\(\int \frac {\cosh (c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\) [126]

Optimal result

Integrand size = 21, antiderivative size = 154 \[ \int \frac {\cosh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {3 b \left (8 a^2+4 a b+b^2\right ) \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a+b)^{7/2} d}+\frac {\sinh (c+d x)}{(a+b)^3 d}+\frac {b^3 \sinh (c+d x)}{4 a (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}+\frac {3 b^2 (4 a+b) \sinh (c+d x)}{8 a^2 (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )} \] Output:

3/8*b*(8*a^2+4*a*b+b^2)*arctan((a+b)^(1/2)*sinh(d*x+c)/a^(1/2))/a^(5/2)/(a 
+b)^(7/2)/d+sinh(d*x+c)/(a+b)^3/d+1/4*b^3*sinh(d*x+c)/a/(a+b)^3/d/(a+(a+b) 
*sinh(d*x+c)^2)^2+3/8*b^2*(4*a+b)*sinh(d*x+c)/a^2/(a+b)^3/d/(a+(a+b)*sinh( 
d*x+c)^2)
 

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {\frac {3 b \left (8 a^2+4 a b+b^2\right ) \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{a^{5/2} (a+b)^{7/2}}+\frac {\sinh (c+d x) \left (8+\frac {3 b^3}{a^2 \left (a+(a+b) \sinh ^2(c+d x)\right )}+\frac {2 b^2 \left (6 a+b+6 (a+b) \sinh ^2(c+d x)\right )}{a \left (a+(a+b) \sinh ^2(c+d x)\right )^2}\right )}{(a+b)^3}}{8 d} \] Input:

Integrate[Cosh[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]
 

Output:

((3*b*(8*a^2 + 4*a*b + b^2)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/( 
a^(5/2)*(a + b)^(7/2)) + (Sinh[c + d*x]*(8 + (3*b^3)/(a^2*(a + (a + b)*Sin 
h[c + d*x]^2)) + (2*b^2*(6*a + b + 6*(a + b)*Sinh[c + d*x]^2))/(a*(a + (a 
+ b)*Sinh[c + d*x]^2)^2)))/(a + b)^3)/(8*d)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 4159, 300, 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.

\(\displaystyle \int \frac {\cosh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sec (i c+i d x) \left (a-b \tan (i c+i d x)^2\right )^3}dx\)

\(\Big \downarrow \) 4159

\(\displaystyle \frac {\int \frac {\left (\sinh ^2(c+d x)+1\right )^3}{\left ((a+b) \sinh ^2(c+d x)+a\right )^3}d\sinh (c+d x)}{d}\)

\(\Big \downarrow \) 300

\(\displaystyle \frac {\int \left (\frac {3 b (a+b)^2 \sinh ^4(c+d x)+3 b (a+b) (2 a+b) \sinh ^2(c+d x)+b \left (3 a^2+3 b a+b^2\right )}{(a+b)^3 \left ((a+b) \sinh ^2(c+d x)+a\right )^3}+\frac {1}{(a+b)^3}\right )d\sinh (c+d x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {3 b^2 (4 a+b) \sinh (c+d x)}{8 a^2 (a+b)^3 \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {3 b \left (8 a^2+4 a b+b^2\right ) \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a+b)^{7/2}}+\frac {b^3 \sinh (c+d x)}{4 a (a+b)^3 \left ((a+b) \sinh ^2(c+d x)+a\right )^2}+\frac {\sinh (c+d x)}{(a+b)^3}}{d}\)

Input:

Int[Cosh[c + d*x]/(a + b*Tanh[c + d*x]^2)^3,x]
 

Output:

((3*b*(8*a^2 + 4*a*b + b^2)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/( 
8*a^(5/2)*(a + b)^(7/2)) + Sinh[c + d*x]/(a + b)^3 + (b^3*Sinh[c + d*x])/( 
4*a*(a + b)^3*(a + (a + b)*Sinh[c + d*x]^2)^2) + (3*b^2*(4*a + b)*Sinh[c + 
 d*x])/(8*a^2*(a + b)^3*(a + (a + b)*Sinh[c + d*x]^2)))/d
 

Defintions of rubi rules used

rule 300

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int 
[PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c 
, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4159

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_ 
))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f 
  Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2 
*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f} 
, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(140)=280\).

