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

Optimal. Leaf size=166 \[ \frac{\cosh ^3(c+d x)}{3 d (a+b)^3}-\frac{(a-2 b) \cosh (c+d x)}{d (a+b)^4}+\frac{b (7 a-4 b) \text{sech}(c+d x)}{8 d (a+b)^4 \left (a-b \text{sech}^2(c+d x)+b\right )}+\frac{a b \text{sech}(c+d x)}{4 d (a+b)^3 \left (a-b \text{sech}^2(c+d x)+b\right )^2}+\frac{5 \sqrt{b} (3 a-4 b) \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{8 d (a+b)^{9/2}} \]

[Out]

(5*(3*a - 4*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(8*(a + b)^(9/2)*d) - ((a - 2*b)*Cosh[c +
d*x])/((a + b)^4*d) + Cosh[c + d*x]^3/(3*(a + b)^3*d) + (a*b*Sech[c + d*x])/(4*(a + b)^3*d*(a + b - b*Sech[c
+ d*x]^2)^2) + ((7*a - 4*b)*b*Sech[c + d*x])/(8*(a + b)^4*d*(a + b - b*Sech[c + d*x]^2))

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Rubi [A]  time = 0.286174, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3664, 456, 1259, 1261, 208} \[ \frac{\cosh ^3(c+d x)}{3 d (a+b)^3}-\frac{(a-2 b) \cosh (c+d x)}{d (a+b)^4}+\frac{b (7 a-4 b) \text{sech}(c+d x)}{8 d (a+b)^4 \left (a-b \text{sech}^2(c+d x)+b\right )}+\frac{a b \text{sech}(c+d x)}{4 d (a+b)^3 \left (a-b \text{sech}^2(c+d x)+b\right )^2}+\frac{5 \sqrt{b} (3 a-4 b) \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{8 d (a+b)^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(3*a - 4*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(8*(a + b)^(9/2)*d) - ((a - 2*b)*Cosh[c +
d*x])/((a + b)^4*d) + Cosh[c + d*x]^3/(3*(a + b)^3*d) + (a*b*Sech[c + d*x])/(4*(a + b)^3*d*(a + b - b*Sech[c
+ d*x]^2)^2) + ((7*a - 4*b)*b*Sech[c + d*x])/(8*(a + b)^4*d*(a + b - b*Sech[c + d*x]^2))

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 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1259

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^(
m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*(-d)^(-(m/2) + 1)*e^(2*p)*(q + 1)*
(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x]
, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

