[{"2", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[x, s] == 0 In[161]:= eq14 = \[Alpha]1*(q[x, s] /. x -> lh) - (D[q[x, s], x] /. x -> lh) == 0 Out[161]= \[Alpha]1 q[lh, s] - \!$$\*SuperscriptBox[\(q$$, \* TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[lh, s] == 0 In[162]:= eq15 = (D[q[x, s], x] /. x -> +\[Infinity]) == 0 Out[162]= \!$$\*SuperscriptBox[\(q$$, \* TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[\[Infinity], s] == 0 In[163]:= eq50 = Assuming[(lh - x) < 0 && s > 0 && (s + \[Theta]c) > 0 && Dp > 0 && Dc > 0 && \[Alpha]1 > 0, DSolve[{eq13, eq14, eq15}, q[x, s], {x, s}, GeneratedParameters -> B4]] // Simplify // Normal Out[163]= {{q[x, s] -> (E^(-(Sqrt[s]/Sqrt[Dp] + Sqrt[s/ Dp]) x) (-A2 (-Dc s + Dp (s + \[Theta]c)) (-4 E^(Sqrt[s/Dp] (lh + x)) Sqrt[Dp/s] s Sqrt[s/Dp] Sqrt[s + \[Theta]c] Sqrt[ Dp s (s + \[Theta]c)] - Dp (E^((2 Sqrt[s] x)/Sqrt[Dp]) s (Sqrt[s/Dp] + \[Alpha]1) (s + \[Theta]c) + E^(2 Sqrt[s/Dp] x) s (Sqrt[s/Dp] + \[Alpha]1) (s + \[Theta]c) - 2 E^(Sqrt[s/Dp] (lh + x)) Sqrt[s/ Dp] (2 s^2 + 2 s \[Theta]c + \[Alpha]1 Sqrt[s + \[Theta]c] Sqrt[ Dp s (s + \[Theta]c)]))) + Sqrt[s/ Dp] (1/(Sqrt[s/Dp] Sqrt[s + \[Theta]c]) Dp pinf (-2 E^(Sqrt[s/Dp] (lh + x)) Sqrt[s/ Dp] (Sqrt[Dp/s] s^4 Sqrt[Dp s (s + \[Theta]c)] + 5 Sqrt[Dp/s] s^3 \[Theta]c Sqrt[ Dp s (s + \[Theta]c)] + 6 Sqrt[Dp/s] s^2 \[Theta]c^2 Sqrt[ Dp s (s + \[Theta]c)] + 2 Sqrt[Dp/s] s \[Theta]c^3 Sqrt[ Dp s (s + \[Theta]c)] + Sqrt[Dp/s] s^(5/2) Sqrt[s + \[Theta]c] Sqrt[ s (s + \[Theta]c)] Sqrt[Dp s (s + \[Theta]c)] - s \[Alpha]1 Sqrt[ Dp^3 s^5 (s + \[Theta]c)] - \[Alpha]1 \[Theta]c \ Sqrt[Dp^3 s^5 (s + \[Theta]c)]) - Dp (s + \[Theta]c) (E^((2 Sqrt[s] x)/Sqrt[Dp]) s (Sqrt[s/Dp] + \[Alpha]1) (s + \[Theta]c)^(5/2) + E^(2 Sqrt[s/Dp] x) s (Sqrt[s/Dp] + \[Alpha]1) (s + \[Theta]c)^(5/2) - 2 E^(Sqrt[s/Dp] (lh + x)) Sqrt[s/ Dp] (2 s^3 Sqrt[s + \[Theta]c] + 4 s^2 \[Theta]c Sqrt[ s + \[Theta]c] + \[Alpha]1 \[Theta]c^2 Sqrt[ Dp s (s + \[Theta]c)] + 2 s \[Theta]c (\[Theta]c Sqrt[ s + \[Theta]c] + \[Alpha]1 Sqrt[ Dp s (s + \[Theta]c)])))) + 1/(Sqrt[Dp s]) Dc (2 E^(Sqrt[s/Dp] (lh + x)) Sqrt[Dp/s] s Sqrt[Dp s] Sqrt[Dp s (s + \[Theta]c)] (pinf s (s Sqrt[ s + \[Theta]c] + 2 \[Theta]c Sqrt[s + \[Theta]c] + Sqrt[s] Sqrt[s (s + \[Theta]c)]) - A1 (s + \[Theta]c) (-2 s + Sqrt[( Dp s (s + \[Theta]c))/Dc])) + Dp^2 s (-2 A1 E^(Sqrt[s/Dp] (lh + x)) s \[Alpha]1 \[Theta]c Sqrt[s + \[Theta]c] + E^((2 Sqrt[s] x)/Sqrt[Dp]) pinf Sqrt[s/Dp] Sqrt[ Dp s] \[Alpha]1 (s + \[Theta]c)^2 + E^(2 Sqrt[s/Dp] x) pinf Sqrt[s/Dp] Sqrt[ Dp s] \[Alpha]1 (s + \[Theta]c)^2 - A1 E^((2 Sqrt[s] x)/Sqrt[ Dp] + (lh - x) Sqrt[(s + \[Theta]c)/ Dc]) (s + \[Theta]c) Sqrt[ Dp s (s + \[Theta]c)] (Sqrt[s/Dp] Sqrt[( s + \[Theta]c)/ Dc] + \[Alpha]1 (-Sqrt[(s/Dp)] + Sqrt[( s + \[Theta]c)/Dc])) + A1 E^(2 Sqrt[s/Dp] x + (lh - x) Sqrt[(s + \[Theta]c)/ Dc]) (s + \[Theta]c) Sqrt[ Dp s (s + \[Theta]c)] (Sqrt[s/Dp] Sqrt[( s + \[Theta]c)/ Dc] + \[Alpha]1 (Sqrt[s/Dp] + Sqrt[( s + \[Theta]c)/Dc]))) + Dp (E^((2 Sqrt[s] x)/Sqrt[Dp]) pinf s^2 Sqrt[ Dp s] (s + \[Theta]c)^2 + E^(2 Sqrt[s/Dp] x) pinf s^2 Sqrt[ Dp s] (s + \[Theta]c)^2 + ( A1 E^((2 Sqrt[s] x)/Sqrt[ Dp] + (lh - x) Sqrt[(s + \[Theta]c)/Dc]) s (Dp s (s + \[Theta]c))^(3/2))/Dp + ( A1 E^(2 Sqrt[s/Dp] x + (lh - x) Sqrt[(s + \[Theta]c)/ Dc]) s (Dp s (s + \[Theta]c))^(3/2))/Dp - 2 E^(Sqrt[s/ Dp] (lh + x)) (A1 (Sqrt[Dp s] Sqrt[Dp s^5] \[Alpha]1 Sqrt[ s + \[Theta]c] + ( 2 s (Dp s (s + \[Theta]c))^(3/2))/Dp) + pinf Sqrt[ Dp s] (2 s^4 + 4 s^3 \[Theta]c + 2 s^2 \[Theta]c^2 + s \[Alpha]1 \[Theta]c Sqrt[s + \[Theta]c] Sqrt[ Dp s (s + \[Theta]c)] + \[Alpha]1 Sqrt[ s + \[Theta]c] Sqrt[ Dp s^5 (s + \[Theta]c)])))))))/(2 Dc Dp^3 (s/Dp)^( 3/2) (Sqrt[s/Dp] + \[Alpha]1) (s + \[Theta]c)^(3/2) Sqrt[ Dp s (s + \[Theta]c)] (Sqrt[s/Dp] - Sqrt[(s + \[Theta]c)/ Dc]) (Sqrt[s/Dp] + Sqrt[(s + \[Theta]c)/Dc]))}} In[164]:= eq51 = Assuming[(lh - x) < 0 && s > 0 && (s + \[Theta]c) > 0 && Dp > 0 && Dc > 0 && \[Alpha]1 > 0, FullSimplify[q[x, s] /. eq50[[1, 1]]]] Out[164]= -(E^(-2 Sqrt[s/Dp] x) (2 A2 s (-Dp E^(Sqrt[s/Dp] (lh + x)) \[Alpha]1 + E^(2 Sqrt[s/Dp] x) (Sqrt[Dp s] + Dp \[Alpha]1)) (s + \[Theta]c) (-Dc s + Dp (s + \[Theta]c)) + Sqrt[s/Dp] (-1/(Sqrt[(s (s + \[Theta]c))/Dp])2 Dp E^( Sqrt[s/Dp] x) pinf (-Dp (E^(lh Sqrt[s/Dp]) - E^( Sqrt[s/Dp] x)) s \[Alpha]1 (s + \[Theta]c)^(7/2) + E^(Sqrt[s/Dp] x) (\[Theta]c^3 Sqrt[Dp s^3 (s + \[Theta]c)] + 3 \[Theta]c^2 Sqrt[Dp s^5 (s + \[Theta]c)] + 3 \[Theta]c Sqrt[Dp s^7 (s + \[Theta]c)] + Sqrt[ Dp s^9 (s + \[Theta]c)])) + 2 Sqrt[Dc] Dp s (s + \[Theta]c) (pinf (Sqrt[Dc] E^(2 Sqrt[s/Dp] x) s + (E^(2 Sqrt[s/Dp] x) - E^( Sqrt[s/Dp] (lh + x))) Sqrt[ Dc Dp s] \[Alpha]1) (s + \[Theta]c) + A1 (-Sqrt[Dp] E^(Sqrt[s/Dp] (lh + x)) s^(3/2) - E^(Sqrt[s/ Dp] (lh + x)) (Sqrt[Dp s] \[Theta]c + \[Alpha]1 Sqrt[ Dc Dp s (s + \[Theta]c)]) + E^(2 Sqrt[s/Dp] x + (lh - x) Sqrt[(s + \[Theta]c)/ Dc]) (s Sqrt[Dc (s + \[Theta]c)] + \[Alpha]1 Sqrt[ Dc Dp s (s + \[Theta]c)]))))))/(2 Dp s^2 (Sqrt[s/ Dp] + \[Alpha]1) (s + \[Theta]c)^2 (-Dc s + Dp (s + \[Theta]c))) In[165]:= eq52 = ExpandAll[eq51, x] /. {(Sqrt[s/Dp] + \[Alpha]1) -> (( Sqrt[s] + Sqrt[ Dp ] \[Alpha]1)/Sqrt[ Dp ]), Sqrt[s/Dp] -> Sqrt[s]/Sqrt[ Dp ]} // Normal Out[165]= ( A2 Sqrt[Dp] E^((lh Sqrt[s])/Sqrt[Dp] - (Sqrt[s] x)/Sqrt[Dp]) \[Alpha]1)/( s (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c)) - ( A2 (Sqrt[Dp s] + Dp \[Alpha]1))/( Sqrt[Dp] s (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c)) - ( Dc pinf Sqrt[ s])/((Sqrt[s] + Sqrt[Dp] \[Alpha]1) (-Dc s + Dp (s + \[Theta]c))) - (Sqrt[Dc] pinf Sqrt[Dc Dp s] \[Alpha]1)/( Sqrt[s] (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (-Dc s + Dp (s + \[Theta]c))) + ( Sqrt[Dc] E^((lh Sqrt[s])/Sqrt[Dp] - (Sqrt[s] x)/Sqrt[Dp]) pinf Sqrt[ Dc Dp s] \[Alpha]1)/( Sqrt[s] (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (-Dc s + Dp (s + \[Theta]c))) + ( A1 Sqrt[Dc] Sqrt[Dp] E^((lh Sqrt[s])/Sqrt[Dp] - (Sqrt[s] x)/Sqrt[ Dp]) s)/((Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s + Dp (s + \[Theta]c))) + (Dp pinf \[Alpha]1 (s + \[Theta]c)^(3/2))/( Sqrt[s] (Sqrt[s] + Sqrt[Dp] \[Alpha]1) Sqrt[(s (s + \[Theta]c))/ Dp] (-Dc s + Dp (s + \[Theta]c))) - ( Dp E^((lh Sqrt[s])/Sqrt[Dp] - (Sqrt[s] x)/Sqrt[Dp]) pinf \[Alpha]1 (s + \[Theta]c)^(3/2))/( Sqrt[s] (Sqrt[s] + Sqrt[Dp] \[Alpha]1) Sqrt[(s (s + \[Theta]c))/ Dp] (-Dc s + Dp (s + \[Theta]c))) + ( A1 Sqrt[Dc] E^((lh Sqrt[s])/Sqrt[Dp] - (Sqrt[s] x)/Sqrt[ Dp]) (Sqrt[Dp s] \[Theta]c + \[Alpha]1 Sqrt[ Dc Dp s (s + \[Theta]c)]))/( Sqrt[s] (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s + Dp (s + \[Theta]c))) - ( A1 Sqrt[Dc] E^( lh Sqrt[(s + \[Theta]c)/Dc] - x Sqrt[(s + \[Theta]c)/ Dc]) (s Sqrt[Dc (s + \[Theta]c)] + \[Alpha]1 Sqrt[ Dc Dp s (s + \[Theta]c)]))/( Sqrt[s] (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s + Dp (s + \[Theta]c))) + ( pinf (\[Theta]c^3 Sqrt[Dp s^3 (s + \[Theta]c)] + 3 \[Theta]c^2 Sqrt[Dp s^5 (s + \[Theta]c)] + 3 \[Theta]c Sqrt[Dp s^7 (s + \[Theta]c)] + Sqrt[ Dp s^9 (s + \[Theta]c)]))/( s^(3/2) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c)^2 Sqrt[( s (s + \[Theta]c))/Dp] (-Dc s + Dp (s + \[Theta]c))) In[166]:= eq53 = Map[FullSimplify, eq52] Out[166]= (A2 Sqrt[Dp] E^((Sqrt[s] (lh - x))/Sqrt[Dp]) \[Alpha]1)/( s (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c)) - ( A2 (Sqrt[Dp s] + Dp \[Alpha]1))/( Sqrt[Dp] s (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c)) - ( Dc pinf Sqrt[ s])/((Sqrt[s] + Sqrt[Dp] \[Alpha]1) (-Dc s + Dp (s + \[Theta]c))) - (Sqrt[Dc] pinf Sqrt[Dc Dp s] \[Alpha]1)/( Sqrt[s] (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (-Dc s + Dp (s + \[Theta]c))) + ( Sqrt[Dc] E^((Sqrt[s] (lh - x))/Sqrt[Dp]) pinf Sqrt[ Dc Dp s] \[Alpha]1)/( Sqrt[s] (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (-Dc s + Dp (s + \[Theta]c))) + ( A1 Sqrt[Dc] Sqrt[Dp] E^((Sqrt[s] (lh - x))/Sqrt[ Dp]) s)/((Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s + Dp (s + \[Theta]c))) + ( pinf Sqrt[s] \[Alpha]1 (s + \[Theta]c)^( 5/2))/((Sqrt[s] + Sqrt[Dp] \[Alpha]1) ((s (s + \[Theta]c))/Dp)^( 3/2) (-Dc s + Dp (s + \[Theta]c))) - ( E^((Sqrt[s] (lh - x))/Sqrt[Dp]) pinf Sqrt[ s] \[Alpha]1 (s + \[Theta]c)^( 5/2))/((Sqrt[s] + Sqrt[Dp] \[Alpha]1) ((s (s + \[Theta]c))/Dp)^( 3/2) (-Dc s + Dp (s + \[Theta]c))) + ( A1 Sqrt[Dc] E^((Sqrt[s] (lh - x))/Sqrt[ Dp]) (Sqrt[Dp s] \[Theta]c + \[Alpha]1 Sqrt[ Dc Dp s (s + \[Theta]c)]))/( Sqrt[s] (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s + Dp (s + \[Theta]c))) - ( A1 Sqrt[Dc] E^((lh - x) Sqrt[(s + \[Theta]c)/ Dc]) (s Sqrt[Dc (s + \[Theta]c)] + \[Alpha]1 Sqrt[ Dc Dp s (s + \[Theta]c)]))/( Sqrt[s] (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s + Dp (s + \[Theta]c))) + ( pinf (\[Theta]c^3 Sqrt[Dp s^3 (s + \[Theta]c)] + 3 \[Theta]c^2 Sqrt[Dp s^5 (s + \[Theta]c)] + 3 \[Theta]c Sqrt[Dp s^7 (s + \[Theta]c)] + Sqrt[ Dp s^9 (s + \[Theta]c)]))/( s^(3/2) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c)^2 Sqrt[( s (s + \[Theta]c))/Dp] (-Dc s + Dp (s + \[Theta]c))) In[167]:= eq54 = Collect[eq52, { E^((lh Sqrt[s])/Sqrt[Dp] - (Sqrt[s] x)/Sqrt[Dp]), E^( lh Sqrt[(s + \[Theta]c)/Dc] - x Sqrt[(s + \[Theta]c)/Dc])}, Simplify] Out[167]= -(( A1 Sqrt[Dc] E^( lh Sqrt[(s + \[Theta]c)/Dc] - x Sqrt[(s + \[Theta]c)/ Dc]) (s Sqrt[Dc (s + \[Theta]c)] + \[Alpha]1 Sqrt[ Dc Dp s (s + \[Theta]c)]))/( Sqrt[s] (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s + Dp (s + \[Theta]c)))) + (E^((lh Sqrt[s])/Sqrt[Dp] - (Sqrt[s] x)/ Sqrt[Dp]) (A1 Sqrt[Dc] s (Sqrt[Dp] s^(3/2) + Sqrt[Dp