3.108 \(\int \frac{\cosh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx\)

Optimal. Leaf size=53 \[ \frac{\sinh (c+d x)}{d (a+b)}+\frac{b \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a+b)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0704951, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3676, 388, 205} \[ \frac{\sinh (c+d x)}{d (a+b)}+\frac{b \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a+b)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(3/2)*d) + Sinh[c + d*x]/((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 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 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{\cosh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\sinh (c+d x)}{(a+b) d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{(a+b) d}\\ &=\frac{b \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{3/2} d}+\frac{\sinh (c+d x)}{(a+b) d}\\ \end{align*}

Mathematica [A]  time = 0.0923188, size = 53, normalized size = 1. \[ \frac{\sinh (c+d x)}{d (a+b)}+\frac{b \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a+b)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.071, size = 315, normalized size = 5.9 \begin{align*} -{\frac{b}{d \left ( a+b \right ) }{\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}}{d \left ( a+b \right ) }{\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{b}{d \left ( a+b \right ) }\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}}{d \left ( a+b \right ) }\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}}}}-2\,{\frac{1}{d \left ( 2\,b+2\,a \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) }}-2\,{\frac{1}{d \left ( 2\,b+2\,a \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x\right )}}{2 \,{\left (a d e^{c} + b d e^{c}\right )}} + \frac{1}{2} \, \int \frac{4 \,{\left (b e^{\left (3 \, d x + 3 \, c\right )} + b e^{\left (d x + c\right )}\right )}}{a^{2} + 2 \, a b + b^{2} +{\left (a^{2} e^{\left (4 \, c\right )} + 2 \, a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{2} e^{\left (2 \, c\right )} - b^{2} 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)/(a+b*tanh(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.46219, size = 2040, normalized size = 38.49 \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)/(a+b*tanh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/2*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 -
sqrt(-a^2 - a*b)*(b*cosh(d*x + c) + b*sinh(d*x + c))*log(((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*si
nh(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))*sq
rt(-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)/((a^3 + 2*a^2*b + a*b^2)*d*cosh(d*x +
c) + (a^3 + 2*a^2*b + a*b^2)*d*sinh(d*x + c)), 1/2*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*
sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + 2*sqrt(a^2 + a*b)*(b*cosh(d*x + c) + b*sinh(d*x + c))*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)) + 2*sqrt(a^2 + a*b)*(b*cosh(
d*x + c) + b*sinh(d*x + c))*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x + c))/a) - a^2 - a*b)/((a^3 +
2*a^2*b + a*b^2)*d*cosh(d*x + c) + (a^3 + 2*a^2*b + a*b^2)*d*sinh(d*x + c))]

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

[Out]

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

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Giac [C]  time = 1.58398, size = 6017, normalized size = 113.53 \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)/(a+b*tanh(d*x+c)^2),x, algorithm="giac")

[Out]

1/4*(2*(3*(a^2*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*re
al_part(arccos(-a/(a + b) + b/(a + b))))^3 - 9*(a^2*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*cos(1/2*real_pa
rt(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*b*e^(4*c) + 2*a*
b^2*e^(4*c) + b^3*e^(4*c))*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*(a^2*b*e^(4*c) + 2*a*b^2*e^(4*
c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^
(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*cosh(1/2*imag_part(arcco
s(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) - (a^2*b*e^(4*c) + 2*a*b^2*e^(4
*c) + b^3*e^(4*c))*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^2 + 2*a*b + b^2)/(a^2*e^(4*c) + 2*a*b*e^(4*c) + b^2*e^(4*c)))^(1/4)*cos(1/2*arccos(-(
a - b)/(a + b))) + e^(d*x))/(((a^2 + 2*a*b + b^2)/(a^2*e^(4*c) + 2*a*b*e^(4*c) + b^2*e^(4*c)))^(1/4)*sin(1/2*a
rccos(-(a - b)/(a + b)))))/(2*(a*e^(2*c) + b*e^(2*c))^2*a*b + (a^2*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b)*abs(-a*e^
(2*c) - b*e^(2*c))) + 2*(3*(a^2*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c)
)*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*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)))) + 3*(a^2
*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*(a^2*b*e^(4*c
) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arc
cos(-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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*s
inh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (a^2*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*
c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arcc
os(-a/(a + b) + b/(a + b)))))*arctan(-(((a^2 + 2*a*b + b^2)/(a^2*e^(4*c) + 2*a*b*e^(4*c) + b^2*e^(4*c)))^(1/4)
*cos(1/2*arccos(-(a - b)/(a + b))) - e^(d*x))/(((a^2 + 2*a*b + b^2)/(a^2*e^(4*c) + 2*a*b*e^(4*c) + b^2*e^(4*c)
))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(2*(a*e^(2*c) + b*e^(2*c))^2*a*b + (a^2*e^(2*c) - b^2*e^(2*c))*sq
rt(-a*b)*abs(-a*e^(2*c) - b*e^(2*c))) + ((a^2*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*cos(1/2*real_part(arc
cos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 - 3*(a^2*b*e^(4*c) + 2*a
*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*
c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*
e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*(a^2*b
*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_par
t(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(arc
cos(-a/(a + b) + b/(a + b))))^2 - (a^2*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^
(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*
e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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 + 2*a*b + b^2)/(a^2*e^(4*c) + 2*a*b*e^(4*c) + b^2*e^
(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a^2 + 2*a*b + b^2)/(a^2*e^(4*c) + 2*a*b*e^(4*c
) + b^2*e^(4*c))) + e^(2*d*x))/(2*(a*e^(2*c) + b*e^(2*c))^2*a*b + (a^2*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b)*abs(-
a*e^(2*c) - b*e^(2*c))) - ((a^2*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) +
b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*
c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*re
al_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(a^2*b*e^(4
*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(a
rccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(a^2*b*e^(4*c) + 2*a
*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^
(4*c))*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*b*e^(4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*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*b*e^(
4*c) + 2*a*b^2*e^(4*c) + b^3*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(ar
ccos(-a/(a + b) + b/(a + b)))))*log(-2*((a^2 + 2*a*b + b^2)/(a^2*e^(4*c) + 2*a*b*e^(4*c) + b^2*e^(4*c)))^(1/4)
*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a^2 + 2*a*b + b^2)/(a^2*e^(4*c) + 2*a*b*e^(4*c) + b^2*e^(4*
c))) + e^(2*d*x))/(2*(a*e^(2*c) + b*e^(2*c))^2*a*b + (a^2*e^(2*c) - b^2*e^(2*c))*sqrt(-a*b)*abs(-a*e^(2*c) - b
*e^(2*c))) + 2*e^(d*x + 6*c)/(a*e^(5*c) + b*e^(5*c)) - 2*e^(-d*x)/(a*e^c + b*e^c))/d