6.8.18 6.5
6.8.18.1 [1864] Problem 1
problem number 1864
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]];
\[\left \{\left \{w(x,y,z)\to \exp \left (\frac {c \sqrt {\cos ^2(\gamma x)} \sec (\gamma x) \sin ^{k+1}(\gamma x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {k+1}{2},\frac {k+3}{2},\sin ^2(\gamma x)\right )}{\gamma k+\gamma }\right ) c_1\left (\frac {b \sqrt {\sin ^2(\beta x)} \csc (\beta x) \cos ^{m+1}(\beta x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(\beta x)\right )}{\beta m+\beta }+z,y-\frac {a \sqrt {\cos ^2(\lambda x)} \sec (\lambda x) \sin ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(\lambda x)\right )}{\lambda n+\lambda }\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = f_{1} \left (-a \int \sin \left (\lambda x \right )^{n}d x +y , -b \int \cos \left (\beta x \right )^{m}d x +z \right ) {\mathrm e}^{c \int \sin \left (\gamma x \right )^{k}d x}\]
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6.8.18.2 [1865] Problem 2
problem number 1865
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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right )}{\lambda n+\lambda }+y,z-\int _1^xb \sin ^m\left (\frac {\beta \left (a \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{n+1}(\lambda x)+\lambda (n+1) y-a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}\right )}{\lambda (n+1)}\right )dK[1]\right ) \exp \left (\int _1^x\left (c \cos ^k\left (\frac {\gamma \left (a \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{n+1}(\lambda x)+\lambda (n+1) y-a \cos ^{n+1}(\lambda K[2]) \csc (\lambda K[2]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[2])\right ) \sqrt {\sin ^2(\lambda K[2])}\right )}{\lambda (n+1)}\right )+s \sin ^r\left (\mu \left (z-\int _1^xb \sin ^m\left (\frac {\beta \left (a \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{n+1}(\lambda x)+\lambda (n+1) y-a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}\right )}{\lambda (n+1)}\right )dK[1]+\int _1^{K[2]}b \sin ^m\left (\frac {\beta \left (a \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{n+1}(\lambda x)+\lambda (n+1) y-a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}\right )}{\lambda (n+1)}\right )dK[1]\right )\right )\right )dK[2]\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = f_{1} \left (-b \int \cos \left (\lambda x \right )^{n}d x +y , -b \int _{}^{x}{\left (-\sin \left (\beta \left (b \int \cos \left (\lambda x \right )^{n}d x -b \int \cos \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} -y \right )\right )\right )}^{m}d \textit {\_g} +z \right ) {\mathrm e}^{\int _{}^{x}\left (c {\cos \left (\gamma \left (b \int \cos \left (\lambda x \right )^{n}d x -b \int \cos \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} -y \right )\right )}^{k}+s {\sin \left (\mu \left (-b \int _{}^{x}{\left (-\sin \left (\beta \left (b \int \cos \left (\lambda x \right )^{n}d x -b \int \cos \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} -y \right )\right )\right )}^{m}d \textit {\_g} +z +b \int {\left (-\sin \left (\beta \left (b \int \cos \left (\lambda x \right )^{n}d x -b \int \cos \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} -y \right )\right )\right )}^{m}d \textit {\_g} \right )\right )}^{r}\right )d \textit {\_g}}\]
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6.8.18.3 [1866] Problem 3
problem number 1866
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]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {a \sqrt {\sin ^2(\lambda x)} \csc (\lambda x) \cos ^{n+1}(\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right )}{\lambda n+\lambda }+y,z-\int _1^xb \tan ^m\left (\frac {\beta \left (a \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{n+1}(\lambda x)+\lambda (n+1) y-a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}\right )}{\lambda (n+1)}\right )dK[1]\right ) \exp \left (\int _1^x\left (c \cos ^k\left (\frac {\gamma \left (a \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{n+1}(\lambda x)+\lambda (n+1) y-a \cos ^{n+1}(\lambda K[2]) \csc (\lambda K[2]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[2])\right ) \sqrt {\sin ^2(\lambda K[2])}\right )}{\lambda (n+1)}\right )+s \tan ^k\left (\mu \left (z-\int _1^xb \tan ^m\left (\frac {\beta \left (a \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{n+1}(\lambda x)+\lambda (n+1) y-a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}\right )}{\lambda (n+1)}\right )dK[1]+\int _1^{K[2]}b \tan ^m\left (\frac {\beta \left (a \csc (\lambda x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda x)\right ) \sqrt {\sin ^2(\lambda x)} \cos ^{n+1}(\lambda x)+\lambda (n+1) y-a \cos ^{n+1}(\lambda K[1]) \csc (\lambda K[1]) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(\lambda K[1])\right ) \sqrt {\sin ^2(\lambda K[1])}\right )}{\lambda (n+1)}\right )dK[1]\right )\right )\right )dK[2]\right )\right \}\right \}\]
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'));
