### 3.29 $$\int \frac{\text{csch}(c+d x)}{a+b \tanh ^2(c+d x)} \, dx$$

Optimal. Leaf size=55 $\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{a d \sqrt{a+b}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a d}$

[Out]

-(ArcTanh[Cosh[c + d*x]]/(a*d)) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]*d)

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Rubi [A]  time = 0.078118, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.19, Rules used = {3664, 391, 207, 208} $\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{a d \sqrt{a+b}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(ArcTanh[Cosh[c + d*x]]/(a*d)) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]*d)

Rule 3664

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a - b + b*ff^2*x^2)^p)/x^(
m + 1), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

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 207

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

Rule 208

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

Rubi steps

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

Mathematica [C]  time = 0.199405, size = 123, normalized size = 2.24 $\frac{\sqrt{a+b} \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+i \sqrt{b} \tan ^{-1}\left (\frac{-\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )+i \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )}{a d \sqrt{a+b}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(I*Sqrt[b]*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + I*Sqrt[b]*ArcTan[((-I)*Sqrt[a + b]
+ Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + Sqrt[a + b]*Log[Tanh[(c + d*x)/2]])/(a*Sqrt[a + b]*d)

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Maple [A]  time = 0.063, size = 69, normalized size = 1.3 \begin{align*}{\frac{b}{da}{\it Artanh} \left ({\frac{1}{4} \left ( 2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,a+4\,b \right ){\frac{1}{\sqrt{ab+{b}^{2}}}}} \right ){\frac{1}{\sqrt{ab+{b}^{2}}}}}+{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d/a*b/(a*b+b^2)^(1/2)*arctanh(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+2*a+4*b)/(a*b+b^2)^(1/2))+1/d/a*ln(tanh(1/2*d*x
+1/2*c))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.39541, size = 1673, normalized size = 30.42 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [C]  time = 1.52559, size = 5242, normalized size = 95.31 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*(3*(a^2*b*e^(2*c) + a*b^2*e^(2*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^(2*c) +
a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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*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^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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_p
art(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(a^2*b*e^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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^(
2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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 + 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*e^(2*c) + (a^2*e^(2*c) - a*b*e^(2*c))*sqrt(-a*b)*abs(a)) + 2*
(3*(a^2*b*e^(2*c) + a*b^2*e^(2*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))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) - (a^2*b*e^(2*c) + a*b^2*e^
(2*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^(2*c) + a*b^2*e^(2*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_par
t(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(arc
cos(-a/(a + b) + b/(a + b)))) + 3*(a^2*b*e^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_pa
rt(arccos(-a/(a + b) + b/(a + b))))*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))))^2 - 3*(a^2*b*e^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_pa
rt(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (a^2*b*e^(2*c) + a
*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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 + 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*e^(2*c) + (a^2*e^(2*c) - a*b*e^(2*c))*sqrt(-a*b)*abs(a)) + ((a^2*b*e
^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*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))))^2 - 3*(
a^2*b*e^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(
2*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^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^
(2*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^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arcc
os(-a/(a + b) + b/(a + b)))) + (a^2*b*e^(2*c) + a*b^2*e^(2*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 + 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*e^(2*c) + (a^2*e^(2*c) - a*b*e^(2*c))*sqrt(-a*b)*abs(a)) - ((a^2*b*e^(2*c) + a*b^2*e^(2*c))*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))))^3 - 3*(a^2*b*e^(2*c)
+ a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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*im
ag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(a^2*b*e^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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*ima
g_part(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(a^2*b*e^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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^(2*c) + a*b^2*e^(2*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
^(2*c) + a*b^2*e^(2*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 + 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*e^(2*c) + (a^2*e^(2*c) - a*b*e^(2*c
))*sqrt(-a*b)*abs(a)) + 4*log(e^(d*x + c) + 1)/a - 4*log(abs(e^(d*x + c) - 1))/a)/d