6.2.19 6.5

6.2.19.1 [693] problem number 1
6.2.19.2 [694] problem number 2
6.2.19.3 [695] problem number 3
6.2.19.4 [696] problem number 4
6.2.19.5 [697] problem number 5
6.2.19.6 [698] problem number 6
6.2.19.7 [699] problem number 7
6.2.19.8 [700] problem number 8
6.2.19.9 [701] problem number 9
6.2.19.10 [702] problem number 10
6.2.19.11 [703] problem number 11

6.2.19.1 [693] problem number 1

problem number 693

Added January 20, 2019.

Problem 2.6.5.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + a \sin ^k(\lambda x) \cos ^n(\mu y) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*Sin[lambda*x]^k*Cos[mu*y]^n*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {\sin ^2(\mu y)} \csc (\mu y) \cos ^{1-n}(\mu y) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(\mu y)\right )}{\mu (n-1)}-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{k+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\sin ^2(\lambda x)\right )}{k \lambda +\lambda }\right )\right \}\right \}\]

Maple

restart; 
pde :=   diff(w(x,y),x)+ a*sin(lambda*x)^k*cos(mu*y)^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -\int \! \left ( \sin \left ( \lambda \,x \right ) \right ) ^{k}\,{\rm d}x+\int \!{\frac { \left ( \cos \left ( \mu \,y \right ) \right ) ^{-n}}{a}}\,{\rm d}y \right ) \] Has unresolved integrals

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6.2.19.2 [694] problem number 2

problem number 694

Added January 20, 2019.

Problem 2.6.5.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( y^2-y \tan x+a(1-a) \cot ^2 x \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 - y*Tan[x] + a*(1 - a)*Cot[x]^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (-\frac {\left (-\sin ^2(x)\right )^{\frac {1}{2} i \sqrt {a-1} \sqrt {a} \sqrt {\frac {1}{a-a^2}-4}} \left (i \sqrt {a-1} \sqrt {a} \sqrt {\frac {1}{a-a^2}-4} \cos (x)+2 y \sin (x)+\cos (x)\right )}{-i \sqrt {a-1} \sqrt {a} \sqrt {\frac {1}{a-a^2}-4} \cos (x)+2 y \sin (x)+\cos (x)}\right )\right \}\right \}\]

Maple

restart; 
pde :=   diff(w(x,y),x)+ (y^2-y *tan(x)+a*(1-a)*cot(x)^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac { \left ( \sin \left ( x \right ) \right ) ^{2\,a-1} \left ( y\sin \left ( x \right ) +\cos \left ( x \right ) a \right ) }{ \left ( a-1 \right ) \cos \left ( x \right ) -y\sin \left ( x \right ) }} \right ) \]

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6.2.19.3 [695] problem number 3

problem number 695

Added January 20, 2019.

Problem 2.6.5.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( y^2-m y \tan x+b^2 \cos ^{2 m} x \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 - m*y*Tan[x] + b^2*Cos[x]^(2*m))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\frac {\sqrt {b^2} \sqrt {\sin ^2(x)} \csc (x) \cos ^{m+1}(x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(x)\right )}{m+1}+\tan ^{-1}\left (\frac {y \cos ^{-m}(x)}{\sqrt {b^2}}\right )\right )\right \}\right \}\]

Maple

restart; 
pde :=   diff(w(x,y),x)+ (y^2-m*y*tan(x)+b^2*cos(x)^(2*m) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {\frac {3\,\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( 1/3\, \left ( \cos \left ( x \right ) -1 \right ) \left ( \cos \left ( x \right ) +1 \right ) \left ( m-1 \right ) \hypergeom \left ( [3/2,-m/2+3/2],[5/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) +\hypergeom \left ( [1/2,-m/2+1/2],[3/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ) b \left ( \cos \left ( x \right ) \right ) ^{2}\cos \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ) \hypergeom \left ( [1/2,-m/2+1/2],[3/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ) +3\,y\sin \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ) \hypergeom \left ( [1/2,-m/2+1/2],[3/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ) \left ( \cos \left ( x \right ) \right ) ^{m}}{-3\,y\cos \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ) \hypergeom \left ( [1/2,-m/2+1/2],[3/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ) \left ( \cos \left ( x \right ) \right ) ^{m}+3\,\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}}\sin \left ( b\sqrt { \left ( \cos \left ( x \right ) \right ) ^{2\,m-2}} \left ( \cos \left ( x \right ) \right ) ^{-m+1}\sin \left ( x \right ) \hypergeom \left ( [1/2,-m/2+1/2],[3/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ) \left ( 1/3\, \left ( \cos \left ( x \right ) -1 \right ) \left ( \cos \left ( x \right ) +1 \right ) \left ( m-1 \right ) \hypergeom \left ( [3/2,-m/2+3/2],[5/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) +\hypergeom \left ( [1/2,-m/2+1/2],[3/2], \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \right ) b \left ( \cos \left ( x \right ) \right ) ^{2}}} \right ) \] Mathematica answer is simpler

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6.2.19.4 [696] problem number 4

problem number 696

Added January 20, 2019.

