Integrand size = 24, antiderivative size = 237 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=-\frac {\left (6 \sqrt {a}-5 \sqrt {b}\right ) \sqrt {b} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}+\frac {\left (6 \sqrt {a}+5 \sqrt {b}\right ) \sqrt {b} \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}-\frac {\coth (c+d x)}{a^2 d}+\frac {b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-2 a \tanh ^2(c+d x)+(a-b) \tanh ^4(c+d x)\right )} \] Output:
-1/8*(6*a^(1/2)-5*b^(1/2))*b^(1/2)*arctanh((a^(1/2)-b^(1/2))^(1/2)*tanh(d* x+c)/a^(1/4))/a^(9/4)/(a^(1/2)-b^(1/2))^(3/2)/d+1/8*(6*a^(1/2)+5*b^(1/2))* b^(1/2)*arctanh((a^(1/2)+b^(1/2))^(1/2)*tanh(d*x+c)/a^(1/4))/a^(9/4)/(a^(1 /2)+b^(1/2))^(3/2)/d-coth(d*x+c)/a^2/d+1/4*b*tanh(d*x+c)*(a-(a+b)*tanh(d*x +c)^2)/a^2/(a-b)/d/(a-2*a*tanh(d*x+c)^2+(a-b)*tanh(d*x+c)^4)
Time = 2.78 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.15 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\frac {\frac {\left (6 a \sqrt {b}-5 \sqrt {a} b\right ) \arctan \left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {\left (6 a \sqrt {b}+5 \sqrt {a} b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {a+\sqrt {a} \sqrt {b}}}-8 \sqrt {a} \coth (c+d x)+\frac {4 \sqrt {a} b (2 a+b-b \cosh (2 (c+d x))) \sinh (2 (c+d x))}{(a-b) (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}}{8 a^{5/2} d} \] Input:
Integrate[Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^2,x]
Output:
(((6*a*Sqrt[b] - 5*Sqrt[a]*b)*ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/S qrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])*Sqrt[-a + Sqrt[a]*Sqrt[b] ]) + ((6*a*Sqrt[b] + 5*Sqrt[a]*b)*ArcTanh[((Sqrt[a] + Sqrt[b])*Tanh[c + d* x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])*Sqrt[a + Sqrt[a]*Sqrt [b]]) - 8*Sqrt[a]*Coth[c + d*x] + (4*Sqrt[a]*b*(2*a + b - b*Cosh[2*(c + d* x)])*Sinh[2*(c + d*x)])/((a - b)*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Co sh[4*(c + d*x)])))/(8*a^(5/2)*d)
Time = 0.79 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {3042, 25, 3696, 1673, 27, 2195, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\sin (i c+i d x)^2 \left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\sin (i c+i d x)^2 \left (a-b \sin (i c+i d x)^4\right )^2}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\coth ^2(c+d x) \left (1-\tanh ^2(c+d x)\right )^4}{\left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1673 |
| \(\displaystyle \frac {\frac {b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}-\frac {\int -\frac {2 \coth ^2(c+d x) \left (\frac {b \left (4 a^2-b a-b^2\right ) \tanh ^4(c+d x)}{a-b}-\frac {a (8 a-7 b) b \tanh ^2(c+d x)}{a-b}+4 a b\right )}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{8 a^2 b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\coth ^2(c+d x) \left (\frac {b \left (4 a^2-b a-b^2\right ) \tanh ^4(c+d x)}{a-b}-\frac {a (8 a-7 b) b \tanh ^2(c+d x)}{a-b}+4 a b\right )}{(a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a}d\tanh (c+d x)}{4 a^2 b}+\frac {b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{d}\) |
| \(\Big \downarrow \) 2195 |
| \(\displaystyle \frac {\frac {\int \left (\frac {\left ((7 a-5 b) \tanh ^2(c+d x)-a\right ) b^2}{(a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}+4 \coth ^2(c+d x) b\right )d\tanh (c+d x)}{4 a^2 b}+\frac {b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {-\frac {b^{3/2} \left (6 \sqrt {a}-5 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {b^{3/2} \left (6 \sqrt {a}+5 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-4 b \coth (c+d x)}{4 a^2 b}+\frac {b \tanh (c+d x) \left (a-(a+b) \tanh ^2(c+d x)\right )}{4 a^2 (a-b) \left ((a-b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )}}{d}\) |
