6.7.9 4.3

6.7.9.1 [1645] Problem 1

problem number 1645

Added June 20, 2019.

Problem Chapter 7.4.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y + b w_z = c \tanh ^k(\lambda x)+s \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*D[w[x, y,z], y] +  b*D[w[x,y,z],z]== c*Tanh[lambda*x]^k+s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1(y-a x,z-b x)+\frac {c \tanh ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\tanh ^2(\lambda x)\right )}{k \lambda +\lambda }+s x\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+ a*diff(w(x,y,z),y)+ b*diff(w(x,y,z),z)=c*tanh(lambda*x)^k+s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int \!c \left ( \tanh \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+sx+{\it \_F1} \left ( -ax+y,-xb+z \right ) \]

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6.7.9.2 [1646] Problem 2

problem number 1646

Added June 20, 2019.

Problem Chapter 7.4.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y + c \tanh (\lambda x) w_z = k \tanh (\beta y)+s \tanh (\gamma z) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +  c*Tanh[lambda*x]*D[w[x,y,z],z]== k*Tanh[beta*y]+s*Tanh[gamma*z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {k \tanh \left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )+s \tanh \left (\frac {\gamma (a \lambda z-c \log (\cosh (\lambda x))+c \log (\cosh (\lambda K[1])))}{a \lambda }\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a},z-\frac {c \log (\cosh (\lambda x))}{a \lambda }\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := a*diff(w(x,y,z),x)+ b*diff(w(x,y,z),y)+ c*tanh(lambda*x)*diff(w(x,y,z),z)=k*tanh(beta*y)+s*tanh(gamma*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 ( {\frac {ay-xb}{a}},{\frac {2\,za\lambda +c\ln \left ( \tanh \left ( x\lambda \right ) -1 \right ) +c\ln \left ( \tanh \left ( x\lambda \right ) +1 \right ) }{2\,a\lambda }} \right ) +\int ^{x}\!{\frac {1}{a} \left ( \left ( k-s \right ) \sinh \left ( {\frac {\gamma \,c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) -1 \right ) +\gamma \,c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) +1 \right ) -\gamma \,c\ln \left ( \tanh \left ( x\lambda \right ) -1 \right ) -\gamma \,c\ln \left ( \tanh \left ( x\lambda \right ) +1 \right ) +2\, \left ( -az\gamma +\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) \right ) \lambda }{2\,a\lambda }} \right ) -\sinh \left ( {\frac {\gamma \,c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) -1 \right ) +\gamma \,c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) +1 \right ) -\gamma \,c\ln \left ( \tanh \left ( x\lambda \right ) -1 \right ) -\gamma \,c\ln \left ( \tanh \left ( x\lambda \right ) +1 \right ) -2\, \left ( az\gamma +\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) \right ) \lambda }{2\,a\lambda }} \right ) \left ( k+s \right ) \right ) \left ( \cosh \left ( {\frac {\gamma \,c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) -1 \right ) +\gamma \,c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) +1 \right ) -\gamma \,c\ln \left ( \tanh \left ( x\lambda \right ) -1 \right ) -\gamma \,c\ln \left ( \tanh \left ( x\lambda \right ) +1 \right ) -2\, \left ( az\gamma +\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) \right ) \lambda }{2\,a\lambda }} \right ) +\cosh \left ( {\frac {\gamma \,c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) -1 \right ) +\gamma \,c\ln \left ( \tanh \left ( {\it \_a}\,\lambda \right ) +1 \right ) -\gamma \,c\ln \left ( \tanh \left ( x\lambda \right ) -1 \right ) -\gamma \,c\ln \left ( \tanh \left ( x\lambda \right ) +1 \right ) +2\, \left ( -az\gamma +\beta \, \left ( ay-b \left ( x-{\it \_a} \right ) \right ) \right ) \lambda }{2\,a\lambda }} \right ) \right ) ^{-1}}{d{\it \_a}}\]

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6.7.9.3 [1647] Problem 3

problem number 1647

Added June 19, 2019.