s] \[Theta]c + \[Alpha]1 Sqrt[ Dc Dp s (s + \[Theta]c)]) + \[Alpha]1 (A2 Sqrt[Dp] Sqrt[ s] (-Dc s + Dp (s + \[Theta]c)) + pinf (s + \[Theta]c) (Sqrt[Dc] s Sqrt[Dc Dp s] - Dp^2 Sqrt[s + \[Theta]c] Sqrt[(s (s + \[Theta]c))/ Dp]))))/(s^( 3/2) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s + Dp (s + \[Theta]c))) + (-Sqrt[Dc] Sqrt[Dp] pinf s^2 Sqrt[ Dc Dp s] \[Alpha]1 (s + \[Theta]c)^3 + Dc s^(5/2) (s + \[Theta]c)^2 (A2 Sqrt[Dp s] + A2 Dp \[Alpha]1 - Sqrt[Dp] pinf Sqrt[s] (s + \[Theta]c)) + 1/(Sqrt[s + \[Theta]c]) Dp (-A2 s^(3/2) (Sqrt[Dp s] + Dp \[Alpha]1) (s + \[Theta]c)^( 7/2) + Sqrt[Dp] pinf Sqrt[(s (s + \[Theta]c))/ Dp] (Dp s \[Alpha]1 (s + \[Theta]c)^4 + Sqrt[s + \[Theta]c] (\[Theta]c^3 Sqrt[ Dp s^3 (s + \[Theta]c)] + 3 \[Theta]c^2 Sqrt[Dp s^5 (s + \[Theta]c)] + 3 \[Theta]c Sqrt[Dp s^7 (s + \[Theta]c)] + Sqrt[ Dp s^9 (s + \[Theta]c)]))))/(Sqrt[Dp] s^( 5/2) (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c)^3 (-Dc s + Dp (s + \[Theta]c))) In[168]:= eq55 = Assuming[s > 0 && (s + \[Theta]c) > 0 && Dp > 0 && Dc > 0, FullSimplify[eq54[[1]]] + FullSimplify[eq54[[2]]] + FullSimplify[eq54[[3]]]] // Normal Out[168]= (-A2 + pinf (s + \[Theta]c))/(s (s + \[Theta]c)) - ( A1 E^((lh - x) Sqrt[(s + \[Theta]c)/Dc]) Sqrt[Dc/ s] (s Sqrt[Dc (s + \[Theta]c)] + \[Alpha]1 Sqrt[ Dc Dp s (s + \[Theta]c)]))/((Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s + Dp (s + \[Theta]c))) + (E^((s (lh - x))/Sqrt[ Dp s]) (-Sqrt[ Dp s] \[Alpha]1 (- 
9 months ago
7 Replies
 Gianluca Gorni 1 Vote This way it becomes reasonably simple in my opinion: sol = Out[163]; solSimplified = Assuming[(lh - x) < 0 && s > 0 && (s + \[Theta]c) > 0 && Dp > 0 && Dc > 0 && \[Alpha]1 > 0, Simplify@ PowerExpand@Simplify[Simplify[PowerExpand[Numerator@sol]]/ FullSimplify[Expand[Denominator@sol]]]]; solSimplified // Apart // PowerExpand // FullSimplify 
9 months ago
 Dear Gianlua, Would you like to explain the difference Simplify@ and Simplify[ ]? Thanks.
6 months ago
 They are the same. You can write f@x or f[x], they have the same effect. The syntax with @ can only be used with functions of one variable: Simplify[Sqrt[x^2], x>0] cannot be written with @. When you can use @ you can save brackets and the code is simpler. For example, in f@g@h@x you can remove g@ without caring for the brackets. It is more difficult to remove g from f[g[h[x]]].
