3.111 \(\int \frac{\text{sech}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\)
Optimal. Leaf size=55 \[ \frac{\sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b d}-\frac{\tan ^{-1}(\sinh (c+d x))}{b d} \]
[Out]
-(ArcTan[Sinh[c + d*x]]/(b*d)) + (Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b*d)
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Rubi [A] time = 0.0729596, antiderivative size = 55, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used =
{3676, 391, 203, 205} \[ \frac{\sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b d}-\frac{\tan ^{-1}(\sinh (c+d x))}{b d} \]
Antiderivative was successfully verified.
[In]
Int[Sech[c + d*x]^3/(a + b*Tanh[c + d*x]^2),x]
[Out]
-(ArcTan[Sinh[c + d*x]]/(b*d)) + (Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/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 391
Int[1/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^n),
x], x] - Dist[d/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0]
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}^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{b d}+\frac{(a+b) \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{b d}\\ &=-\frac{\tan ^{-1}(\sinh (c+d x))}{b d}+\frac{\sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a+b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} b d}\\ \end{align*}
Mathematica [A] time = 0.194912, size = 55, normalized size = 1. \[ -\frac{\frac{\sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{a+b}}\right )}{\sqrt{a}}+2 \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{b d} \]
Antiderivative was successfully verified.
[In]
Integrate[Sech[c + d*x]^3/(a + b*Tanh[c + d*x]^2),x]
[Out]
-(((Sqrt[a + b]*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/Sqrt[a] + 2*ArcTan[Tanh[(c + d*x)/2]])/(b*d))
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Maple [B] time = 0.073, size = 494, normalized size = 9. \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)^3/(a+b*tanh(d*x+c)^2),x)
[Out]
-1/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)
)+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*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))+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/((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))-2/d/b*arctan(tanh(1/2*d*x+1/2*c))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, \arctan \left (e^{\left (d x + c\right )}\right )}{b d} + 8 \, \int \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 )}}{4 \,{\left (a b + b^{2} +{\left (a b e^{\left (4 \, c\right )} + b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a b e^{\left (2 \, c\right )} - b^{2} 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),x, algorithm="maxima")
[Out]
-2*arctan(e^(d*x + c))/(b*d) + 8*integrate(1/4*((a*e^(3*c) + b*e^(3*c))*e^(3*d*x) + (a*e^c + b*e^c)*e^(d*x))/(
a*b + b^2 + (a*b*e^(4*c) + b^2*e^(4*c))*e^(4*d*x) + 2*(a*b*e^(2*c) - b^2*e^(2*c))*e^(2*d*x)), x)
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Fricas [B] time = 2.58427, size = 1494, normalized size = 27.16 \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),x, algorithm="fricas")
[Out]
[1/2*(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)*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*(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)*c
osh(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)) - 4*arctan(cosh(d*x + c) + sinh(d*x + c)))/(b*d), (sqrt((a + b)/a)*arct
an(1/2*sqrt((a + b)/a)*(cosh(d*x + c) + sinh(d*x + c))) + 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)*cos
h(d*x + c)^2 + 3*a - b)*sinh(d*x + c))*sqrt((a + b)/a)/(a + b)) - 2*arctan(cosh(d*x + c) + sinh(d*x + c)))/(b*
d)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{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(sech(d*x+c)**3/(a+b*tanh(d*x+c)**2),x)
[Out]
Integral(sech(c + d*x)**3/(a + b*tanh(c + d*x)**2), x)
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Giac [C] time = 1.79286, size = 4938, normalized size = 89.78 \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),x, algorithm="giac")
[Out]
1/4*(2*(3*(a^2*b + 2*a*b^2 + b^3)*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)))) - (a^2*b + 2*a*b^2 + b^3)*co
sh(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 + 2*a*b^2 + b^3)*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*b + 2*a*b^2 + b^3)*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_pa
rt(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(a^2*b + 2*a*b^2
+ b^3)*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 + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*cosh(1/2*imag_part(a
rccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) - (a^2*b + 2*a*b^2 + b^3)*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 + b^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*b + b^2)/(a
*b*e^(4*c) + b^2*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(2*a*b^3 + (a*b - b^2)*sqrt(-a*b)*abs(b))
+ 2*(3*(a^2*b + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*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*b + 2*a*b^2 + b^3)*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 + 2*a*b^2 +
b^3)*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 + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*cos(1/2*r
eal_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_p
art(arccos(-a/(a + b) + b/(a + b))))^3 + (a^2*b + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*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)))) - (a^2*b + 2*a*b^2 + b^3)*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 + b^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*b + b^2)/(a*
b*e^(4*c) + b^2*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(2*a*b^3 + (a*b - b^2)*sqrt(-a*b)*abs(b))
+ ((a^2*b + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*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
+ 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*cos(1/2*real_pa
rt(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(ar
ccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(a^2*b + 2*a*b^2 + b^
3)*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))))*si
nh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(a^2*b + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*cos(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))))^3 + 3*(a^2*b + 2*a*
b^2 + b^3)*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 + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*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
+ b^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*b + b^2)/(a*b*e^
(4*c) + b^2*e^(4*c))) + e^(2*d*x))/(2*a*b^3 + (a*b - b^2)*sqrt(-a*b)*abs(b)) - ((a^2*b + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*cos(1/2*real_pa
rt(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 + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*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))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_par
t(arccos(-a/(a + b) + b/(a + b))))^2 - (a^2*b + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*cos(1/2*real_part(arcc
os(-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 + 2*a*b^2 + b^3)*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 + 2*a*b^2 + b^3)*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 + b^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*b + b^2)/(a*b*e^(4*c) + b^2*e^(4*c))) + e^(2*d*x
))/(2*a*b^3 + (a*b - b^2)*sqrt(-a*b)*abs(b)) - 8*arctan(e^(d*x + c))/b)/d