Problem 2.6.5.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( y^2+m y \cot x+b^2 \sin ^m x \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + m*y*Cot[x] + b^2*Sin[x]^m)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=   diff(w(x,y),x)+ (y^2+m*y*cot(x)+b^2*sin(x)^m )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

server hangs Server hangs

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6.2.19.5 [697] problem number 5

problem number 697

Added January 20, 2019.

Problem 2.6.5.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( y^2-2 \lambda ^2 \tan ^2(\lambda x)-2 \lambda ^2 \cot ^2(\lambda x) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 - 2*lambda^2*Tan[lambda*x]^2 - 2*lambda^2*Cot[lambda*x]^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=   diff(w(x,y),x)+ (y^2-2*lambda^2*tan(lambda*x)^2-2*lambda^2*cot(lambda*x)^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( 8\,{\frac {\sqrt {2\,\cos \left ( 2\,\lambda \,x \right ) -2} \left ( y\sin \left ( 2\,\lambda \,x \right ) -2\,\lambda \,\cos \left ( 2\,\lambda \,x \right ) \right ) }{8\,\sqrt {2\,\cos \left ( 2\,\lambda \,x \right ) -2} \left ( y\sin \left ( 2\,\lambda \,x \right ) -2\,\lambda \,\cos \left ( 2\,\lambda \,x \right ) \right ) \ln \left ( \cos \left ( \lambda \,x \right ) +1/2\,\sqrt {2\,\cos \left ( 2\,\lambda \,x \right ) -2} \right ) -4\,y\sin \left ( \lambda \,x \right ) \left ( \cos \left ( 2\,\lambda \,x \right ) \right ) ^{2}+2\,y\sin \left ( 5\,\lambda \,x \right ) -y\sin \left ( 7\,\lambda \,x \right ) +y\sin \left ( \lambda \,x \right ) -14\,\cos \left ( \lambda \,x \right ) \lambda +14\,\cos \left ( 3\,\lambda \,x \right ) \lambda +2\,\cos \left ( 5\,\lambda \,x \right ) \lambda -2\,\cos \left ( 7\,\lambda \,x \right ) \lambda }} \right ) \]

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6.2.19.6 [698] problem number 6

problem number 698

Added January 20, 2019.

Problem 2.6.5.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( y^2+\lambda (a+b)+2 a b+a(\lambda -a) \tan ^2(\lambda x)+ b(\lambda -b) \cot ^2(\lambda x) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*(a + b) + 2*a*b + a*(lambda - a)*Tan[lambda*x]^2 + b*(lambda - b)*Cot[lambda*x]^2)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=   diff(w(x,y),x)+ ( y^2+lambda*(a+b)+2*a*b+a*(lambda -a)*tan(lambda*x)^2+ b*(lambda -b)*cot(lambda*x)^2)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( 2\,{ \left ( \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2}a- \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}b-\sin \left ( \lambda \,x \right ) y\cos \left ( \lambda \,x \right ) \right ) \left ( a-3/2\,\lambda \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{{\frac {a}{\lambda }}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{{\frac {b}{\lambda }}} \left ( \cos \left ( \lambda \,x \right ) \right ) ^{{\frac {a-\lambda }{\lambda }}} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{{\frac {b-\lambda }{\lambda }}} \left ( -4\,\lambda \, \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2} \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( b-\lambda +a \right ) \hypergeom \left ( [2,{\frac {-b+2\,\lambda -a}{\lambda }}],[-1/2\,{\frac {2\,a-5\,\lambda }{\lambda }}], \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2} \right ) +2\, \left ( \left ( \lambda -b \right ) \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2}+\sin \left ( \lambda \,x \right ) y\cos \left ( \lambda \,x \right ) + \left ( \sin \left ( \lambda \,x \right ) \right ) ^{2} \left ( a-\lambda \right ) \right ) \hypergeom \left ( [1,{\frac {-b+\lambda -a}{\lambda }}],[-1/2\,{\frac {2\,a-3\,\lambda }{\lambda }}], \left ( \cos \left ( \lambda \,x \right ) \right ) ^{2} \right ) \left ( a-3/2\,\lambda \right ) \right ) ^{-1}} \right ) \]

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6.2.19.7 [699] problem number 7

problem number 699

Added January 20, 2019.

