Added Oct 18, 2019.
Problem Chapter 8.6.5.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \sin ^n(\lambda x) w_y + b \cos ^m(\beta x) w_z = c \sin ^k(\gamma x) w \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Sin[lambda*x]^n*D[w[x, y,z], y] + b*Cos[beta*x]^m*D[w[x,y,z],z]== c*Sin[gamma*x]^k*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \sqrt {\cos ^2(\gamma x)} \sec (\gamma x) \sin ^{k+1}(\gamma x) \, _2F_1\left (\frac {1}{2},\frac {k+1}{2};\frac {k+3}{2};\sin ^2(\gamma x)\right )}{\gamma k+\gamma }\right ) c_1\left (\frac {b \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{m+1}(\beta x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(\beta x)\right )}{\beta m+\beta }+z,y-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*sin(lambda*x)^n*diff(w(x,y,z),y)+ b*cos(beta*x)^m*diff(w(x,y,z),z)= c*sin(gamma*x)^k*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \sin \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \cos \left ( \beta \,x \right ) \right ) ^{m}\,{\rm d}x+z \right ) {{\rm e}^{\int \!c \left ( \sin \left ( \gamma \,x \right ) \right ) ^{k}\,{\rm d}x}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.5.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \cos ^n(\lambda x)w_y + b \sin ^m(\beta y) w_z = \left ( c \cos ^k(\gamma y)+s \sin ^r(\mu z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cos[lambda*x]^n*D[w[x, y,z], y] + b*Sin[beta*y]^m*D[w[x,y,z],z]== (c*Cos[gamma*y]^k+s*Sin[mu*z]^r)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+b*cos(lambda*x)^n*diff(w(x,y,z),y)+ b*sin(beta*y)^m*diff(w(x,y,z),z)= (c*cos(gamma*y)^k+s*sin(mu*z)^r)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!b \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!b \left ( \sin \left ( \beta \, \left ( \int \! \left ( \cos \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b}b-\int \!b \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) {{\rm e}^{\int ^{x}\! \left ( \sin \left ( \mu \, \left ( \int \!b \left ( \sin \left ( \beta \, \left ( b\int \! \left ( \cos \left ( \lambda \,{\it \_g} \right ) \right ) ^{n}\,{\rm d}{\it \_g}-\int \!b \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}\,{\rm d}{\it \_g}-\int ^{x}\!b \left ( \sin \left ( \beta \, \left ( \int \! \left ( \cos \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b}b-\int \!b \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \right ) ^{m}{d{\it \_b}}+z \right ) \right ) \right ) ^{r}s+c \left ( \cos \left ( \gamma \, \left ( -\int \!b \left ( \cos \left ( \lambda \,{\it \_g} \right ) \right ) ^{n}\,{\rm d}{\it \_g}+\int \!b \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x-y \right ) \right ) \right ) ^{k}{d{\it \_g}}}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.5.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ w_x + a \cos ^n(\lambda x) w_y + b \tan ^m(\beta y) w_z = \left ( c \cos ^k(\gamma y) + s \tan ^k(\mu z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + a*Cos[lambda*x]^n*D[w[x, y,z], y] + b*Tan[beta*y]^m*D[w[x,y,z],z]== (c*Cos[gamma*y]^k+s*Tan[mu*z]^k)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := diff(w(x,y,z),x)+a*cos(lambda*x)^n*diff(w(x,y,z),y)+ b*tan(beta*y)^m*diff(w(x,y,z),z)= (c*cos(gamma*y)^k+s*tan(mu*z)^k)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int ^{x}\!b \left ( {\frac {-\tan \left ( \beta \, \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) -\tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) }{\tan \left ( \beta \, \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) -1}} \right ) ^{m}{d{\it \_b}}+z \right ) {{\rm e}^{\int ^{x}\!c \left ( \cos \left ( \gamma \, \left ( -\int \!a \left ( \cos \left ( \lambda \,{\it \_g} \right ) \right ) ^{n}\,{\rm d}{\it \_g}+\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x-y \right ) \right ) \right ) ^{k}+s \left ( {\sin \left ( \mu \, \left ( \int \!b \left ( {\frac {-\tan \left ( \beta \, \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) -\tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_g} \right ) \right ) ^{n}\,{\rm d}{\it \_g} \right ) }{\tan \left ( \beta \, \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_g} \right ) \right ) ^{n}\,{\rm