Problem Chapter 7.4.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a \tanh ^n(\beta x) w_y + c \tanh ^k(\lambda x) w_z = c \tanh ^m(\gamma x)+s \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + a*Tanh[beta*x]^n*D[w[x, y,z], y] +  b*Tanh[lambda*x]^k*D[w[x,y,z],z]== c*Tanh[gamma*x]^m+s; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to c_1\left (y-\frac {a \tanh ^{n+1}(\beta x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\tanh ^2(\beta x)\right )}{\beta n+\beta },z-\frac {b \tanh ^{k+1}(\lambda x) \, _2F_1\left (1,\frac {k+1}{2};\frac {k+3}{2};\tanh ^2(\lambda x)\right )}{k \lambda +\lambda }\right )+\frac {c \tanh ^{m+1}(\gamma x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};\tanh ^2(\gamma x)\right )}{\gamma m+\gamma }+s x\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := diff(w(x,y,z),x)+ a*tanh(beta*x)^n*diff(w(x,y,z),y)+ b*tanh(lambda*x)^k*diff(w(x,y,z),z)=c*tanh(gamma*x)^m+s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int \!c \left ( \tanh \left ( x\gamma \right ) \right ) ^{m}\,{\rm d}x+sx+{\it \_F1} \left ( -\int \!a \left ( \tanh \left ( \beta \,x \right ) \right ) ^{n}\,{\rm d}x+y,-\int \!b \left ( \tanh \left ( x\lambda \right ) \right ) ^{k}\,{\rm d}x+z \right ) \]

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6.7.9.4 [1648] Problem 4

problem number 1648

Added June 19, 2019.

Problem Chapter 7.4.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \tanh (\beta y) w_y + c \tanh (\lambda x) w_z = k \tanh (\gamma z) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*Tanh[beta*y]*D[w[x, y,z], y] +  c*Tanh[lambda*x]*D[w[x,y,z],z]== k*Tanh[gamma*z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde := a*diff(w(x,y,z),x)+ b*tanh(beta*y)*diff(w(x,y,z),y)+ c*tanh(lambda*x)*diff(w(x,y,z),z)=k*tanh(gamma*z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) ={\it \_F1} \left ( {\frac {2\,b\beta \,x+a\ln \left ( \tanh \left ( \beta \,y \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,y \right ) \right ) +a\ln \left ( \tanh \left ( \beta \,y \right ) -1 \right ) }{2\,\beta \,b}},{\frac {za\lambda -c\ln \left ( \cosh \left ( x\lambda \right ) \right ) }{a\lambda }} \right ) +\int ^{y}\!{\frac {k}{b\tanh \left ( \beta \,{\it \_a} \right ) }\sinh \left ( {\frac {\gamma }{a\lambda } \left ( za\lambda -c\ln \left ( \cosh \left ( x\lambda \right ) \right ) +c\ln \left ( \cosh \left ( {\frac { \left ( -2\,b\beta \,x-a\ln \left ( \tanh \left ( \beta \,y \right ) +1 \right ) +2\,a\ln \left ( \tanh \left ( \beta \,y \right ) \right ) -a\ln \left ( \tanh \left ( \beta \,y \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) -1 \right ) \right ) \lambda }{2\,\beta \,b}} \right ) \right ) \right ) } \right ) \left ( \cosh \left ( {\frac {\gamma }{a\lambda } \left ( za\lambda -c\ln \left ( \cosh \left ( x\lambda \right ) \right ) +c\ln \left ( \cosh \left ( {\frac { \left ( -2\,b\beta \,x-a\ln \left ( \tanh \left ( \beta \,y \right ) +1 \right ) +2\,a\ln \left ( \tanh \left ( \beta \,y \right ) \right ) -a\ln \left ( \tanh \left ( \beta \,y \right ) -1 \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) +1 \right ) -2\,a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) \right ) +a\ln \left ( \tanh \left ( \beta \,{\it \_a} \right ) -1 \right ) \right ) \lambda }{2\,\beta \,b}} \right ) \right ) \right ) } \right ) \right ) ^{-1}}{d{\it \_a}}\]

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6.7.9.5 [1649] Problem 5

problem number 1649

Added June 19, 2019.