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)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1+x^2}{x^4 \left (a+b-b x^2\right )^3} \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=\frac{a b \text{sech}(c+d x)}{4 (a+b)^3 d \left (a+b-b \text{sech}^2(c+d x)\right )^2}+\frac{b \operatorname{Subst}\left (\int \frac{-\frac{4}{b (a+b)}+\frac{4 a x^2}{b (a+b)^2}+\frac{3 a x^4}{(a+b)^3}}{x^4 \left (a+b-b x^2\right )^2} \, dx,x,\text{sech}(c+d x)\right )}{4 d}\\ &=\frac{a b \text{sech}(c+d x)}{4 (a+b)^3 d \left (a+b-b \text{sech}^2(c+d x)\right )^2}+\frac{(7 a-4 b) b \text{sech}(c+d x)}{8 (a+b)^4 d \left (a+b-b \text{sech}^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{-8 b (a+b)+8 (a-b) b x^2+\frac{(7 a-4 b) b^2 x^4}{a+b}}{x^4 \left (a+b-b x^2\right )} \, dx,x,\text{sech}(c+d x)\right )}{8 b (a+b)^3 d}\\ &=\frac{a b \text{sech}(c+d x)}{4 (a+b)^3 d \left (a+b-b \text{sech}^2(c+d x)\right )^2}+\frac{(7 a-4 b) b \text{sech}(c+d x)}{8 (a+b)^4 d \left (a+b-b \text{sech}^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \left (-\frac{8 b}{x^4}+\frac{8 (a-2 b) b}{(a+b) x^2}+\frac{5 (3 a-4 b) b^2}{(a+b) \left (a+b-b x^2\right )}\right ) \, dx,x,\text{sech}(c+d x)\right )}{8 b (a+b)^3 d}\\ &=-\frac{(a-2 b) \cosh (c+d x)}{(a+b)^4 d}+\frac{\cosh ^3(c+d x)}{3 (a+b)^3 d}+\frac{a b \text{sech}(c+d x)}{4 (a+b)^3 d \left (a+b-b \text{sech}^2(c+d x)\right )^2}+\frac{(7 a-4 b) b \text{sech}(c+d x)}{8 (a+b)^4 d \left (a+b-b \text{sech}^2(c+d x)\right )}+\frac{(5 (3 a-4 b) b) \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\text{sech}(c+d x)\right )}{8 (a+b)^4 d}\\ &=\frac{5 (3 a-4 b) \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{8 (a+b)^{9/2} d}-\frac{(a-2 b) \cosh (c+d x)}{(a+b)^4 d}+\frac{\cosh ^3(c+d x)}{3 (a+b)^3 d}+\frac{a b \text{sech}(c+d x)}{4 (a+b)^3 d \left (a+b-b \text{sech}^2(c+d x)\right )^2}+\frac{(7 a-4 b) b \text{sech}(c+d x)}{8 (a+b)^4 d \left (a+b-b \text{sech}^2(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 1.81168, size = 227, normalized size = 1.37 \[ \frac{-\frac{6 \cosh (c+d x) \left (\left (-27 a^2 b+6 a^3-11 a b^2+22 b^3\right ) \cosh (2 (c+d x))-24 a^2 b+3 a^3+30 a b^2+3 (a-3 b) (a+b)^2 \cosh ^2(2 (c+d x))-13 b^3\right )}{(a+b)^4 ((a+b) \cosh (2 (c+d x))+a-b)^2}+\frac{2 \cosh (3 (c+d x))}{(a+b)^3}+\frac{15 i \sqrt{b} (3 a-4 b) \left (\tan ^{-1}\left (\frac{-\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )+\tan ^{-1}\left (\frac{\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )\right )}{(a+b)^{9/2}}}{24 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((15*I)*(3*a - 4*b)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sq
rt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]))/(a + b)^(9/2) - (6*Cosh[c + d*x]*(3*a^3 - 24*a^2*b + 30*a*b^
2 - 13*b^3 + (6*a^3 - 27*a^2*b - 11*a*b^2 + 22*b^3)*Cosh[2*(c + d*x)] + 3*(a - 3*b)*(a + b)^2*Cosh[2*(c + d*x)
]^2))/((a + b)^4*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (2*Cosh[3*(c + d*x)])/(a + b)^3)/(24*d)

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Maple [B]  time = 0.109, size = 341, normalized size = 2.1 \begin{align*}{\frac{1}{d} \left ({\frac{1}{3\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-{\frac{1}{2\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{a-5\,b}{2\, \left ( a+b \right ) ^{4}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-2\,{\frac{b}{ \left ( a+b \right ) ^{4}} \left ({\frac{1}{ \left ( \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) ^{2}} \left ( -1/8\, \left ( 9\,a+20\,b \right ) a \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-1/8\,{\frac{ \left ( 27\,{a}^{3}+66\,{a}^{2}b+56\,a{b}^{2}-16\,{b}^{3} \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}}{a}}+ \left ( -{\frac{27\,{a}^{2}}{8}}-11/2\,ab+2\,{b}^{2} \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-{\frac{9\,{a}^{2}}{8}}+1/4\,ab \right ) }-{\frac{15\,a-20\,b}{16\,\sqrt{ab+{b}^{2}}}{\it Artanh} \left ( 1/4\,{\frac{2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,a+4\,b}{\sqrt{ab+{b}^{2}}}} \right ) } \right ) }-{\frac{1}{3\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{2\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}-{\frac{-a+5\,b}{2\, \left ( a+b \right ) ^{4}} \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)^3,x)

[Out]