Input:
Int[Csch[c + d*x]^2/(a - b*Sinh[c + d*x]^4)^2,x]
Output:
((-1/2*((6*Sqrt[a] - 5*Sqrt[b])*b^(3/2)*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*T anh[c + d*x])/a^(1/4)])/(a^(1/4)*(Sqrt[a] - Sqrt[b])^(3/2)) + ((6*Sqrt[a] + 5*Sqrt[b])*b^(3/2)*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/ 4)])/(2*a^(1/4)*(Sqrt[a] + Sqrt[b])^(3/2)) - 4*b*Coth[c + d*x])/(4*a^2*b) + (b*Tanh[c + d*x]*(a - (a + b)*Tanh[c + d*x]^2))/(4*a^2*(a - b)*(a - 2*a* Tanh[c + d*x]^2 + (a - b)*Tanh[c + d*x]^4)))/d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_), x_Symbol] :> With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q , a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d + e*x^ 2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a *b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5 ) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x], x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && ILtQ[m/2, 0]
Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2) ^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] & & IntegerQ[m/2] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.40 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.38
| method | result | size |
| derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}}-\frac {16 b \left (\frac {-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{32 \left (a -b \right )}+\frac {\left (a +4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{32 a -32 b}+\frac {\left (a +4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{32 a -32 b}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-a \,\textit {\_R}^{6}+\left (27 a -20 b \right ) \textit {\_R}^{4}+\left (-27 a +20 b \right ) \textit {\_R}^{2}+a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{256 a -256 b}\right )}{a^{2}}-\frac {1}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(326\) |
| default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}}-\frac {16 b \left (\frac {-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{32 \left (a -b \right )}+\frac {\left (a +4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{32 a -32 b}+\frac {\left (a +4 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{32 a -32 b}-\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 \left (a -b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a -4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a +6 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -16 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-a \,\textit {\_R}^{6}+\left (27 a -20 b \right ) \textit {\_R}^{4}+\left (-27 a +20 b \right ) \textit {\_R}^{2}+a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{256 a -256 b}\right )}{a^{2}}-\frac {1}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(326\) |
| risch | \(\text {Expression too large to display}\) | \(1097\) |
Input:
int(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/2/a^2*tanh(1/2*d*x+1/2*c)-16*b/a^2*((-1/32*a/(a-b)*tanh(1/2*d*x+1/ 2*c)^7+1/32*(a+4*b)/(a-b)*tanh(1/2*d*x+1/2*c)^5+1/32*(a+4*b)/(a-b)*tanh(1/ 2*d*x+1/2*c)^3-1/32*a/(a-b)*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^8*a- 4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2* c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)+1/256/(a-b)*sum((-a*_R^6+(27*a-20*b)*_R^ 4+(-27*a+20*b)*_R^2+a)/(_R^7*a-3*_R^5*a+3*_R^3*a-8*_R^3*b-_R*a)*ln(tanh(1/ 2*d*x+1/2*c)-_R),_R=RootOf(a*_Z^8-4*a*_Z^6+(6*a-16*b)*_Z^4-4*a*_Z^2+a)))-1 /2/a^2/tanh(1/2*d*x+1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 8824 vs. \(2 (188) = 376\).
