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

Optimal. Leaf size=36 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d \sqrt{a+b}} \]

[Out]

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

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Rubi [A]  time = 0.0443204, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3676, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0370236, size = 36, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.059, size = 235, normalized size = 6.5 \begin{align*} -{\frac{1}{d}{\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}{d}{\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}{d}\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}{d}\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)/(a+b*tanh(d*x+c)^2),x)

[Out]

-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*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/((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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.39758, size = 1385, normalized size = 38.47 \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)/(a+b*tanh(d*x+c)^2),x, algorithm="fricas")

[Out]

[-1/2*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))/((a^2 + a*b)*d), (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)) + sqrt(a^2 + a*b)*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sin
h(d*x + c))/a))/((a^2 + a*b)*d)]

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

[Out]

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

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Giac [C]  time = 1.56991, size = 6607, normalized size = 183.53 \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)/(a+b*tanh(d*x+c)^2),x, algorithm="giac")

[Out]

1/4*(2*(3*(2*a^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arcco
s(-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*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^
2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^
(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arcco
s(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(2*a^2*b*e^(2*c) + 2*a*b
^2*e^(2*c) - (a^2*e^(2*c) - b^2*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*im
ag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e
^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2
*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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)))) - (2*a^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*sin(1/2*real_part(arc
cos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))))*arctan((((a + b)/(a*e^(4*c)
+ b*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) + e^(d*x))/(((a + b)/(a*e^(4*c) + b*e^(4*c)))^(1/4)*sin
(1/2*arccos(-(a - b)/(a + b)))))/(a^3*b*e^(2*c) + 2*a^2*b^2*e^(2*c) + a*b^3*e^(2*c)) + 2*(3*(2*a^2*b*e^(2*c) +
2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c
) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) -
b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^
2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*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)))) - (2*a^2*b*e^(2*c)
+ 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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 + b)/(a*e^(4*c) + b*e^(4*c)))^(1/4)*cos(1/
2*arccos(-(a - b)/(a + b))) - e^(d*x))/(((a + b)/(a*e^(4*c) + b*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b
)))))/(a^3*b*e^(2*c) + 2*a^2*b^2*e^(2*c) + a*b^3*e^(2*c)) + ((2*a^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c)
- b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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*r
eal_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*
e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arcco
s(-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*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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*imag_part(arccos(-a/(a + b) + b/(a + b)
)))^2 - (2*a^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b
^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*
b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(
2*c) - b^2*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 + b)/(a*e^(4*c) + b*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d
*x) + sqrt((a + b)/(a*e^(4*c) + b*e^(4*c))) + e^(2*d*x))/(a^3*b*e^(2*c) + 2*a^2*b^2*e^(2*c) + a*b^3*e^(2*c)) -
((2*a^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(
2*c) - (a^2*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*ima
g_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*e^
(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(
2*c) - (a^2*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*i
mag_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*e
^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - (2*a^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) -
b^2*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^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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*ima
g_part(arccos(-a/(a + b) + b/(a + b))))^3 + (2*a^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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)
))) - (2*a^2*b*e^(2*c) + 2*a*b^2*e^(2*c) - (a^2*e^(2*c) - b^2*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 + b)/(a*e^(4*c) + b*e^
(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a + b)/(a*e^(4*c) + b*e^(4*c))) + e^(2*d*x))/(
a^3*b*e^(2*c) + 2*a^2*b^2*e^(2*c) + a*b^3*e^(2*c)))/d