### 3.26 $$\int \frac{\sinh ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx$$

Optimal. Leaf size=75 $\frac{\cosh ^3(c+d x)}{3 d (a+b)}-\frac{a \cosh (c+d x)}{d (a+b)^2}+\frac{a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{d (a+b)^{5/2}}$

[Out]

(a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/((a + b)^(5/2)*d) - (a*Cosh[c + d*x])/((a + b)^2*d) +
Cosh[c + d*x]^3/(3*(a + b)*d)

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Rubi [A]  time = 0.128102, antiderivative size = 75, 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 = {3664, 453, 325, 208} $\frac{\cosh ^3(c+d x)}{3 d (a+b)}-\frac{a \cosh (c+d x)}{d (a+b)^2}+\frac{a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{d (a+b)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^2),x]

[Out]

(a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/((a + b)^(5/2)*d) - (a*Cosh[c + d*x])/((a + b)^2*d) +
Cosh[c + d*x]^3/(3*(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 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

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 ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1+x^2}{x^4 \left (a+b-b x^2\right )} \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=\frac{\cosh ^3(c+d x)}{3 (a+b) d}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b-b x^2\right )} \, dx,x,\text{sech}(c+d x)\right )}{(a+b) d}\\ &=-\frac{a \cosh (c+d x)}{(a+b)^2 d}+\frac{\cosh ^3(c+d x)}{3 (a+b) d}+\frac{(a b) \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\text{sech}(c+d x)\right )}{(a+b)^2 d}\\ &=\frac{a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{(a+b)^{5/2} d}-\frac{a \cosh (c+d x)}{(a+b)^2 d}+\frac{\cosh ^3(c+d x)}{3 (a+b) d}\\ \end{align*}

Mathematica [C]  time = 0.538918, size = 135, normalized size = 1.8 $\frac{(a+b)^{3/2} \cosh (3 (c+d x))-3 (3 a-b) \sqrt{a+b} \cosh (c+d x)+12 i a \sqrt{b} \left (\tan ^{-1}\left (\frac{-\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )+\tan ^{-1}\left (\frac{\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )\right )}{12 d (a+b)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^2),x]

[Out]

((12*I)*a*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sqrt[a + b] +
Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]) - 3*(3*a - b)*Sqrt[a + b]*Cosh[c + d*x] + (a + b)^(3/2)*Cosh[3*(c + d*x)
])/(12*(a + b)^(5/2)*d)

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Maple [B]  time = 0.069, size = 202, normalized size = 2.7 \begin{align*}{\frac{1}{d} \left ( -8\,{\frac{1}{ \left ( 16\,a+16\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+{\frac{16}{48\,a+48\,b} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-{\frac{a-b}{2\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{ab}{ \left ( a+b \right ) ^{2}}{\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{16}{48\,a+48\,b} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-8\,{\frac{1}{ \left ( 16\,a+16\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{2}}}-{\frac{-a+b}{2\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^3/(a+b*tanh(d*x+c)^2),x)

[Out]