Time = 0.63 (sec) , antiderivative size = 8824, normalized size of antiderivative = 37.23 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {Timed out} \] Input:
integrate(csch(d*x+c)**2/(a-b*sinh(d*x+c)**4)**2,x)
Output:
Timed out
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^2,x, algorithm="maxima")
Output:
1/2*(4*a*b - 5*b^2 + (6*a*b*e^(8*c) - 5*b^2*e^(8*c))*e^(8*d*x) - 2*(13*a*b *e^(6*c) - 10*b^2*e^(6*c))*e^(6*d*x) - 2*(32*a^2*e^(4*c) - 47*a*b*e^(4*c) + 15*b^2*e^(4*c))*e^(4*d*x) - 2*(7*a*b*e^(2*c) - 10*b^2*e^(2*c))*e^(2*d*x) )/(a^3*b*d - a^2*b^2*d - (a^3*b*d*e^(10*c) - a^2*b^2*d*e^(10*c))*e^(10*d*x ) + 5*(a^3*b*d*e^(8*c) - a^2*b^2*d*e^(8*c))*e^(8*d*x) + 2*(8*a^4*d*e^(6*c) - 13*a^3*b*d*e^(6*c) + 5*a^2*b^2*d*e^(6*c))*e^(6*d*x) - 2*(8*a^4*d*e^(4*c ) - 13*a^3*b*d*e^(4*c) + 5*a^2*b^2*d*e^(4*c))*e^(4*d*x) - 5*(a^3*b*d*e^(2* c) - a^2*b^2*d*e^(2*c))*e^(2*d*x)) - 4*integrate(1/4*((6*a*b*e^(6*c) - 5*b ^2*e^(6*c))*e^(6*d*x) - 2*(8*a*b*e^(4*c) - 5*b^2*e^(4*c))*e^(4*d*x) + (6*a *b*e^(2*c) - 5*b^2*e^(2*c))*e^(2*d*x))/(a^3*b - a^2*b^2 + (a^3*b*e^(8*c) - a^2*b^2*e^(8*c))*e^(8*d*x) - 4*(a^3*b*e^(6*c) - a^2*b^2*e^(6*c))*e^(6*d*x ) - 2*(8*a^4*e^(4*c) - 11*a^3*b*e^(4*c) + 3*a^2*b^2*e^(4*c))*e^(4*d*x) - 4 *(a^3*b*e^(2*c) - a^2*b^2*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2}}{{\left (b \sinh \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \] Input:
integrate(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^2,x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,d x \] Input:
int(1/(sinh(c + d*x)^2*(a - b*sinh(c + d*x)^4)^2),x)
Output:
int(1/(sinh(c + d*x)^2*(a - b*sinh(c + d*x)^4)^2), x)
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx=\text {too large to display} \] Input:
int(csch(d*x+c)^2/(a-b*sinh(d*x+c)^4)^2,x)
Output:
( - 112742891520*e**(14*c + 10*d*x)*int(e**(4*d*x)/(448*e**(20*c + 20*d*x) *a**2*b**2 + 24*e**(20*c + 20*d*x)*a*b**3 - 15*e**(20*c + 20*d*x)*b**4 - 4 480*e**(18*c + 18*d*x)*a**2*b**2 - 240*e**(18*c + 18*d*x)*a*b**3 + 150*e** (18*c + 18*d*x)*b**4 - 14336*e**(16*c + 16*d*x)*a**3*b + 19392*e**(16*c + 16*d*x)*a**2*b**2 + 1560*e**(16*c + 16*d*x)*a*b**3 - 675*e**(16*c + 16*d*x )*b**4 + 86016*e**(14*c + 14*d*x)*a**3*b - 49152*e**(14*c + 14*d*x)*a**2*b **2 - 5760*e**(14*c + 14*d*x)*a*b**3 + 1800*e**(14*c + 14*d*x)*b**4 + 1146 88*e**(12*c + 12*d*x)*a**4 - 208896*e**(12*c + 12*d*x)*a**3*b + 78720*e**( 12*c + 12*d*x)*a**2*b**2 + 12240*e**(12*c + 12*d*x)*a*b**3 - 3150*e**(12*c + 12*d*x)*b**4 - 229376*e**(10*c + 10*d*x)*a**4 + 274432*e**(10*c + 10*d* x)*a**3*b - 89856*e**(10*c + 10*d*x)*a**2*b**2 - 15648*e**(10*c + 10*d*x)* a*b**3 + 3780*e**(10*c + 10*d*x)*b**4 + 114688*e**(8*c + 8*d*x)*a**4 - 208 896*e**(8*c + 8*d*x)*a**3*b + 78720*e**(8*c + 8*d*x)*a**2*b**2 + 12240*e** (8*c + 8*d*x)*a*b**3 - 3150*e**(8*c + 8*d*x)*b**4 + 86016*e**(6*c + 6*d*x) *a**3*b - 49152*e**(6*c + 6*d*x)*a**2*b**2 - 5760*e**(6*c + 6*d*x)*a*b**3 + 1800*e**(6*c + 6*d*x)*b**4 - 14336*e**(4*c + 4*d*x)*a**3*b + 19392*e**(4 *c + 4*d*x)*a**2*b**2 + 1560*e**(4*c + 4*d*x)*a*b**3 - 675*e**(4*c + 4*d*x )*b**4 - 4480*e**(2*c + 2*d*x)*a**2*b**2 - 240*e**(2*c + 2*d*x)*a*b**3 + 1 50*e**(2*c + 2*d*x)*b**4 + 448*a**2*b**2 + 24*a*b**3 - 15*b**4),x)*a**8*b* d + 163074539520*e**(14*c + 10*d*x)*int(e**(4*d*x)/(448*e**(20*c + 20*d...