1/d*(1/3/(a+b)^3/(tanh(1/2*d*x+1/2*c)+1)^3-1/2/(a+b)^3/(tanh(1/2*d*x+1/2*c)+1)^2-1/2*(a-5*b)/(a+b)^4/(tanh(1/2
*d*x+1/2*c)+1)-2*b/(a+b)^4*((-1/8*(9*a+20*b)*a*tanh(1/2*d*x+1/2*c)^6-1/8*(27*a^3+66*a^2*b+56*a*b^2-16*b^3)/a*t
anh(1/2*d*x+1/2*c)^4+(-27/8*a^2-11/2*a*b+2*b^2)*tanh(1/2*d*x+1/2*c)^2-9/8*a^2+1/4*a*b)/(tanh(1/2*d*x+1/2*c)^4*
a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2-5/16*(3*a-4*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)))-1/3/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)^3-1/2/(a+b)^3/(tanh(1/2*d*x+
1/2*c)-1)^2-1/2/(a+b)^4*(-a+5*b)/(tanh(1/2*d*x+1/2*c)-1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**3,x)

[Out]

Timed out

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Giac [C]  time = 3.14298, size = 8107, normalized size = 48.84 \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)^3,x, algorithm="giac")

[Out]

-1/96*(30*(3*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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)))) - (6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*
b - 7*a*b^2 + 4*b^3)*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arcco
s(-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*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-a*b))*cosh(1/2*imag_part(a
rccos(-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(arcc
os(-a/(a + b) + b/(a + b)))) + 9*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-a*b))*cos(1/2*real_p
art(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(a
rccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(6*a^2*b^2 - 8*a*b^3
- (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*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))))^2 - 3*(6*a^2*b^2 - 8
*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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 + (6*a^2*b^2
- 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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 - (6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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)))) + (
6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*
a*b^4 + b^5)/(a^5*e^(4*c) + 5*a^4*b*e^(4*c) + 10*a^3*b^2*e^(4*c) + 10*a^2*b^3*e^(4*c) + 5*a*b^4*e^(4*c) + b^5*
e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) + e^(d*x))/(((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*
b^4 + b^5)/(a^5*e^(4*c) + 5*a^4*b*e^(4*c) + 10*a^3*b^2*e^(4*c) + 10*a^2*b^3*e^(4*c) + 5*a*b^4*e^(4*c) + b^5*e^
(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(a^6*b + 5*a^5*b^2 + 10*a^4*b^3 + 10*a^3*b^4 + 5*a^2*b^5 + a
*b^6) + 30*(3*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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)))) - (6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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*(6*a^2*b^2 - 8*a*b^3 - (3*a^2
*b - 7*a*b^2 + 4*b^3)*sqrt(-a*b))*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))))^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*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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(arc
cos(-a/(a + b) + b/(a + b)))) + 9*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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*(6*a^2*b^2 - 8*a*b^
3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-a*b))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(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))))^2 - 3*(6*a^2*b^2 -
8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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 + (6*a^2*b^2
- 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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 - (6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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)))) +
(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 +
5*a*b^4 + b^5)/(a^5*e^(4*c) + 5*a^4*b*e^(4*c) + 10*a^3*b^2*e^(4*c) + 10*a^2*b^3*e^(4*c) + 5*a*b^4*e^(4*c) + b^
5*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) - e^(d*x))/(((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*
a*b^4 + b^5)/(a^5*e^(4*c) + 5*a^4*b*e^(4*c) + 10*a^3*b^2*e^(4*c) + 10*a^2*b^3*e^(4*c) + 5*a*b^4*e^(4*c) + b^5*
e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(a^6*b + 5*a^5*b^2 + 10*a^4*b^3 + 10*a^3*b^4 + 5*a^2*b^5 +