1/d*(-8/(16*a+16*b)/(tanh(1/2*d*x+1/2*c)+1)^2+16/3/(tanh(1/2*d*x+1/2*c)+1)^3/(16*a+16*b)-1/2*(a-b)/(a+b)^2/(ta
nh(1/2*d*x+1/2*c)+1)+a*b/(a+b)^2/(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))-16/3/(tanh(1/2*d*x+1/2*c)-1)^3/(16*a+16*b)-8/(16*a+16*b)/(tanh(1/2*d*x+1/2*c)-1)^2-1/2/(a+b)^2*(-a+b)/(tan
h(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 ({\left (a e^{\left (6 \, c\right )} + b e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - 3 \,{\left (3 \, a e^{\left (4 \, c\right )} - b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 3 \,{\left (3 \, a e^{\left (2 \, c\right )} - b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + a + b\right )} e^{\left (-3 \, d x\right )}}{24 \,{\left (a^{2} d e^{\left (3 \, c\right )} + 2 \, a b d e^{\left (3 \, c\right )} + b^{2} d e^{\left (3 \, c\right )}\right )}} - \frac{1}{8} \, \int \frac{16 \,{\left (a b e^{\left (3 \, d x + 3 \, c\right )} - a b e^{\left (d x + c\right )}\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} +{\left (a^{3} e^{\left (4 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{3} e^{\left (2 \, c\right )} + a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )} - b^{3} 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)^3/(a+b*tanh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/24*((a*e^(6*c) + b*e^(6*c))*e^(6*d*x) - 3*(3*a*e^(4*c) - b*e^(4*c))*e^(4*d*x) - 3*(3*a*e^(2*c) - b*e^(2*c))*
e^(2*d*x) + a + b)*e^(-3*d*x)/(a^2*d*e^(3*c) + 2*a*b*d*e^(3*c) + b^2*d*e^(3*c)) - 1/8*integrate(16*(a*b*e^(3*d
*x + 3*c) - a*b*e^(d*x + c))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3 + (a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c)
+ b^3*e^(4*c))*e^(4*d*x) + 2*(a^3*e^(2*c) + a^2*b*e^(2*c) - a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.45116, size = 3729, normalized size = 49.72 \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)^3/(a+b*tanh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/24*((a + b)*cosh(d*x + c)^6 + 6*(a + b)*cosh(d*x + c)*sinh(d*x + c)^5 + (a + b)*sinh(d*x + c)^6 - 3*(3*a -
b)*cosh(d*x + c)^4 + 3*(5*(a + b)*cosh(d*x + c)^2 - 3*a + b)*sinh(d*x + c)^4 + 4*(5*(a + b)*cosh(d*x + c)^3 -
3*(3*a - b)*cosh(d*x + c))*sinh(d*x + c)^3 - 3*(3*a - b)*cosh(d*x + c)^2 + 3*(5*(a + b)*cosh(d*x + c)^4 - 6*(3
*a - b)*cosh(d*x + c)^2 - 3*a + b)*sinh(d*x + c)^2 + 12*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)^2*sinh(d*x + c)
+ 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3)*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)) + 6*((a + b)*cosh(d*x + c)^5 - 2*(3*a - b)*cosh(d*x + c)^3 - (3*a - b)*cosh(d*x + c))*sinh(d*x +
c) + a + b)/((a^2 + 2*a*b + b^2)*d*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*d*cosh(d*x + c)^2*sinh(d*x + c) + 3
*(a^2 + 2*a*b + b^2)*d*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2 + 2*a*b + b^2)*d*sinh(d*x + c)^3), 1/24*((a + b)*c
osh(d*x + c)^6 + 6*(a + b)*cosh(d*x + c)*sinh(d*x + c)^5 + (a + b)*sinh(d*x + c)^6 - 3*(3*a - b)*cosh(d*x + c)
^4 + 3*(5*(a + b)*cosh(d*x + c)^2 - 3*a + b)*sinh(d*x + c)^4 + 4*(5*(a + b)*cosh(d*x + c)^3 - 3*(3*a - b)*cosh
(d*x + c))*sinh(d*x + c)^3 - 3*(3*a - b)*cosh(d*x + c)^2 + 3*(5*(a + b)*cosh(d*x + c)^4 - 6*(3*a - b)*cosh(d*x
+ c)^2 - 3*a + b)*sinh(d*x + c)^2 + 24*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)^2*sinh(d*x + c) + 3*a*cosh(d*x
+ c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3)*sqrt(-b/(a + b))*arctan(1/2*((a + b)*cosh(d*x + c)^3 + 3*(a + b)*cos
h(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) - 24*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)^2*sinh(d*x + c) + 3*a*
cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3)*sqrt(-b/(a + b))*arctan(1/2*((a + b)*cosh(d*x + c) + (a + b
)*sinh(d*x + c))*sqrt(-b/(a + b))/b) + 6*((a + b)*cosh(d*x + c)^5 - 2*(3*a - b)*cosh(d*x + c)^3 - (3*a - b)*co
sh(d*x + c))*sinh(d*x + c) + a + b)/((a^2 + 2*a*b + b^2)*d*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b + b^2)*d*cosh(d*x
+ c)^2*sinh(d*x + c) + 3*(a^2 + 2*a*b + b^2)*d*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2 + 2*a*b + b^2)*d*sinh(d*x
+ c)^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{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(sinh(d*x+c)**3/(a+b*tanh(d*x+c)**2),x)

