### 3.113 $$\int \frac{\text{sech}^5(c+d x)}{a+b \tanh ^2(c+d x)} \, dx$$

Optimal. Leaf size=86 $\frac{(a+b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^2 d}-\frac{(2 a+3 b) \tan ^{-1}(\sinh (c+d x))}{2 b^2 d}-\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 b d}$

[Out]

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

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Rubi [A]  time = 0.120484, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.217, Rules used = {3676, 414, 522, 203, 205} $\frac{(a+b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b^2 d}-\frac{(2 a+3 b) \tan ^{-1}(\sinh (c+d x))}{2 b^2 d}-\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 b d}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[c + d*x]^5/(a + b*Tanh[c + d*x]^2),x]

[Out]

-((2*a + 3*b)*ArcTan[Sinh[c + d*x]])/(2*b^2*d) + ((a + b)^(3/2)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(
Sqrt[a]*b^2*d) - (Sech[c + d*x]*Tanh[c + d*x])/(2*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 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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

Mathematica [A]  time = 0.557462, size = 79, normalized size = 0.92 $-\frac{2 (2 a+3 b) \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+\frac{2 (a+b)^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right )}{\sqrt{a}}+b \tanh (c+d x) \text{sech}(c+d x)}{2 b^2 d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[c + d*x]^5/(a + b*Tanh[c + d*x]^2),x]

[Out]

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

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Maple [B]  time = 0.076, size = 836, normalized size = 9.7 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^5/(a+b*tanh(d*x+c)^2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{e^{\left (3 \, d x + 3 \, c\right )} - e^{\left (d x + c\right )}}{b d e^{\left (4 \, d x + 4 \, c\right )} + 2 \, b d e^{\left (2 \, d x + 2 \, c\right )} + b d} - \frac{{\left (2 \, a e^{c} + 3 \, b e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )}}{b^{2} d} + 32 \, \int \frac{{\left (a^{2} e^{\left (3 \, c\right )} + 2 \, a b e^{\left (3 \, c\right )} + b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (a^{2} e^{c} + 2 \, a b e^{c} + b^{2} e^{c}\right )} e^{\left (d x\right )}}{16 \,{\left (a b^{2} + b^{3} +{\left (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 b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^5/(a+b*tanh(d*x+c)^2),x, algorithm="maxima")

[Out]

-(e^(3*d*x + 3*c) - e^(d*x + c))/(b*d*e^(4*d*x + 4*c) + 2*b*d*e^(2*d*x + 2*c) + b*d) - (2*a*e^c + 3*b*e^c)*arc
tan(e^(d*x + c))*e^(-c)/(b^2*d) + 32*integrate(1/16*((a^2*e^(3*c) + 2*a*b*e^(3*c) + b^2*e^(3*c))*e^(3*d*x) + (
a^2*e^c + 2*a*b*e^c + b^2*e^c)*e^(d*x))/(a*b^2 + b^3 + (a*b^2*e^(4*c) + b^3*e^(4*c))*e^(4*d*x) + 2*(a*b^2*e^(2
*c) - b^3*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.82544, size = 4316, normalized size = 50.19 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^5/(a+b*tanh(d*x+c)^2),x, algorithm="fricas")

[Out]

[-1/2*(2*b*cosh(d*x + c)^3 + 6*b*cosh(d*x + c)*sinh(d*x + c)^2 + 2*b*sinh(d*x + c)^3 - ((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)*sqrt(-(a + b)/a)*log(((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*s
inh(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*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh
(d*x + c)^2 + a*sinh(d*x + c)^3 - a*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c))*sqrt(-(a + b)/a)
+ 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)) + 2*((2*a + 3*b)*cosh(d*x + c)^4 + 4*(2*a + 3*b)*cosh(d*x + c)*sin
h(d*x + c)^3 + (2*a + 3*b)*sinh(d*x + c)^4 + 2*(2*a + 3*b)*cosh(d*x + c)^2 + 2*(3*(2*a + 3*b)*cosh(d*x + c)^2
+ 2*a + 3*b)*sinh(d*x + c)^2 + 4*((2*a + 3*b)*cosh(d*x + c)^3 + (2*a + 3*b)*cosh(d*x + c))*sinh(d*x + c) + 2*a
+ 3*b)*arctan(cosh(d*x + c) + sinh(d*x + c)) - 2*b*cosh(d*x + c) + 2*(3*b*cosh(d*x + c)^2 - b)*sinh(d*x + c))
/(b^2*d*cosh(d*x + c)^4 + 4*b^2*d*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*d*sinh(d*x + c)^4 + 2*b^2*d*cosh(d*x + c
)^2 + b^2*d + 2*(3*b^2*d*cosh(d*x + c)^2 + b^2*d)*sinh(d*x + c)^2 + 4*(b^2*d*cosh(d*x + c)^3 + b^2*d*cosh(d*x
+ c))*sinh(d*x + c)), -(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 - ((a + b)*c
osh(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)*sqrt((a + b)/a)*arctan(1/2*sqrt((a + b)/a)*(cosh(d*x + c) + sinh(d*x + c))) - ((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)*sqrt((a + b)/a)*arctan(1/2*((a + b)*cosh(d*x + c)^3 + 3*(a + b)*cosh(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 + b)/a)/(a + b)) + ((2*a + 3*b)*cosh(d*x + c)^4 + 4*(2*a + 3*b)*cosh(d*x + c)*sinh(d*x + c)^3 +
(2*a + 3*b)*sinh(d*x + c)^4 + 2*(2*a + 3*b)*cosh(d*x + c)^2 + 2*(3*(2*a + 3*b)*cosh(d*x + c)^2 + 2*a + 3*b)*s
inh(d*x + c)^2 + 4*((2*a + 3*b)*cosh(d*x + c)^3 + (2*a + 3*b)*cosh(d*x + c))*sinh(d*x + c) + 2*a + 3*b)*arctan
(cosh(d*x + c) + sinh(d*x + c)) - b*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 - b)*sinh(d*x + c))/(b^2*d*cosh(d*x +
c)^4 + 4*b^2*d*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*d*sinh(d*x + c)^4 + 2*b^2*d*cosh(d*x + c)^2 + b^2*d + 2*(3
*b^2*d*cosh(d*x + c)^2 + b^2*d)*sinh(d*x + c)^2 + 4*(b^2*d*cosh(d*x + c)^3 + b^2*d*cosh(d*x + c))*sinh(d*x + c
))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{5}{\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(sech(d*x+c)**5/(a+b*tanh(d*x+c)**2),x)

