\(\int \frac {\text {sech}^4(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\) [121]

Optimal result

Integrand size = 23, antiderivative size = 77 \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=-\frac {(a-b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2} d}+\frac {(a+b) \tanh (c+d x)}{2 a b d \left (a+b \tanh ^2(c+d x)\right )} \] Output:

-1/2*(a-b)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))/a^(3/2)/b^(3/2)/d+1/2*(a+b) 
*tanh(d*x+c)/a/b/d/(a+b*tanh(d*x+c)^2)
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.08 \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {(-a+b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )+\frac {\sqrt {a} \sqrt {b} (a+b) \sinh (2 (c+d x))}{a-b+(a+b) \cosh (2 (c+d x))}}{2 a^{3/2} b^{3/2} d} \] Input:

Integrate[Sech[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]
 

Output:

((-a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] + (Sqrt[a]*Sqrt[b]*(a + 
b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]))/(2*a^(3/2)*b^(3 
/2)*d)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4158, 298, 218}

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.

\(\displaystyle \int \frac {\text {sech}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sec (i c+i d x)^4}{\left (a-b \tan (i c+i d x)^2\right )^2}dx\)

\(\Big \downarrow \) 4158

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

\(\Big \downarrow \) 298

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {(a+b) \tanh (c+d x)}{2 a b \left (a+b \tanh ^2(c+d x)\right )}-\frac {(a-b) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}}{d}\)

Input:

Int[Sech[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^2,x]
 

Output:

(-1/2*((a - b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(3/2)*b^(3/2)) 
+ ((a + b)*Tanh[c + d*x])/(2*a*b*(a + b*Tanh[c + d*x]^2)))/d
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 3042

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

rule 4158

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ 
)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim 
p[ff/(c^(m - 1)*f)   Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)^n)^ 
p, x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && I 
ntegerQ[m/2] && (IntegersQ[n, p] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] 
 || EqQ[n^2, 16])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(258\) vs. \(2(65)=130\).

Time = 127.20 (sec) , antiderivative size = 259, normalized size of antiderivative = 3.36

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-\frac {\left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a b}-\frac {\left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a b}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}-\frac {\left (a -b \right ) \left (\frac {\left (-a -\sqrt {\left (a +b \right ) b}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (a -\sqrt {\left (a +b \right ) b}+b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{b}}{d}\)	\(259\)
default	\(\frac {-\frac {2 \left (-\frac {\left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a b}-\frac {\left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a b}\right )}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a}-\frac {\left (a -b \right ) \left (\frac {\left (-a -\sqrt {\left (a +b \right ) b}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (a -\sqrt {\left (a +b \right ) b}+b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{b}}{d}\)	\(259\)
risch	\(-\frac {{\mathrm e}^{2 d x +2 c} a -{\mathrm e}^{2 d x +2 c} b +a +b}{a b d \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{4 \sqrt {-a b}\, d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{4 \sqrt {-a b}\, d a}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{4 \sqrt {-a b}\, d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{4 \sqrt {-a b}\, d a}\)	\(326\)

Input:

int(sech(d*x+c)^4/(a+tanh(d*x+c)^2*b)^2,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-2*(-1/2*(a+b)/a/b*tanh(1/2*d*x+1/2*c)^3-1/2*(a+b)/a/b*tanh(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)-(a-b)/b*(1/2*(-a-((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-((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))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 569 vs. \(2 (65) = 130\).

Time = 0.13 (sec) , antiderivative size = 1443, normalized size of antiderivative = 18.74 \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(sech(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")
 

Output:

[-1/4*(4*a^2*b + 4*a*b^2 + 4*(a^2*b - a*b^2)*cosh(d*x + c)^2 + 8*(a^2*b - 
a*b^2)*cosh(d*x + c)*sinh(d*x + c) + 4*(a^2*b - a*b^2)*sinh(d*x + c)^2 - ( 
(a^2 - b^2)*cosh(d*x + c)^4 + 4*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 
+ (a^2 - b^2)*sinh(d*x + c)^4 + 2*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2 + 2* 
(3*(a^2 - b^2)*cosh(d*x + c)^2 + a^2 - 2*a*b + b^2)*sinh(d*x + c)^2 + a^2 
- b^2 + 4*((a^2 - b^2)*cosh(d*x + c)^3 + (a^2 - 2*a*b + b^2)*cosh(d*x + c) 
)*sinh(d*x + c))*sqrt(-a*b)*log(((a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*( 
a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sin 
h(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*co 
sh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 - 6*a*b + b^2 + 4*((a^2 + 
 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c) - 
 4*((a + b)*cosh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c)*sinh(d*x + c) + (a + 
 b)*sinh(d*x + c)^2 + a - b)*sqrt(-a*b))/((a + b)*cosh(d*x + c)^4 + 4*(a + 
 b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 + 2*(a - b)*co 
sh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 + 4* 
((a + b)*cosh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b))) 
/((a^3*b^2 + a^2*b^3)*d*cosh(d*x + c)^4 + 4*(a^3*b^2 + a^2*b^3)*d*cosh(d*x 
 + c)*sinh(d*x + c)^3 + (a^3*b^2 + a^2*b^3)*d*sinh(d*x + c)^4 + 2*(a^3*b^2 
 - a^2*b^3)*d*cosh(d*x + c)^2 + 2*(3*(a^3*b^2 + a^2*b^3)*d*cosh(d*x + c)^2 
 + (a^3*b^2 - a^2*b^3)*d)*sinh(d*x + c)^2 + (a^3*b^2 + a^2*b^3)*d + 4*(...
 