[Out]

Integral(sinh(c + d*x)**3/(a + b*tanh(c + d*x)**2), x)

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Giac [C]  time = 1.88798, size = 5603, normalized size = 74.71 \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)^3/(a+b*tanh(d*x+c)^2),x, algorithm="giac")

[Out]

-1/24*(12*(3*(2*a*b + sqrt(-a*b)*(a - b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_p
art(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) - (2*a*b + sqrt(-a*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))))^3 - 9*(2*a*b + sqrt(-a*b)*(a - b))*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))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(ar
ccos(-a/(a + b) + b/(a + b)))) + 3*(2*a*b + sqrt(-a*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))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))
)) + 9*(2*a*b + sqrt(-a*b)*(a - b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(ar
ccos(-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*b + sqrt(-a*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))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 -
3*(2*a*b + sqrt(-a*b)*(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*b + sqrt(-a*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))))^3 - (
2*a*b + sqrt(-a*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*a*b + sqrt(-a*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)))))*arctan((((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*
e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) + e^(d*x))/(((a^3 + 3*a^2*b
+ 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b
)/(a + b)))))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 12*(3*(2*a*b + sqrt(-a*b)*(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*b + sqrt(-a*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))))^3 - 9*(2*a*b + sqrt(-a*b)*(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*b + sqrt(-a*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))))^3*sinh(1/2
*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(2*a*b + sqrt(-a*b)*(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*b + sqrt(-a*b)*(a - b))*cosh(1/2*i
mag_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_p
art(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a*b + sqrt(-a*b)*(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*b + sqrt(-a*b)*(a - b))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_pa
rt(arccos(-a/(a + b) + b/(a + b))))^3 - (2*a*b + sqrt(-a*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*a*b + sqrt(-a*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)))))*arctan(-(((a^3 + 3*a^2*
b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a -
b)/(a + b))) - e^(d*x))/(((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) +
b^3*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + (9*a*e^(2*d*x + 2*c)
- 3*b*e^(2*d*x + 2*c) - a - b)*e^(-3*d*x)/(a^2*e^(3*c) + 2*a*b*e^(3*c) + b^2*e^(3*c)) + 6*((2*a*b + sqrt(-a*b
)*(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*b + sqrt(-a*b)*(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*real_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a*b + sqrt(-a*
b)*(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*b + sqrt(-a*b)*(a - b))*cos(1/2*real_par
t(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(arc
cos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(2*a*b + sqrt(-a*b)*(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*b + sqrt(-a*b)*(a - b))*cos(1/2*real_part(arc
cos(-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*b + sqrt(-a*b)*(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*b + sqrt(-a*b)*(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*b + sqrt(-a*b)*(a - b))*c
os(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*b
+ sqrt(-a*b)*(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^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3
*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c)
+ 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c))) + e^(2*d*x))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 6*((2*a*b
+ sqrt(-a*b)*(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*b + sqrt(-a*b)*(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*real_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(2*a*b
+ sqrt(-a*b)*(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*b + sqrt(-a*b)*(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*re
al_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(2*a*b + sq
rt(-a*b)*(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*b + sqrt(-a*b)*(a - b))*cos(1/2*re
al_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*b + sqrt(-a*b
)*(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*b + sqrt(-a*b)*(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*b + sqrt(-a*b)
*(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*b + sqrt(-a*b)*(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^3 + 3*a^2*b + 3*a*b^2 + b^3)/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e
^(4*c) + b^3*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a^3 + 3*a^2*b + 3*a*b^2 + b^3)/
(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c))) + e^(2*d*x))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3)
- (a^2*e^(3*d*x + 24*c) + 2*a*b*e^(3*d*x + 24*c) + b^2*e^(3*d*x + 24*c) - 9*a^2*e^(d*x + 22*c) - 6*a*b*e^(d*x
+ 22*c) + 3*b^2*e^(d*x + 22*c))/(a^3*e^(21*c) + 3*a^2*b*e^(21*c) + 3*a*b^2*e^(21*c) + b^3*e^(21*c)))/d