[Out]

Integral(sech(c + d*x)**5/(a + b*tanh(c + d*x)**2), x)

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Giac [C]  time = 1.84435, size = 5747, normalized size = 66.83 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^5/(a+b*tanh(d*x+c)^2),x, algorithm="giac")

[Out]

-1/4*(4*(2*a*e^c + 3*b*e^c)*arctan(e^(d*x + c))*e^(-c)/b^2 + 4*(e^(3*d*x + 3*c) - e^(d*x + c))/(b*(e^(2*d*x +
2*c) + 1)^2) - 2*(3*(2*a^2*b + 2*a*b^2 - (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*sqrt(-a*b))*cos(1/2*real_p
art(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^2*b + 2*a*b^2 -
(a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)
*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^2*b + 2*a*b^2 - (a^2 - b^2)*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^
2*b + 2*a*b^2 - (a^2 - b^2)*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*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a^2*b + 2*a*b^2 -
(a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*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)))) - (2*a^2*b + 2*a*b^2 - (a^2 - b^2)*sqrt(-a*b))*s
in(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*b^2 + b^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*b^2 + b
^3)/(a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(a*b^3) - 2*(3*(2*a^2*b + 2*a*b^2
- (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*
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^2*b + 2*a*b^2 - (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*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_pa
rt(arccos(-a/(a + b) + b/(a + b)))) + 9*(2*a^2*b + 2*a*b^2 - (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2
)*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^2*b + 2*a*b^2 - (a^2 - b^2)*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*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a^2*b + 2*a*b^2 - (a^2 - b^2)*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^2*b + 2*a
*b^2 - (a^2 - b^2)*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)))) - (2*a^2*b + 2*a*b^2 - (a^2 - b^2)*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*b^2 + b^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*b^2 + b^3)/(a*b^2*e^(4*c) + b^3*e^(4*c)))^(1
/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(a*b^3) - ((2*a^2*b + 2*a*b^2 - (a^2 - b^2)*sqrt(-a*b))*cos(1/2*real_p
art(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 - 3*(2*a^2*b + 2*
a*b^2 - (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2
- b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*im
ag_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(2*a^2*b +
2*a*b^2 - (a^2 - b^2)*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*imag_part(arccos(-a/(
a + b) + b/(a + b))))^2 - (2*a^2*b + 2*a*b^2 - (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*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^2*b + 2*
a*b^2 - (a^2 - b^2)*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*b^2 + b^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*b^2 + b^3)/(a*b^2*e^(4*c) + b^3*e^(4*c))) + e^(2*d*x))/(a*b^3) + ((2*a^2*b + 2*a*b^2
- (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*im
ag_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^2*b + 2*a*b^2 - (a^2 - b^2)*sqrt(-a*b))*cos(1/2*real_part(arcco
s(-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^2*b + 2*a*b^2 - (a^2 - b^2)*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*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - (2*a^2*b + 2*a*b^2 - (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*rea
l_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a^2*b + 2
*a*b^2 - (a^2 - b^2)*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^2*b + 2*a*b^2 - (a^2 - b^2)*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*b^2 + b^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*b^2 + b^3)/(a*b^2*e^(4*c) + b^3*e^(4*c)))
+ e^(2*d*x))/(a*b^3))/d