Problem Chapter 7.4.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b \tanh (\beta y) w_y + c \tanh (\gamma z) w_z = k \tanh (\lambda x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*Tanh[beta*y]*D[w[x, y,z], y] +  c*Tanh[gamma*z]*D[w[x,y,z],z]== k*Tanh[lambda*x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \frac {k \log (\cosh (\lambda x))}{a \lambda }+c_1\left (\frac {1}{2} \left (\frac {\log (\sinh (\beta y))}{\beta }-\frac {b x}{a}\right ),\frac {b \log \left (\sinh ^2(\gamma z)\right )}{\gamma }-\frac {2 c \log (\sinh (\beta y))}{\beta }\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := a*diff(w(x,y,z),x)+ b*tanh(beta*y)*diff(w(x,y,z),y)+ c*tanh(gamma*z)*diff(w(x,y,z),z)=k*tanh(lambda*x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
 

\[w \left ( x,y,z \right ) ={{\it \_C2}\,{\it \_C1}\,{{\rm e}^{{\it \_c}_{{1}}x}}{\it \_C3} \left ( {\frac {\sinh \left ( \beta \,y \right ) +\cosh \left ( \beta \,y \right ) }{\cosh \left ( \beta \,y \right ) }} \right ) ^{-{\frac {{\it \_c}_{{2}}}{2\,\beta }}} \left ( {\frac {\sinh \left ( \beta \,y \right ) }{\cosh \left ( \beta \,y \right ) }} \right ) ^{{\frac {{\it \_c}_{{2}}}{\beta }}} \left ( -{\frac {-\sinh \left ( \beta \,y \right ) +\cosh \left ( \beta \,y \right ) }{\cosh \left ( \beta \,y \right ) }} \right ) ^{-{\frac {{\it \_c}_{{2}}}{2\,\beta }}} \left ( {\frac {\sinh \left ( \gamma \,z \right ) +\cosh \left ( \gamma \,z \right ) }{\cosh \left ( \gamma \,z \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{2\,c\gamma }}} \left ( {\frac {\sinh \left ( \gamma \,z \right ) +\cosh \left ( \gamma \,z \right ) }{\cosh \left ( \gamma \,z \right ) }} \right ) ^{{\frac {b{\it \_c}_{{2}}}{2\,c\gamma }}} \left ( -{\frac {-\sinh \left ( \gamma \,z \right ) +\cosh \left ( \gamma \,z \right ) }{\cosh \left ( \gamma \,z \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{2\,c\gamma }}} \left ( -{\frac {-\sinh \left ( \gamma \,z \right ) +\cosh \left ( \gamma \,z \right ) }{\cosh \left ( \gamma \,z \right ) }} \right ) ^{{\frac {b{\it \_c}_{{2}}}{2\,c\gamma }}} \left ( \left ( {\frac {\sinh \left ( \gamma \,z \right ) }{\cosh \left ( \gamma \,z \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{c\gamma }}} \right ) ^{-1} \left ( \left ( {\frac {\sinh \left ( \gamma \,z \right ) }{\cosh \left ( \gamma \,z \right ) }} \right ) ^{{\frac {b{\it \_c}_{{2}}}{c\gamma }}} \right ) ^{-1}}+{\frac {\ln \left ( {{\rm e}^{2\,x\lambda }}+1 \right ) k}{a\lambda }}-{\frac {\ln \left ( {{\rm e}^{x\lambda }} \right ) k}{a\lambda }}+{\frac {{\it \_C1}\,k}{{\it \_C2}\,a}}\]

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6.7.9.6 [1650] Problem 6

problem number 1650

Added June 19, 2019.