d}{\it \_g} \right ) -1}} \right ) ^{m}\,{\rm d}{\it \_g}-\int ^{x}\!b \left ( {\frac {-\tan \left ( \beta \, \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) -\tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) }{\tan \left ( \beta \, \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) -1}} \right ) ^{m}{d{\it \_b}}+z \right ) \right ) \left ( \cos \left ( \mu \, \left ( \int \!b \left ( {\frac {-\tan \left ( \beta \, \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) -\tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_g} \right ) \right ) ^{n}\,{\rm d}{\it \_g} \right ) }{\tan \left ( \beta \, \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_g} \right ) \right ) ^{n}\,{\rm d}{\it \_g} \right ) -1}} \right ) ^{m}\,{\rm d}{\it \_g}-\int ^{x}\!b \left ( {\frac {-\tan \left ( \beta \, \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) -\tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) }{\tan \left ( \beta \, \left ( -\int \!a \left ( \cos \left ( \lambda \,x \right ) \right ) ^{n}\,{\rm d}x+y \right ) \right ) \tan \left ( \beta \,a\int \! \left ( \cos \left ( \lambda \,{\it \_b} \right ) \right ) ^{n}\,{\rm d}{\it \_b} \right ) -1}} \right ) ^{m}{d{\it \_b}}+z \right ) \right ) \right ) ^{-1}} \right ) ^{k}{d{\it \_g}}}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.5.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 \sin ^{n_1}(\lambda _1 x) w_x + b_1 \cot ^{m_1}(\beta _1 y) w_y + c_1 \cos ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \cos ^{n_2}(\lambda _2 x) + b_2 \sin ^{m_2}(\beta _2 y) + c_2 \cos ^{k_2}(\gamma _2 z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Sin[lambda1*z]^n1*D[w[x, y,z], x] + b1*Cot[beta1*y]^m1*D[w[x, y,z], y] + c1*Cos[gamma1*z]^k1*D[w[x,y,z],z]== (a2*Cos[lambda2*z]^n2 + b2*Sin[beta2*y]^m2 + c2*Cos[gamma2*z]^k2)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a1*sin(lambda1*z)^n1*diff(w(x,y,z),x)+ b1*cot(beta1*y)^m1*diff(w(x,y,z),y)+ c1*cos(gamma1*z)^k1*diff(w(x,y,z),z)= (a2*cos(lambda2*z)^n2 + b2*sin(beta2*y)^m2 + c2*cos(gamma2*z)^k2)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \! \left ( {\frac {\cos \left ( \beta 1\,y \right ) }{\sin \left ( \beta 1\,y \right ) }} \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \cos \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z,-\int ^{y}\!{\frac {{\it a1}}{{\it b1}} \left ( \sin \left ( \lambda 1\,\RootOf \left ( \int \! \left ( {\frac {\cos \left ( \beta 1\,{\it \_f} \right ) }{\sin \left ( \beta 1\,{\it \_f} \right ) }} \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cos \left ( \gamma 1\,{\it \_b} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_b}}-\int \! \left ( {\frac {\cos \left ( \beta 1\,y \right ) }{\sin \left ( \beta 1\,y \right ) }} \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \cos \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it n1}} \left ( {\frac {\cos \left ( \beta 1\,{\it \_f} \right ) }{\sin \left ( \beta 1\,{\it \_f} \right ) }} \right ) ^{-{\it m1}}}{d{\it \_f}}+x \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}}{{\it b1}} \left ( {\it a2}\, \left ( \cos \left ( \lambda 2\,\RootOf \left ( -\int \! \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}+\int ^{{\it \_Z}}\!{\frac { \left ( \cos \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}+\int \! \left ( {\frac {\cos \left ( \beta 1\,y \right ) }{\sin \left ( \beta 1\,y \right ) }} \right ) ^{-{\it m1}}\,{\rm d}y-\int \!{\frac {{\it b1}\, \left ( \cos \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it n2}}+{\it b2}\, \left ( \sin \left ( \beta 2\,{\it \_f} \right ) \right ) ^{{\it m2}}+{\it c2}\, \left ( \cos \left ( \gamma 2\,\RootOf \left ( -\int \! \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}+\int ^{{\it \_Z}}\!{\frac { \left ( \cos \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}+\int \! \left ( {\frac {\cos \left ( \beta 1\,y \right ) }{\sin \left ( \beta 1\,y \right ) }} \right ) ^{-{\it m1}}\,{\rm d}y-\int \!{\frac {{\it b1}\, \left ( \cos \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{{\it k2}} \right ) }{d{\it \_f}}}}\]
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Added Oct 18, 2019.