Time = 11.77 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.44

method	result	size
derivativedivides	\(\frac {-\frac {1}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b \left (\frac {-\frac {b \left (12 a +5 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a}-\frac {3 \left (4 a^{2}+15 a b +4 b^{2}\right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a^{2}}+\frac {3 \left (4 a^{2}+15 a b +4 b^{2}\right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{2}}+\frac {b \left (12 a +5 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}+\frac {3 \left (8 a^{2}+4 a b +b^{2}\right ) \left (\frac {\left (\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{8 a}\right )}{\left (a +b \right )^{3}}-\frac {1}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(375\)
default	\(\frac {-\frac {1}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b \left (\frac {-\frac {b \left (12 a +5 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 a}-\frac {3 \left (4 a^{2}+15 a b +4 b^{2}\right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 a^{2}}+\frac {3 \left (4 a^{2}+15 a b +4 b^{2}\right ) b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 a^{2}}+\frac {b \left (12 a +5 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}+\frac {3 \left (8 a^{2}+4 a b +b^{2}\right ) \left (\frac {\left (\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{8 a}\right )}{\left (a +b \right )^{3}}-\frac {1}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(375\)
risch	\(\frac {{\mathrm e}^{d x +c}}{2 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) d}-\frac {{\mathrm e}^{-d x -c}}{2 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right ) d}+\frac {\left (12 \,{\mathrm e}^{6 d x +6 c} a^{2}+15 \,{\mathrm e}^{6 d x +6 c} a b +3 \,{\mathrm e}^{6 d x +6 c} b^{2}+12 \,{\mathrm e}^{4 d x +4 c} a^{2}-25 \,{\mathrm e}^{4 d x +4 c} a b -9 \,{\mathrm e}^{4 d x +4 c} b^{2}-12 \,{\mathrm e}^{2 d x +2 c} a^{2}+25 \,{\mathrm e}^{2 d x +2 c} b a +9 b^{2} {\mathrm e}^{2 d x +2 c}-12 a^{2}-15 a b -3 b^{2}\right ) {\mathrm e}^{d x +c} b^{2}}{4 \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )^{2} \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right ) d \,a^{2}}-\frac {3 b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d a}-\frac {3 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d \,a^{2}}+\frac {3 b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d a}+\frac {3 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d \,a^{2}}\)	\(649\)

Input:

int(cosh(d*x+c)/(a+tanh(d*x+c)^2*b)^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)+2/(a+b)^3*b*((-1/8*b*(12*a+5*b)/a* 
tanh(1/2*d*x+1/2*c)^7-3/8*(4*a^2+15*a*b+4*b^2)/a^2*b*tanh(1/2*d*x+1/2*c)^5 
+3/8*(4*a^2+15*a*b+4*b^2)/a^2*b*tanh(1/2*d*x+1/2*c)^3+1/8*b*(12*a+5*b)/a*t 
anh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*b 
*tanh(1/2*d*x+1/2*c)^2+a)^2+3/8/a*(8*a^2+4*a*b+b^2)*(1/2*(((a+b)*b)^(1/2)+ 
b)/a/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2 
*d*x+1/2*c)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1/2))-1/2*(((a+b)*b)^(1/2)-b)/a 
/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d* 
x+1/2*c)/((2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/2))))-1/(a+b)^3/(tanh(1/2*d*x+1/ 
2*c)+1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6267 vs. \(2 (140) = 280\).

Time = 0.33 (sec) , antiderivative size = 11399, normalized size of antiderivative = 74.02 \[ \int \frac {\cosh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(cosh(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \] Input:

integrate(cosh(d*x+c)/(a+b*tanh(d*x+c)**2)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cosh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \] Input:

integrate(cosh(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
 

Output:

-1/4*(2*a^4 + 4*a^3*b + 2*a^2*b^2 - 2*(a^4*e^(10*c) + 2*a^3*b*e^(10*c) + a 
^2*b^2*e^(10*c))*e^(10*d*x) - (6*a^4*e^(8*c) - 4*a^3*b*e^(8*c) + 2*a^2*b^2 
*e^(8*c) + 15*a*b^3*e^(8*c) + 3*b^4*e^(8*c))*e^(8*d*x) - (4*a^4*e^(6*c) - 
8*a^3*b*e^(6*c) + 32*a^2*b^2*e^(6*c) - 25*a*b^3*e^(6*c) - 9*b^4*e^(6*c))*e 
^(6*d*x) + (4*a^4*e^(4*c) - 8*a^3*b*e^(4*c) + 32*a^2*b^2*e^(4*c) - 25*a*b^ 
3*e^(4*c) - 9*b^4*e^(4*c))*e^(4*d*x) + (6*a^4*e^(2*c) - 4*a^3*b*e^(2*c) + 
2*a^2*b^2*e^(2*c) + 15*a*b^3*e^(2*c) + 3*b^4*e^(2*c))*e^(2*d*x))/((a^7*d*e 
^(9*c) + 5*a^6*b*d*e^(9*c) + 10*a^5*b^2*d*e^(9*c) + 10*a^4*b^3*d*e^(9*c) + 
 5*a^3*b^4*d*e^(9*c) + a^2*b^5*d*e^(9*c))*e^(9*d*x) + 4*(a^7*d*e^(7*c) + 3 
*a^6*b*d*e^(7*c) + 2*a^5*b^2*d*e^(7*c) - 2*a^4*b^3*d*e^(7*c) - 3*a^3*b^4*d 
*e^(7*c) - a^2*b^5*d*e^(7*c))*e^(7*d*x) + 2*(3*a^7*d*e^(5*c) + 7*a^6*b*d*e 
^(5*c) + 6*a^5*b^2*d*e^(5*c) + 6*a^4*b^3*d*e^(5*c) + 7*a^3*b^4*d*e^(5*c) + 
 3*a^2*b^5*d*e^(5*c))*e^(5*d*x) + 4*(a^7*d*e^(3*c) + 3*a^6*b*d*e^(3*c) + 2 
*a^5*b^2*d*e^(3*c) - 2*a^4*b^3*d*e^(3*c) - 3*a^3*b^4*d*e^(3*c) - a^2*b^5*d 
*e^(3*c))*e^(3*d*x) + (a^7*d*e^c + 5*a^6*b*d*e^c + 10*a^5*b^2*d*e^c + 10*a 
^4*b^3*d*e^c + 5*a^3*b^4*d*e^c + a^2*b^5*d*e^c)*e^(d*x)) + 1/2*integrate(3 
/2*((8*a^2*b*e^(3*c) + 4*a*b^2*e^(3*c) + b^3*e^(3*c))*e^(3*d*x) + (8*a^2*b 
*e^c + 4*a*b^2*e^c + b^3*e^c)*e^(d*x))/(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3* 
b^3 + a^2*b^4 + (a^6*e^(4*c) + 4*a^5*b*e^(4*c) + 6*a^4*b^2*e^(4*c) + 4*a^3 
*b^3*e^(4*c) + a^2*b^4*e^(4*c))*e^(4*d*x) + 2*(a^6*e^(2*c) + 2*a^5*b*e^...
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\cosh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cosh(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable 
to make series expansion Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \] Input:

int(cosh(c + d*x)/(a + b*tanh(c + d*x)^2)^3,x)
 

Output:

int(cosh(c + d*x)/(a + b*tanh(c + d*x)^2)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 3187, normalized size of antiderivative = 20.69 \[ \int \frac {\cosh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:

int(cosh(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x)
 

Output:

(24*e**(9*c + 9*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) - 
sqrt(b))/sqrt(a))*a**4*b + 60*e**(9*c + 9*d*x)*sqrt(a)*sqrt(a + b)*atan((e 
**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**3*b**2 + 51*e**(9*c + 9*d*x 
)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a 
**2*b**3 + 18*e**(9*c + 9*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt 
(a + b) - sqrt(b))/sqrt(a))*a*b**4 + 3*e**(9*c + 9*d*x)*sqrt(a)*sqrt(a + b 
)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b**5 + 96*e**(7*c + 7 
*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a 
))*a**4*b + 48*e**(7*c + 7*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqr 
t(a + b) - sqrt(b))/sqrt(a))*a**3*b**2 - 84*e**(7*c + 7*d*x)*sqrt(a)*sqrt( 
a + b)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2*b**3 - 48*e 
**(7*c + 7*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) - sqrt( 
b))/sqrt(a))*a*b**4 - 12*e**(7*c + 7*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c 
+ d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*b**5 + 144*e**(5*c + 5*d*x)*sqrt(a) 
*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**4*b - 2 
4*e**(5*c + 5*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) - sq 
rt(b))/sqrt(a))*a**3*b**2 + 114*e**(5*c + 5*d*x)*sqrt(a)*sqrt(a + b)*atan( 
(e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2*b**3 + 60*e**(5*c + 5*d 
*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a)) 
*a*b**4 + 18*e**(5*c + 5*d*x)*sqrt(a)*sqrt(a + b)*atan((e**(c + d*x)*sq...