a*b^6) + 4*(9*a*e^(2*d*x + 2*c) - 27*b*e^(2*d*x + 2*c) - a - b)*e^(-3*d*x)/(a^4*e^(3*c) + 4*a^3*b*e^(3*c) + 6
*a^2*b^2*e^(3*c) + 4*a*b^3*e^(3*c) + b^4*e^(3*c)) + 15*((6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqr
t(-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*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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*real_part(arccos(-a/(a
+ b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b
- 7*a*b^2 + 4*b^3)*sqrt(-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*(6*a^2*b^2 - 8*a*b^3 - (3*
a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*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))))^2*sinh(1/2*imag_part(arccos(-
a/(a + b) + b/(a + b))))^2 - (6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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*(6*a^2*b^2 - 8*a*
b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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 - (6*a^2*b^2 - 8
*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*i
mag_part(arccos(-a/(a + b) + b/(a + b)))) + (6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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
^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)/(a^5*e^(4*c) + 5*a^4*b*e^(4*c) + 10*a^3*b^2*e^(4*c) +
10*a^2*b^3*e^(4*c) + 5*a*b^4*e^(4*c) + b^5*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a
^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)/(a^5*e^(4*c) + 5*a^4*b*e^(4*c) + 10*a^3*b^2*e^(4*c) +
10*a^2*b^3*e^(4*c) + 5*a*b^4*e^(4*c) + b^5*e^(4*c))) + e^(2*d*x))/(a^6*b + 5*a^5*b^2 + 10*a^4*b^3 + 10*a^3*b^4
+ 5*a^2*b^5 + a*b^6) - 15*((6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-a*b))*cos(1/2*real_part(a
rccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 - 3*(6*a^2*b^2 - 8*a*b
^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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*(6*a^2*b^2 -
8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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*(6*a^2*
b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-a*b))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cos
h(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*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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))))*si
nh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqr
t(-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)))
)*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 -
(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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*(6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4
*b^3)*sqrt(-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 - (6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2
+ 4*b^3)*sqrt(-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)))) + (6*a^2*b^2 - 8*a*b^3 - (3*a^2*b - 7*a*b^2 + 4*b^3)*sqrt(-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^5 + 5*a^4*b + 10*a^3*b^2 +
10*a^2*b^3 + 5*a*b^4 + b^5)/(a^5*e^(4*c) + 5*a^4*b*e^(4*c) + 10*a^3*b^2*e^(4*c) + 10*a^2*b^3*e^(4*c) + 5*a*b^
4*e^(4*c) + b^5*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a^5 + 5*a^4*b + 10*a^3*b^2 +
10*a^2*b^3 + 5*a*b^4 + b^5)/(a^5*e^(4*c) + 5*a^4*b*e^(4*c) + 10*a^3*b^2*e^(4*c) + 10*a^2*b^3*e^(4*c) + 5*a*b^
4*e^(4*c) + b^5*e^(4*c))) + e^(2*d*x))/(a^6*b + 5*a^5*b^2 + 10*a^4*b^3 + 10*a^3*b^4 + 5*a^2*b^5 + a*b^6) - 4*(
a^6*e^(3*d*x + 48*c) + 6*a^5*b*e^(3*d*x + 48*c) + 15*a^4*b^2*e^(3*d*x + 48*c) + 20*a^3*b^3*e^(3*d*x + 48*c) +
15*a^2*b^4*e^(3*d*x + 48*c) + 6*a*b^5*e^(3*d*x + 48*c) + b^6*e^(3*d*x + 48*c) - 9*a^6*e^(d*x + 46*c) - 18*a^5*
b*e^(d*x + 46*c) + 45*a^4*b^2*e^(d*x + 46*c) + 180*a^3*b^3*e^(d*x + 46*c) + 225*a^2*b^4*e^(d*x + 46*c) + 126*a
*b^5*e^(d*x + 46*c) + 27*b^6*e^(d*x + 46*c))/(a^9*e^(45*c) + 9*a^8*b*e^(45*c) + 36*a^7*b^2*e^(45*c) + 84*a^6*b
^3*e^(45*c) + 126*a^5*b^4*e^(45*c) + 126*a^4*b^5*e^(45*c) + 84*a^3*b^6*e^(45*c) + 36*a^2*b^7*e^(45*c) + 9*a*b^
8*e^(45*c) + b^9*e^(45*c)) - 24*(9*a^2*b*e^(7*d*x + 7*c) + 5*a*b^2*e^(7*d*x + 7*c) - 4*b^3*e^(7*d*x + 7*c) + 2
7*a^2*b*e^(5*d*x + 5*c) - 13*a*b^2*e^(5*d*x + 5*c) + 4*b^3*e^(5*d*x + 5*c) + 27*a^2*b*e^(3*d*x + 3*c) - 13*a*b
^2*e^(3*d*x + 3*c) + 4*b^3*e^(3*d*x + 3*c) + 9*a^2*b*e^(d*x + c) + 5*a*b^2*e^(d*x + c) - 4*b^3*e^(d*x + c))/((
a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*
e^(2*d*x + 2*c) + a + b)^2))/d