6 months ago
 Kapio Letto 2 Votes You can for pure functions, for example: Simplify[#, x > 0] &@Sqrt[x^2] Sqrt[x^2] // Simplify[#, x > 0] & and do things like f@Sequence[x, y, z] (*f[x,y,z]*) f@@{x, y, z} (*f[x,y,z]*) f @@@ {{5, 6}, {9, 7}, {2, 5}} (*{f[5,6],f[9,7],f[2,5]}*) Not necessarily always convenient or shorter, but can come handy sometimes.
 Moderation Team 2 Votes @Zhonghui Ou, the question about @ is very basic and you could try searching docs for relevant info first as the rules of the forum suggest: http://wolfr.am/READ-1ST There is, for example, this tutorial: How to | Use Shorthand Notations:http://reference.wolfram.com/language/howto/UseShorthandNotations.htmlThis forum has written useful tips for beginners:http://community.wolfram.com/groups/-/m/t/1070264see Sander Huisman answer there about short notations (and other tips too). We would also suggest: Do not overload screen with too much not-needed code, it makes it hard to read: Avoid using Greek letters - they make code much longer Out cells are not always needed - given Input people can just run your code Try to minimize your example to the essence of your issue
 The simplest form can be gotten by two steps with the help of Mr. Gorni, In[79]:= eq51 = PowerExpand[q[x, s] /. eq50[[1, 1]]] // FullSimplify // Normal Out[79]= -(E^(-((2 Sqrt[s] x)/Sqrt[ Dp])) (A2 (-Sqrt[Dp] E^((Sqrt[s] (lh + x))/Sqrt[Dp]) \[Alpha]1 + E^((2 Sqrt[s] x)/Sqrt[ Dp]) (Sqrt[s] + Sqrt[Dp] \[Alpha]1)) (-Dc s + Dp (s + \[Theta]c)) - pinf (-Sqrt[Dp] E^((Sqrt[s] (lh + x))/Sqrt[Dp]) \[Alpha]1 + E^((2 Sqrt[s] x)/Sqrt[ Dp]) (Sqrt[s] + Sqrt[Dp] \[Alpha]1)) (s + \[Theta]c) (-Dc s + Dp (s + \[Theta]c)) + A1 (Dc s (-Sqrt[Dp] E^((Sqrt[s] (lh + x))/Sqrt[Dp]) \[Alpha]1 + E^((2 Sqrt[s] x)/Sqrt[ Dp] + ((lh - x) Sqrt[s + \[Theta]c])/Sqrt[ Dc]) (Sqrt[s] + Sqrt[Dp] \[Alpha]1)) Sqrt[ s + \[Theta]c] - Sqrt[Dc] Sqrt[Dp] E^((Sqrt[s] (lh + x))/Sqrt[Dp]) s (s + \[Theta]c))))/(s (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s + Dp (s + \[Theta]c))) In[80]:= eq52 = ExpandAll[eq51, x] // Cancel Out[80]= pinf/s - ( Sqrt[Dp] E^((lh Sqrt[s])/Sqrt[Dp] - (Sqrt[s] x)/Sqrt[Dp]) pinf \[Alpha]1)/(s (Sqrt[s] + Sqrt[Dp] \[Alpha]1)) - A2/( s (s + \[Theta]c)) + ( A2 Sqrt[Dp] E^((lh Sqrt[s])/Sqrt[Dp] - (Sqrt[s] x)/Sqrt[Dp]) \[Alpha]1)/( s (Sqrt[s] + Sqrt[Dp] \[Alpha]1) (s + \[Theta]c)) - ( A1 Sqrt[Dc] Sqrt[Dp] E^((lh Sqrt[s])/Sqrt[Dp] - (Sqrt[s] x)/Sqrt[ Dp]))/((Sqrt[s] + Sqrt[Dp] \[Alpha]1) (Dc s - Dp s - Dp \[Theta]c)) + ( A1 Dc E^((lh Sqrt[s + \[Theta]c])/Sqrt[Dc] - (x Sqrt[s + \[Theta]c])/ Sqrt[Dc]))/(Sqrt[s + \[Theta]c] (Dc s - Dp s - Dp \[Theta]c)) - ( A1 Dc Sqrt[Dp] E^((lh Sqrt[s])/Sqrt[Dp] - (Sqrt[s] x)/Sqrt[ Dp]) \[Alpha]1)/((Sqrt[s] + Sqrt[Dp] \[Alpha]1) Sqrt[ s + \[Theta]c] (Dc s - Dp s - Dp \[Theta]c))