Sympy [F]

\[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {\operatorname {sech}^{4}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \] Input:

integrate(sech(d*x+c)**4/(a+b*tanh(d*x+c)**2)**2,x)
 

Output:

Integral(sech(c + d*x)**4/(a + b*tanh(c + d*x)**2)**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.65 \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a + b}{{\left (a^{2} b + a b^{2} + 2 \, {\left (a^{2} b - a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} b + a b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {{\left (a - b\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{2 \, \sqrt {a b} a b d} \] Input:

integrate(sech(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")
 

Output:

((a - b)*e^(-2*d*x - 2*c) + a + b)/((a^2*b + a*b^2 + 2*(a^2*b - a*b^2)*e^( 
-2*d*x - 2*c) + (a^2*b + a*b^2)*e^(-4*d*x - 4*c))*d) + 1/2*(a - b)*arctan( 
1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a*b*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (65) = 130\).

Time = 0.35 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.86 \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=-\frac {\frac {{\left (a - b\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} a b} + \frac {2 \, {\left (a e^{\left (2 \, d x + 2 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} a b}}{2 \, d} \] Input:

integrate(sech(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")
 

Output:

-1/2*((a - b)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/s 
qrt(a*b))/(sqrt(a*b)*a*b) + 2*(a*e^(2*d*x + 2*c) - b*e^(2*d*x + 2*c) + a + 
 b)/((a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^ 
(2*d*x + 2*c) + a + b)*a*b))/d
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^4\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \] Input:

int(1/(cosh(c + d*x)^4*(a + b*tanh(c + d*x)^2)^2),x)
 

Output:

int(1/(cosh(c + d*x)^4*(a + b*tanh(c + d*x)^2)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 637, normalized size of antiderivative = 8.27 \[ \int \frac {\text {sech}^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {-e^{4 d x +4 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a^{2}+e^{4 d x +4 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) b^{2}-2 e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a^{2}+4 e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a b -2 e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) b^{2}-\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) a^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}-\sqrt {b}}{\sqrt {a}}\right ) b^{2}+e^{4 d x +4 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a^{2}-e^{4 d x +4 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) b^{2}+2 e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a^{2}-4 e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a b +2 e^{2 d x +2 c} \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) b^{2}+\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) a^{2}-\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {e^{d x +c} \sqrt {a +b}+\sqrt {b}}{\sqrt {a}}\right ) b^{2}+e^{4 d x +4 c} a^{2} b +e^{4 d x +4 c} a \,b^{2}-a^{2} b -a \,b^{2}}{2 a^{2} b^{2} d \left (e^{4 d x +4 c} a +e^{4 d x +4 c} b +2 e^{2 d x +2 c} a -2 e^{2 d x +2 c} b +a +b \right )} \] Input:

int(sech(d*x+c)^4/(a+b*tanh(d*x+c)^2)^2,x)
 

Output:

( - e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt 
(b))/sqrt(a))*a**2 + e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*s 
qrt(a + b) - sqrt(b))/sqrt(a))*b**2 - 2*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*a 
tan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a**2 + 4*e**(2*c + 2*d*x 
)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt(b))/sqrt(a))*a*b - 
 2*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt( 
b))/sqrt(a))*b**2 - sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt( 
b))/sqrt(a))*a**2 + sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) - sqrt( 
b))/sqrt(a))*b**2 + e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sq 
rt(a + b) + sqrt(b))/sqrt(a))*a**2 - e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*atan 
((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*b**2 + 2*e**(2*c + 2*d*x)*s 
qrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b))/sqrt(a))*a**2 - 4 
*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqrt(a + b) + sqrt(b) 
)/sqrt(a))*a*b + 2*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqr 
t(a + b) + sqrt(b))/sqrt(a))*b**2 + sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqr 
t(a + b) + sqrt(b))/sqrt(a))*a**2 - sqrt(b)*sqrt(a)*atan((e**(c + d*x)*sqr 
t(a + b) + sqrt(b))/sqrt(a))*b**2 + e**(4*c + 4*d*x)*a**2*b + e**(4*c + 4* 
d*x)*a*b**2 - a**2*b - a*b**2)/(2*a**2*b**2*d*(e**(4*c + 4*d*x)*a + e**(4* 
c + 4*d*x)*b + 2*e**(2*c + 2*d*x)*a - 2*e**(2*c + 2*d*x)*b + a + b))