Problem Chapter 8.6.5.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\) \[ a_1 \tan ^{n_1}(\lambda _1 x) w_x + b_1 \cot ^{m_1}(\beta _1 y) w_y + c_1 \cot ^{k_1}(\gamma _1 z) w_z = \left ( a_2 \cot ^{n_2}(\lambda _2 x) + b_2 \tan ^{m_2}(\beta _2 y) + c_2 \cot ^{k_2}(\gamma _2 z) \right ) w \]
Mathematica ✗
ClearAll["Global`*"]; pde = a1*Tan[lambda1*z]^n1*D[w[x, y,z], x] + b1*Cot[beta1*y]^m1*D[w[x, y,z], y] + c1*Cot[gamma1*z]^k1*D[w[x,y,z],z]== (a2*Cot[lambda2*z]^n2 + b2*Tan[beta2*y]^m2 + c2*Cot[gamma2*z]^k2)*w[x,y,z]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x,y,z}], 60*10]];
Failed
Maple ✓
restart; local gamma; pde := a1*tan(lambda1*z)^n1*diff(w(x,y,z),x)+ b1*cot(beta1*y)^m1*diff(w(x,y,z),y)+ c1*cot(gamma1*z)^k1*diff(w(x,y,z),z)= (a2*cot(lambda2*z)^n2 + b2*tan(beta2*y)^m2 + c2*cot(gamma2*z)^k2)*w(x,y,z); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left ( x,y,z \right ) ={\it \_F1} \left ( -\int \! \left ( \cot \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \cot \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z,-\int ^{y}\!{\frac {{\it a1}}{{\it b1}} \left ( {\sin \left ( \lambda 1\,\RootOf \left ( \int \! \left ( {\frac {\cos \left ( \beta 1\,{\it \_f} \right ) }{\sin \left ( \beta 1\,{\it \_f} \right ) }} \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac {{\it b1}}{{\it c1}} \left ( {\frac {\cos \left ( \gamma 1\,{\it \_b} \right ) }{\sin \left ( \gamma 1\,{\it \_b} \right ) }} \right ) ^{-{\it k1}}}{d{\it \_b}}-\int \! \left ( \cot \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \cot \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \left ( \cos \left ( \lambda 1\,\RootOf \left ( \int \! \left ( {\frac {\cos \left ( \beta 1\,{\it \_f} \right ) }{\sin \left ( \beta 1\,{\it \_f} \right ) }} \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac {{\it b1}}{{\it c1}} \left ( {\frac {\cos \left ( \gamma 1\,{\it \_b} \right ) }{\sin \left ( \gamma 1\,{\it \_b} \right ) }} \right ) ^{-{\it k1}}}{d{\it \_b}}-\int \! \left ( \cot \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \cot \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{-1}} \right ) ^{{\it n1}} \left ( {\frac {\cos \left ( \beta 1\,{\it \_f} \right ) }{\sin \left ( \beta 1\,{\it \_f} \right ) }} \right ) ^{-{\it m1}}}{d{\it \_f}}+x \right ) {{\rm e}^{\int ^{y}\!{\frac { \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}}{{\it b1}} \left ( {\it a2}\, \left ( {\cos \left ( \lambda 2\,\RootOf \left ( \int \! \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cot \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \cot \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \cot \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \left ( \sin \left ( \lambda 2\,\RootOf \left ( \int \! \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cot \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \cot \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \cot \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{-1}} \right ) ^{{\it n2}}+{\it b2}\, \left ( {\frac {\sin \left ( \beta 2\,{\it \_f} \right ) }{\cos \left ( \beta 2\,{\it \_f} \right ) }} \right ) ^{{\it m2}}+{\it c2}\, \left ( {\cos \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cot \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \cot \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \cot \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \left ( \sin \left ( \gamma 2\,\RootOf \left ( \int \! \left ( \cot \left ( \beta 1\,{\it \_f} \right ) \right ) ^{-{\it m1}}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac { \left ( \cot \left ( \gamma 1\,{\it \_a} \right ) \right ) ^{-{\it k1}}{\it b1}}{{\it c1}}}{d{\it \_a}}-\int \! \left ( \cot \left ( \beta 1\,y \right ) \right ) ^{-{\it m1}}\,{\rm d}y+\int \!{\frac {{\it b1}\, \left ( \cot \left ( \gamma 1\,z \right ) \right ) ^{-{\it k1}}}{{\it c1}}}\,{\rm d}z \right ) \right ) \right ) ^{-1}} \right ) ^{{\it k2}} \right ) }{d{\it \_f}}}}\]
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