### 3.28 $$\int \frac{\sinh (c+d x)}{a+b \tanh ^2(c+d x)} \, dx$$

Optimal. Leaf size=53 $\frac{\cosh (c+d x)}{d (a+b)}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{d (a+b)^{3/2}}$

[Out]

-((Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/((a + b)^(3/2)*d)) + Cosh[c + d*x]/((a + b)*d)

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Rubi [A]  time = 0.0626703, 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 = {3664, 325, 208} $\frac{\cosh (c+d x)}{d (a+b)}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{d (a+b)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/((a + b)^(3/2)*d)) + Cosh[c + d*x]/((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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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

Mathematica [C]  time = 0.242032, size = 107, normalized size = 2.02 $\frac{\sqrt{a+b} \cosh (c+d x)-i \sqrt{b} \left (\tan ^{-1}\left (\frac{-\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )+\tan ^{-1}\left (\frac{\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )\right )}{d (a+b)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sinh[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]] + ArcTan[((-I)*Sqrt[a + b] + Sqr
t[a]*Tanh[(c + d*x)/2])/Sqrt[b]]) + Sqrt[a + b]*Cosh[c + d*x])/((a + b)^(3/2)*d)

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Maple [B]  time = 0.055, size = 104, normalized size = 2. \begin{align*}{\frac{1}{d} \left ( -{\frac{b}{a+b}{\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}}}}}-4\,{\frac{1}{ \left ( 4\,a+4\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) }}+4\,{\frac{1}{ \left ( 4\,a+4\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) }} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-b/(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))-4/(4*a+4*b)/(ta
nh(1/2*d*x+1/2*c)-1)+4/(4*a+4*b)/(tanh(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(sinh(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 = 2.20077, size = 1906, normalized size = 35.96 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [C]  time = 1.55928, 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(sinh(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