\[w \left (x , y , z\right ) = f_{1} \left (-a \int \cos \left (\lambda x \right )^{n}d x +y , -b \int _{}^{x}{\left (-\frac {\tan \left (\beta \left (a \int \cos \left (\lambda x \right )^{n}d x -y \right )\right )-\tan \left (\beta a \int \cos \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} \right )}{1+\tan \left (\beta \left (a \int \cos \left (\lambda x \right )^{n}d x -y \right )\right ) \tan \left (\beta a \int \cos \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} \right )}\right )}^{m}d \textit {\_g} +z \right ) {\mathrm e}^{\int _{}^{x}\left (c {\cos \left (\gamma \left (a \int \cos \left (\lambda x \right )^{n}d x -a \int \cos \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} -y \right )\right )}^{k}+s {\tan \left (\mu \left (-b \int _{}^{x}{\left (-\frac {\tan \left (\beta \left (a \int \cos \left (\lambda x \right )^{n}d x -y \right )\right )-\tan \left (\beta a \int \cos \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} \right )}{1+\tan \left (\beta \left (a \int \cos \left (\lambda x \right )^{n}d x -y \right )\right ) \tan \left (\beta a \int \cos \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} \right )}\right )}^{m}d \textit {\_g} +z +b \int {\left (-\frac {\tan \left (\beta \left (a \int \cos \left (\lambda x \right )^{n}d x -y \right )\right )-\tan \left (\beta a \int \cos \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} \right )}{1+\tan \left (\beta \left (a \int \cos \left (\lambda x \right )^{n}d x -y \right )\right ) \tan \left (\beta a \int \cos \left (\lambda \textit {\_g} \right )^{n}d \textit {\_g} \right )}\right )}^{m}d \textit {\_g} \right )\right )}^{k}\right )d \textit {\_g}}\]
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6.8.18.4 [1867] Problem 4
problem number 1867
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 ) = f_{1} \left (-\int \cot \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y +\frac {\operatorname {b1} \int \cos \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z}{\operatorname {c1}}, -\frac {\operatorname {a1} \int _{}^{y}{\sin \left (\lambda \operatorname {1} \operatorname {RootOf}\left (\int \cot \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} \operatorname {c1} -\int \cot \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y \operatorname {c1} -\operatorname {b1} \int _{}^{\textit {\_Z}}\cos \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} +\operatorname {b1} \int \cos \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z \right )\right )}^{\operatorname {n1}} \cot \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f}}{\operatorname {b1}}+x \right ) {\mathrm e}^{\frac {\int _{}^{y}\left (\operatorname {a2} {\cos \left (\lambda \operatorname {2} \operatorname {RootOf}\left (\int \cot \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} \operatorname {c1} -\int \cot \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y \operatorname {c1} -\operatorname {b1} \int _{}^{\textit {\_Z}}\cos \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} +\operatorname {b1} \int \cos \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z \right )\right )}^{\operatorname {n2}}+\operatorname {b2} \sin \left (\beta \operatorname {2} \textit {\_f} \right )^{\operatorname {m2}}+\operatorname {c2} {\cos \left (\gamma \operatorname {2} \operatorname {RootOf}\left (\int \cot \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} \operatorname {c1} -\int \cot \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y \operatorname {c1} -\operatorname {b1} \int _{}^{\textit {\_Z}}\cos \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} +\operatorname {b1} \int \cos \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z \right )\right )}^{\operatorname {k2}}\right ) \cot \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f}}{\operatorname {b1}}}\]
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6.8.18.5 [1868] Problem 5
problem number 1868
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 ) = f_{1} \left (-\int \cot \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y +\frac {\operatorname {b1} \int \cot \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z}{\operatorname {c1}}, -\frac {\operatorname {a1} \int _{}^{y}{\tan \left (\lambda \operatorname {1} \operatorname {RootOf}\left (\operatorname {b1} \int \cot \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z -\int \cot \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y \operatorname {c1} +\int \cot \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} \operatorname {c1} -\operatorname {b1} \int _{}^{\textit {\_Z}}\cot \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} \right )\right )}^{\operatorname {n1}} \cot \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f}}{\operatorname {b1}}+x \right ) {\mathrm e}^{\frac {\int _{}^{y}\left (\operatorname {a2} {\cot \left (\lambda \operatorname {2} \operatorname {RootOf}\left (\operatorname {b1} \int \cot \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z -\int \cot \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y \operatorname {c1} +\int \cot \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} \operatorname {c1} -\operatorname {b1} \int _{}^{\textit {\_Z}}\cot \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} \right )\right )}^{\operatorname {n2}}+\operatorname {b2} \tan \left (\beta \operatorname {2} \textit {\_f} \right )^{\operatorname {m2}}+\operatorname {c2} {\cot \left (\gamma \operatorname {2} \operatorname {RootOf}\left (\operatorname {b1} \int \cot \left (\gamma \operatorname {1} z \right )^{-\operatorname {k1}}d z -\int \cot \left (\beta \operatorname {1} y \right )^{-\operatorname {m1}}d y \operatorname {c1} +\int \cot \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f} \operatorname {c1} -\operatorname {b1} \int _{}^{\textit {\_Z}}\cot \left (\gamma \operatorname {1} \textit {\_f} \right )^{-\operatorname {k1}}d \textit {\_f} \right )\right )}^{\operatorname {k2}}\right ) \cot \left (\beta \operatorname {1} \textit {\_f} \right )^{-\operatorname {m1}}d \textit {\_f}}{\operatorname {b1}}}\]
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