Problem Chapter 7.4.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a \tanh (\lambda x) w_x + b \tanh (\beta y) w_y + c \tanh (\gamma z) w_z = k \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  a*Tanh[lambda*x]*D[w[x, y,z], x] + b*Tanh[beta*y]*D[w[x, y,z], y] +  c*Tanh[gamma*z]*D[w[x,y,z],z]== k; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

Failed

Maple ✓

restart; 
local gamma; 
pde := a*tanh(lambda*x)*diff(w(x,y,z),x)+ b*tanh(beta*y)*diff(w(x,y,z),y)+ c*tanh(gamma*z)*diff(w(x,y,z),z)=k; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z),'build')),output='realtime'));
 

\[w \left ( x,y,z \right ) ={{\it \_C2}\,{\it \_C1}\,{\it \_C3} \left ( {\frac {\sinh \left ( x\lambda \right ) +\cosh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-{\frac {{\it \_c}_{{1}}}{2\,\lambda }}} \left ( {\frac {\sinh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{{\frac {{\it \_c}_{{1}}}{\lambda }}} \left ( {\frac {\sinh \left ( x\lambda \right ) -\cosh \left ( x\lambda \right ) }{\cosh \left ( x\lambda \right ) }} \right ) ^{-{\frac {{\it \_c}_{{1}}}{2\,\lambda }}} \left ( {\frac {\sinh \left ( \beta \,y \right ) +\cosh \left ( \beta \,y \right ) }{\cosh \left ( \beta \,y \right ) }} \right ) ^{-{\frac {{\it \_c}_{{2}}}{2\,\beta }}} \left ( {\frac {\sinh \left ( \beta \,y \right ) }{\cosh \left ( \beta \,y \right ) }} \right ) ^{{\frac {{\it \_c}_{{2}}}{\beta }}} \left ( {\frac {\sinh \left ( \beta \,y \right ) -\cosh \left ( \beta \,y \right ) }{\cosh \left ( \beta \,y \right ) }} \right ) ^{-{\frac {{\it \_c}_{{2}}}{2\,\beta }}} \left ( {\frac {\sinh \left ( \gamma \,z \right ) +\cosh \left ( \gamma \,z \right ) }{\cosh \left ( \gamma \,z \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{2\,c\gamma }}} \left ( {\frac {\sinh \left ( \gamma \,z \right ) +\cosh \left ( \gamma \,z \right ) }{\cosh \left ( \gamma \,z \right ) }} \right ) ^{{\frac {b{\it \_c}_{{2}}}{2\,c\gamma }}} \left ( -{\frac {-\sinh \left ( \gamma \,z \right ) +\cosh \left ( \gamma \,z \right ) }{\cosh \left ( \gamma \,z \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{2\,c\gamma }}} \left ( -{\frac {-\sinh \left ( \gamma \,z \right ) +\cosh \left ( \gamma \,z \right ) }{\cosh \left ( \gamma \,z \right ) }} \right ) ^{{\frac {b{\it \_c}_{{2}}}{2\,c\gamma }}} \left ( \left ( {\frac {\sinh \left ( \gamma \,z \right ) }{\cosh \left ( \gamma \,z \right ) }} \right ) ^{{\frac {a{\it \_c}_{{1}}}{c\gamma }}} \right ) ^{-1} \left ( \left ( {\frac {\sinh \left ( \gamma \,z \right ) }{\cosh \left ( \gamma \,z \right ) }} \right ) ^{{\frac {b{\it \_c}_{{2}}}{c\gamma }}} \right ) ^{-1}}+{\frac {\ln \left ( {{\rm e}^{x\lambda }}+1 \right ) k}{2\,a\lambda }}-{\frac {\ln \left ( {{\rm e}^{x\lambda }} \right ) k}{a\lambda }}+{\frac {\ln \left ( {{\rm e}^{x\lambda }}-1 \right ) k}{2\,a\lambda }}+{\frac {\ln \left ( {{\rm e}^{2\,x\lambda }}-1 \right ) k}{2\,a\lambda }}+{\frac {{\it \_C1}\,k}{{\it \_C2}\,a}}\]

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6.7.9.7 [1651] Problem 7

problem number 1651

Added June 19, 2019.

