### 3.106 $$\int \frac{\cosh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx$$

Optimal. Leaf size=80 $\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a+b)^{5/2}}+\frac{\sinh ^3(c+d x)}{3 d (a+b)}+\frac{(a+2 b) \sinh (c+d x)}{d (a+b)^2}$

[Out]

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

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Rubi [A]  time = 0.109062, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.13, Rules used = {3676, 390, 205} $\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a+b)^{5/2}}+\frac{\sinh ^3(c+d x)}{3 d (a+b)}+\frac{(a+2 b) \sinh (c+d x)}{d (a+b)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 3676

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> With[{ff = F
reeFactors[Sin[e + f*x], x]}, Dist[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]

Rule 390

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\cosh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a+2 b}{(a+b)^2}+\frac{x^2}{a+b}+\frac{b^2}{(a+b)^2 \left (a+(a+b) x^2\right )}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{(a+2 b) \sinh (c+d x)}{(a+b)^2 d}+\frac{\sinh ^3(c+d x)}{3 (a+b) d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{(a+b)^2 d}\\ &=\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{5/2} d}+\frac{(a+2 b) \sinh (c+d x)}{(a+b)^2 d}+\frac{\sinh ^3(c+d x)}{3 (a+b) d}\\ \end{align*}

Mathematica [A]  time = 0.396, size = 79, normalized size = 0.99 $\frac{-\frac{12 b^2 \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right )}{\sqrt{a} (a+b)^{5/2}}+\frac{3 (3 a+7 b) \sinh (c+d x)}{(a+b)^2}+\frac{\sinh (3 (c+d x))}{a+b}}{12 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-12*b^2*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) + (3*(3*a + 7*b)*Sinh[c + d*x])
/(a + b)^2 + Sinh[3*(c + d*x)]/(a + b))/(12*d)