Problem 2.6.5.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( \lambda \sin (\lambda x) y^2 + a \cos ^n(\lambda x) y-a \cos ^{n-1}(\lambda x) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + a*Cos[lambda*x]^n*y - a*Cos[lambda*x]^(n - 1))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=   diff(w(x,y),x)+ (lambda*sin(lambda*x)* y^2 + a*cos(lambda*x)^n*y-a*cos(lambda*x)^(n-1))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

time expired

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6.2.19.8 [700] problem number 8

problem number 700

Added January 20, 2019.

Problem 2.6.5.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( \lambda \sin (\lambda x) y^2 + a \sin (\lambda x) y-a \tan (\lambda x) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + a*Sin[lambda*x]*y - a*Tan[lambda*x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=   diff(w(x,y),x)+ (lambda*sin(lambda*x)*y^2 + a*sin(lambda*x)*y-a*tan(lambda*x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {(\cos \left ( \lambda \,x \right ) y-1){{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( a{{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( \cos \left ( \lambda \,x \right ) y-1 \right ) \Ei \left ( 1,{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }} \right ) -y\lambda \right ) ^{-1}} \right ) \]

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6.2.19.9 [701] problem number 9

problem number 701

Added January 20, 2019.

Problem 2.6.5.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ w_x + \left ( \lambda \sin (\lambda x) y^2 + a \sin (\lambda x) y-a \tan (\lambda x) \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*Sin[lambda*x]*y^2 + a*Sin[lambda*x]*y - a*Tan[lambda*x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=   diff(w(x,y),x)+ (lambda*sin(lambda*x)*y^2 + a*sin(lambda*x)*y-a*tan(lambda*x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( {(\cos \left ( \lambda \,x \right ) y-1){{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( a{{\rm e}^{{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }}}} \left ( \cos \left ( \lambda \,x \right ) y-1 \right ) \Ei \left ( 1,{\frac {\cos \left ( \lambda \,x \right ) a}{\lambda }} \right ) -y\lambda \right ) ^{-1}} \right ) \]

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6.2.19.10 [702] problem number 10

problem number 702

Added January 20, 2019.

Problem 2.6.5.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + \left ( A e^{\lambda x} \cos (a y) + B e^{\mu x} \sin (a y) + A e^{\lambda x} \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (A*Exp[lambda*x]*Cos[a*y] + B*Exp[mu*x]*Sin[a*y] + A*Exp[lambda*x])*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=   diff(w(x,y),x)+ (A*exp(lambda*x)*cos(a*y) + B*exp(mu*x)*sin(a*y) + A*exp(lambda*x))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( 1/2\,{\frac {1}{ \left ( \cos \left ( 1/2\,ya \right ) \right ) ^{2}a \left ( -\lambda +\mu \right ) } \left ( Aa \left ( \cos \left ( ya \right ) +1 \right ) \int \!{{\rm e}^{{\frac {-B{{\rm e}^{\mu \,x}}a+\lambda \,x\mu }{\mu }}}}\,{\rm d}x-\sin \left ( ya \right ) {{\rm e}^{-{\frac {B{{\rm e}^{\mu \,x}}a}{\mu }}}} \right ) } \right ) \]

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6.2.19.11 [703] problem number 11

problem number 703

Added January 20, 2019.

Problem 2.6.5.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ \sin ^{n+1}(2 x) w_x + \left ( a y^2 \sin ^{2 n}x + b \cos ^{2 n} x \right ) w_y = 0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  Sin[2*x]^(n + 1)*D[w[x, y], x] + (a*y^2*Sin[x]^(2*n) + b*Cos[x]^(2*n))*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$Aborted

Maple

restart; 
pde :=   sin(2*x)^(n+1)*diff(w(x,y),x)+ (a*y^2*sin(x)^(2*n) + b*cos(x)^(2*n) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left ( x,y \right ) ={\it \_F1} \left ( -{\frac { \left ( \cos \left ( x \right ) \right ) ^{-\sqrt {{n}^{2}-a{4}^{-n}b}} \left ( \left ( \sin \left ( x \right ) \right ) ^{\sqrt {{n}^{2}-a{4}^{-n}b}+2\,n}ya+ \left ( \sin \left ( 2\,x \right ) \right ) ^{n} \left ( \sin \left ( x \right ) \right ) ^{\sqrt {{n}^{2}-a{4}^{-n}b}} \left ( \sqrt {{n}^{2}-a{4}^{-n}b}+n \right ) \right ) }{ay \left ( \sin \left ( x \right ) \right ) ^{2\,n}- \left ( \sin \left ( 2\,x \right ) \right ) ^{n}\sqrt {{n}^{2}-a{4}^{-n}b}+ \left ( \sin \left ( 2\,x \right ) \right ) ^{n}n}} \right ) \]

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