### 3.120 $$\int \frac{\text{sech}^3(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx$$

Optimal. Leaf size=72 $\frac{\tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} d \sqrt{a+b}}+\frac{\sinh (c+d x)}{2 a d \left ((a+b) \sinh ^2(c+d x)+a\right )}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0765269, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.13, Rules used = {3676, 199, 205} $\frac{\tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} d \sqrt{a+b}}+\frac{\sinh (c+d x)}{2 a d \left ((a+b) \sinh ^2(c+d x)+a\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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

Mathematica [A]  time = 0.138047, size = 69, normalized size = 0.96 $\frac{\frac{\sqrt{a} \sinh (c+d x)}{(a+b) \sinh ^2(c+d x)+a}+\frac{\tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a+b}}}{2 a^{3/2} d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.101, size = 375, normalized size = 5.2 \begin{align*} -{\frac{1}{da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) ^{-1}}+{\frac{1}{da}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) ^{-1}}-{\frac{1}{2\,da}{\it Artanh} \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }-a-2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }-a-2\,b \right ) a}}}}+{\frac{b}{2\,da}{\it Artanh} \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }-a-2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{b \left ( a+b \right ) }}}{\frac{1}{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }-a-2\,b \right ) a}}}}+{\frac{1}{2\,da}\arctan \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }+a+2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }+a+2\,b \right ) a}}}}+{\frac{b}{2\,da}\arctan \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }+a+2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{b \left ( a+b \right ) }}}{\frac{1}{\sqrt{ \left ( 2\,\sqrt{b \left ( a+b \right ) }+a+2\,b \right ) a}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/a*tanh(1/2*d*x+1/2*c)^3+1
/d/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/a*tanh(1/2*d*x+1/2*c)-1/2/d
/a/((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))+1/2/
d/(b*(a+b))^(1/2)/a/((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))*b+1/2/d/a/((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))+1/2/d/(b*(a+b))^(1/2)/a/((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))*b

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (a e^{\left (3 \, c\right )} + b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (a e^{c} + b e^{c}\right )} e^{\left (d x\right )}}{a^{3} d + 2 \, a^{2} b d + a b^{2} d +{\left (a^{3} d e^{\left (4 \, c\right )} + 2 \, a^{2} b d e^{\left (4 \, c\right )} + a b^{2} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{3} d e^{\left (2 \, c\right )} - a b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + 8 \, \int \frac{e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (d x + c\right )}}{8 \,{\left (a^{2} + a b +{\left (a^{2} e^{\left (4 \, c\right )} + a b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{2} e^{\left (2 \, c\right )} - a b 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)^3/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.27416, size = 4026, normalized size = 55.92 \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)^3/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/4*(4*(a^2 + a*b)*cosh(d*x + c)^3 + 12*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c)^2 + 4*(a^2 + a*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^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)*c
osh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b)) - 4*(a^2 + a*b)*cosh(d*x + c) + 4*(3*(a^2 + a*
b)*cosh(d*x + c)^2 - a^2 - a*b)*sinh(d*x + c))/((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^4 + 4*(a^4 + 2*a^3*b
+ a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^3*b + a^2*b^2)*d*sinh(d*x + c)^4 + 2*(a^4 - a^2*b^2)*
d*cosh(d*x + c)^2 + 2*(3*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^2 + (a^4 - a^2*b^2)*d)*sinh(d*x + c)^2 + (a
^4 + 2*a^3*b + a^2*b^2)*d + 4*((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^3 + (a^4 - a^2*b^2)*d*cosh(d*x + c))*
sinh(d*x + c)), 1/2*(2*(a^2 + a*b)*cosh(d*x + c)^3 + 6*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c)^2 + 2*(a^2 + a*
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^2 + a*b)*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^2 + 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)*sqrt(a^2 + a*b)*arcta
n(1/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x + c))/a) - 2*(a^2 + a*b)*cosh(d*x + c) + 2*(3*(a^2 + a*b)*cosh
(d*x + c)^2 - a^2 - a*b)*sinh(d*x + c))/((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^4 + 4*(a^4 + 2*a^3*b + a^2*
b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^3*b + a^2*b^2)*d*sinh(d*x + c)^4 + 2*(a^4 - a^2*b^2)*d*cosh(
d*x + c)^2 + 2*(3*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^2 + (a^4 - a^2*b^2)*d)*sinh(d*x + c)^2 + (a^4 + 2*
a^3*b + a^2*b^2)*d + 4*((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^3 + (a^4 - a^2*b^2)*d*cosh(d*x + c))*sinh(d*
x + c))]

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [C]  time = 1.9383, size = 4585, normalized size = 63.68 \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)^3/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/8*(2*(3*(a^2 + 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)))) - (a^2 + 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*(a^2 + 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*(a^2 + 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*i
mag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(a^2 + 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*(a^2 + 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*(a^2 + 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 + (a^2 + 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 + (a^2 + 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)))) - (a^2 + a*b)*sin(1/2*rea
l_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))))*arctan((((a^2 + a
*b)/(a^2*e^(4*c) + a*b*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) + e^(d*x))/(((a^2 + a*b)/(a^2*e^(4*c)
+ a*b*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(2*a^3*b + (a^2 - a*b)*sqrt(-a*b)*abs(a)) + 2*(3*(a
^2 + 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)))) - (a^2 + 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*(a^2 + 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*(a^2 + 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(arc
cos(-a/(a + b) + b/(a + b)))) + 9*(a^2 + 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*(a^2 + 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*(a^2
+ 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 + (a^2 + 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 + (a^2 + 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)))) - (a^2 + a*b)*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)))))*arctan(-(((a^2 + a*b)/(a^2*e^
(4*c) + a*b*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) - e^(d*x))/(((a^2 + a*b)/(a^2*e^(4*c) + a*b*e^(4
*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(2*a^3*b + (a^2 - a*b)*sqrt(-a*b)*abs(a)) + ((a^2 + 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*(a^2
+ 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*(a^2 + 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*(a^2 + 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*(a^2 + a*b)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arcco
s(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(a^2 + 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(a
rccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - (a^2 + 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*(a^2 +
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*s
inh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (a^2 + 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)))) - (a^2 + 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^2 + a*b)/(a^2*e^(4*c) + a*b*e^(4
*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a^2 + a*b)/(a^2*e^(4*c) + a*b*e^(4*c))) + e^(2*d
*x))/(2*a^3*b + (a^2 - a*b)*sqrt(-a*b)*abs(a)) - ((a^2 + 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*(a^2 + 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*(a^2 + 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*(a^2 + a*b)*cos(1/2*real_par
t(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(arc
cos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(a^2 + a*b)*cos(1/2*re
al_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_p
art(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(a^2 + 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 - (a^2 + 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*(a^2 + 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 + (a^2 + 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)))) - (a^2 + 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^2 + a*b)/(a^2*e^(4*c) + a*b*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b
)))*e^(d*x) + sqrt((a^2 + a*b)/(a^2*e^(4*c) + a*b*e^(4*c))) + e^(2*d*x))/(2*a^3*b + (a^2 - a*b)*sqrt(-a*b)*abs
(a)) + 8*(e^(3*d*x + 3*c) - e^(d*x + c))/((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))/d