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Maple [B]  time = 0.087, size = 468, normalized size = 5.9 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left ({\left (a e^{\left (6 \, c\right )} + b e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 3 \,{\left (3 \, a e^{\left (4 \, c\right )} + 7 \, b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 3 \,{\left (3 \, a e^{\left (2 \, c\right )} + 7 \, b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - a - b\right )} e^{\left (-3 \, d x\right )}}{24 \,{\left (a^{2} d e^{\left (3 \, c\right )} + 2 \, a b d e^{\left (3 \, c\right )} + b^{2} d e^{\left (3 \, c\right )}\right )}} + \frac{1}{8} \, \int \frac{16 \,{\left (b^{2} e^{\left (3 \, d x + 3 \, c\right )} + b^{2} e^{\left (d x + c\right )}\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} +{\left (a^{3} e^{\left (4 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{3} e^{\left (2 \, c\right )} + a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((a*e^(6*c) + b*e^(6*c))*e^(6*d*x) + 3*(3*a*e^(4*c) + 7*b*e^(4*c))*e^(4*d*x) - 3*(3*a*e^(2*c) + 7*b*e^(2*
c))*e^(2*d*x) - a - b)*e^(-3*d*x)/(a^2*d*e^(3*c) + 2*a*b*d*e^(3*c) + b^2*d*e^(3*c)) + 1/8*integrate(16*(b^2*e^
(3*d*x + 3*c) + b^2*e^(d*x + c))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3 + (a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(
4*c) + b^3*e^(4*c))*e^(4*d*x) + 2*(a^3*e^(2*c) + a^2*b*e^(2*c) - a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.37351, size = 4604, normalized size = 57.55 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*((a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^6 + 6*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^
3 + 2*a^2*b + a*b^2)*sinh(d*x + c)^6 + 3*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c)^4 + 3*(3*a^3 + 10*a^2*b +
7*a*b^2 + 5*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x +
c)^3 + 3*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 - 2*a^2*b - a*b^2 - 3*(3*a^3 + 10*
a^2*b + 7*a*b^2)*cosh(d*x + c)^2 + 3*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^4 - 3*a^3 - 10*a^2*b - 7*a*b^2 +
6*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 12*(b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x +
c)^2*sinh(d*x + c) + 3*b^2*cosh(d*x + c)*sinh(d*x + c)^2 + b^2*sinh(d*x + c)^3)*sqrt(-a^2 - a*b)*log(((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*(3*a + b)*cosh(d*x + c
)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 - 3*a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 - (3*a + b)*cosh(d*
x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x +
c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a^2 - a*b) + 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)*cosh(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)) + 6*((a
^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^5 + 2*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c)^3 - (3*a^3 + 10*a^2*b + 7
*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^3 + 3*(a^4 + 3*a^3*
b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)
*sinh(d*x + c)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*sinh(d*x + c)^3), 1/24*((a^3 + 2*a^2*b + a*b^2)*cosh(
d*x + c)^6 + 6*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^3 + 2*a^2*b + a*b^2)*sinh(d*x + c)^6
+ 3*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c)^4 + 3*(3*a^3 + 10*a^2*b + 7*a*b^2 + 5*(a^3 + 2*a^2*b + a*b^2)*
cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^3 + 3*(3*a^3 + 10*a^2*b + 7*a*b^
2)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 - 2*a^2*b - a*b^2 - 3*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c)^2 + 3
*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^4 - 3*a^3 - 10*a^2*b - 7*a*b^2 + 6*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh
(d*x + c)^2)*sinh(d*x + c)^2 + 24*(b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*b^2*cosh(d*x
+ c)*sinh(d*x + c)^2 + b^2*sinh(d*x + c)^3)*sqrt(a^2 + a*b)*arctan(1/2*((a + b)*cosh(d*x + c)^3 + 3*(a + b)*co
sh(d*x + c)*sinh(d*x + c)^2 + (a + b)*sinh(d*x + c)^3 + (3*a - b)*cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 +
3*a - b)*sinh(d*x + c))/sqrt(a^2 + a*b)) + 24*(b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*
b^2*cosh(d*x + c)*sinh(d*x + c)^2 + b^2*sinh(d*x + c)^3)*sqrt(a^2 + a*b)*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x
+ c) + sinh(d*x + c))/a) + 6*((a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^5 + 2*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*
x + c)^3 - (3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*c
osh(d*x + c)^3 + 3*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^4 + 3*a^3*b + 3*
a^2*b^2 + a*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*sinh(d*x + c)^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh ^{3}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 2.11494, size = 6807, normalized size = 85.09 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(12*(3*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) +
b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/
(a + b)))) - (2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b)
+ b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3 - 9*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) -
b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a
+ b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b)
+ b/(a + b)))) + 3*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/
(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a
+ b) + b/(a + b)))) + 9*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(
-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a
+ b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*
c) - b^3*e^(2*c))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/
(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a*b^3*e^(2*c) + (a*b^2*e
^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arcco
s(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a*b^3*e^(2*c) + (a*b^2*
e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arc
cos(-a/(a + b) + b/(a + b))))^3 + (2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cosh(1/2*imag_p
art(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) - (2*a*b^3*e^(2*c) + (
a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part
(arccos(-a/(a + b) + b/(a + b)))))*arctan((((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3
*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) + e^(d*x))/(((a^3 + 3*a^2*b + 3*a*b^2 +
b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))
))/(a^4*b*e^(2*c) + 3*a^3*b^2*e^(2*c) + 3*a^2*b^3*e^(2*c) + a*b^4*e^(2*c)) + 12*(3*(2*a*b^3*e^(2*c) + (a*b^2*e
^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arcc
os(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) - (2*a*b^3*e^(2*c) + (a*b^2*
e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arc
cos(-a/(a + b) + b/(a + b))))^3 - 9*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_
part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_par
t(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(2*a*b^3*e^(2*c) +
(a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_p
art(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(2*a*b^3*e^(2*c
) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*im
ag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part
(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cosh(1/2*
imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_
part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1
/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*im
ag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*sin(
1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a*
b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*si
n(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) - (2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b)
)*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))))*arct
an(-(((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*c
os(1/2*arccos(-(a - b)/(a + b))) - e^(d*x))/(((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) +
3*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(a^4*b*e^(2*c) + 3*a^3*b^2*e^(2*c)
+ 3*a^2*b^3*e^(2*c) + a*b^4*e^(2*c)) - (9*a*e^(2*d*x + 2*c) + 21*b*e^(2*d*x + 2*c) + a + b)*e^(-3*d*x)/(a^2*e^
(3*c) + 2*a*b*e^(3*c) + b^2*e^(3*c)) + 6*((2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2
*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 - 3*(2*a*b
^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh
(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*
a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3
*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9
*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))
))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*
sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(
-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))
)*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*s
qrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)
)))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2
- (2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b
))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + 3*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*
c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a
+ b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2
*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(
a + b)))) - (2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) +
b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))))*log(2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3
*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) +
sqrt((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c))) + e^(2*
d*x))/(a^4*b*e^(2*c) + 3*a^3*b^2*e^(2*c) + 3*a^2*b^3*e^(2*c) + a*b^4*e^(2*c)) - 6*((2*a*b^3*e^(2*c) + (a*b^2*e
^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arcc
os(-a/(a + b) + b/(a + b))))^3 - 3*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_p
art(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(a
rccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*rea
l_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_
part(arccos(-a/(a + b) + b/(a + b)))) + 9*(2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2
*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real
_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(2*a*b^3*e^(2
*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*
imag_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(2*a*b^3*
e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/
2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*ima
g_part(arccos(-a/(a + b) + b/(a + b))))^2 - (2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1
/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + 3*(2*a
*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*si
n(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*
a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*c
osh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) - (2*a*b^3*e^(2*c) + (a*b^2*e^(2*c) - b^3*e^(2*c))*sqrt(-a*
b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))))*lo
g(-2*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*c
os(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c)
+ 3*a*b^2*e^(4*c) + b^3*e^(4*c))) + e^(2*d*x))/(a^4*b*e^(2*c) + 3*a^3*b^2*e^(2*c) + 3*a^2*b^3*e^(2*c) + a*b^4
*e^(2*c)) + (a^2*e^(3*d*x + 24*c) + 2*a*b*e^(3*d*x + 24*c) + b^2*e^(3*d*x + 24*c) + 9*a^2*e^(d*x + 22*c) + 30*
a*b*e^(d*x + 22*c) + 21*b^2*e^(d*x + 22*c))/(a^3*e^(21*c) + 3*a^2*b*e^(21*c) + 3*a*b^2*e^(21*c) + b^3*e^(21*c)
))/d