Problem Chapter 7.4.3.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a_1 \tanh ^{n_1}(\lambda _1 x) w_x + b_1 \tanh ^{m_1}(\beta _1 y) w_y + c_1 \tanh ^{k_1}(\gamma _1 z) w_z = a_2 \tanh ^{n_2}(\lambda _2 x) + b_2 \tanh ^{m_2}(\beta _2 y) w_y + c_2 \tanh ^{k_2}(\gamma _2 z) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a1*Tanh[lambda1*x]^n1*D[w[x, y,z], x] + b1*Tanh[beta1*x]^m1*D[w[x, y,z], y] +  c1*Tanh[gamma1*x]^k1*D[w[x,y,z],z]== a2*Tanh[lambda1*x]^n2 + b2*Tanh[beta2*x]^m2 +  c2*Tanh[gamma2*x]^k2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

\[\left \{\left \{w(x,y,z)\to \int _1^x\frac {\tanh ^{-\text {n1}}(\text {lambda1} K[3]) \left (\text {c2} \tanh ^{\text {k2}}(\text {gamma2} K[3])+\text {b2} \tanh ^{\text {m2}}(\text {beta2} K[3])+\text {a2} \tanh ^{\text {n2}}(\text {lambda1} K[3])\right )}{\text {a1}}dK[3]+c_1\left (y-\int _1^x\frac {\text {b1} \tanh ^{\text {m1}}(\text {beta1} K[1]) \tanh ^{-\text {n1}}(\text {lambda1} K[1])}{\text {a1}}dK[1],z-\int _1^x\frac {\text {c1} \tanh ^{\text {k1}}(\text {gamma1} K[2]) \tanh ^{-\text {n1}}(\text {lambda1} K[2])}{\text {a1}}dK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
local gamma; 
pde := a1*tanh(lambda1*x)^n1*diff(w(x,y,z),x)+ b1*tanh(beta1*x)^m1*diff(w(x,y,z),y)+ c1*tanh(gamma1*x)^k1*diff(w(x,y,z),z)=a2*tanh(lambda1*x)^n2 + b2*tanh(beta2*x)^m2 +  c2*tanh(gamma2*x)^k2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

\[w \left ( x,y,z \right ) =\int \!{\frac { \left ( \tanh \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}+{\it n2}}{\it a2}+ \left ( \tanh \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}{\it b2}\, \left ( \tanh \left ( \beta 2\,x \right ) \right ) ^{{\it m2}}+ \left ( \tanh \left ( \lambda 1\,x \right ) \right ) ^{-{\it n1}}{\it c2}\, \left ( \tanh \left ( \gamma 2\,x \right ) \right ) ^{{\it k2}}}{{\it a1}}}\,{\rm d}x+{\it \_F1} \left ( {\frac {1}{{\it a1}} \left ( y{\it a1}-{\it b1}\,\int \! \left ( {\frac {\sinh \left ( \beta 1\,x \right ) }{\cosh \left ( \beta 1\,x \right ) }} \right ) ^{{\it m1}} \left ( {\frac {\sinh \left ( \lambda 1\,x \right ) }{\cosh \left ( \lambda 1\,x \right ) }} \right ) ^{-{\it n1}}\,{\rm d}x \right ) },{\frac {1}{{\it a1}} \left ( -{\it c1}\,\int \! \left ( {\frac {\sinh \left ( \gamma 1\,x \right ) }{\cosh \left ( \gamma 1\,x \right ) }} \right ) ^{{\it k1}} \left ( {\frac {\sinh \left ( \lambda 1\,x \right ) }{\cosh \left ( \lambda 1\,x \right ) }} \right ) ^{-{\it n1}}\,{\rm d}x+z{\it a1} \right ) } \right ) \]

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