### 3.47 $$\int \frac{\text{csch}^3(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx$$

Optimal. Leaf size=196 $-\frac{\sqrt{b} \left (15 a^2+40 a b+24 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{8 a^4 d (a+b)^{3/2}}-\frac{b (11 a+12 b) \text{sech}(c+d x)}{8 a^3 d (a+b) \left (a-b \text{sech}^2(c+d x)+b\right )}-\frac{3 b \text{sech}(c+d x)}{4 a^2 d \left (a-b \text{sech}^2(c+d x)+b\right )^2}+\frac{(a+6 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^4 d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a-b \text{sech}^2(c+d x)+b\right )^2}$

[Out]

((a + 6*b)*ArcTanh[Cosh[c + d*x]])/(2*a^4*d) - (Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTanh[(Sqrt[b]*Sech[c + d
*x])/Sqrt[a + b]])/(8*a^4*(a + b)^(3/2)*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d*(a + b - b*Sech[c + d*x]^2)^
2) - (3*b*Sech[c + d*x])/(4*a^2*d*(a + b - b*Sech[c + d*x]^2)^2) - (b*(11*a + 12*b)*Sech[c + d*x])/(8*a^3*(a +
b)*d*(a + b - b*Sech[c + d*x]^2))

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Rubi [A]  time = 0.334025, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.261, Rules used = {3664, 471, 527, 522, 207, 208} $-\frac{\sqrt{b} \left (15 a^2+40 a b+24 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{8 a^4 d (a+b)^{3/2}}-\frac{b (11 a+12 b) \text{sech}(c+d x)}{8 a^3 d (a+b) \left (a-b \text{sech}^2(c+d x)+b\right )}-\frac{3 b \text{sech}(c+d x)}{4 a^2 d \left (a-b \text{sech}^2(c+d x)+b\right )^2}+\frac{(a+6 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^4 d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a-b \text{sech}^2(c+d x)+b\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((a + 6*b)*ArcTanh[Cosh[c + d*x]])/(2*a^4*d) - (Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTanh[(Sqrt[b]*Sech[c + d
*x])/Sqrt[a + b]])/(8*a^4*(a + b)^(3/2)*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d*(a + b - b*Sech[c + d*x]^2)^
2) - (3*b*Sech[c + d*x])/(4*a^2*d*(a + b - b*Sech[c + d*x]^2)^2) - (b*(11*a + 12*b)*Sech[c + d*x])/(8*a^3*(a +
b)*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 471

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

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

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}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (-1+x^2\right )^2 \left (a+b-b x^2\right )^3} \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a+b-b \text{sech}^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{a+b+5 b x^2}{\left (-1+x^2\right ) \left (a+b-b x^2\right )^3} \, dx,x,\text{sech}(c+d x)\right )}{2 a d}\\ &=-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a+b-b \text{sech}^2(c+d x)\right )^2}-\frac{3 b \text{sech}(c+d x)}{4 a^2 d \left (a+b-b \text{sech}^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{2 (a+b) (2 a+3 b)+18 b (a+b) x^2}{\left (-1+x^2\right ) \left (a+b-b x^2\right )^2} \, dx,x,\text{sech}(c+d x)\right )}{8 a^2 (a+b) d}\\ &=-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a+b-b \text{sech}^2(c+d x)\right )^2}-\frac{3 b \text{sech}(c+d x)}{4 a^2 d \left (a+b-b \text{sech}^2(c+d x)\right )^2}-\frac{b (11 a+12 b) \text{sech}(c+d x)}{8 a^3 (a+b) d \left (a+b-b \text{sech}^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{2 (a+b) \left (4 a^2+17 a b+12 b^2\right )+2 b (a+b) (11 a+12 b) x^2}{\left (-1+x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\text{sech}(c+d x)\right )}{16 a^3 (a+b)^2 d}\\ &=-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a+b-b \text{sech}^2(c+d x)\right )^2}-\frac{3 b \text{sech}(c+d x)}{4 a^2 d \left (a+b-b \text{sech}^2(c+d x)\right )^2}-\frac{b (11 a+12 b) \text{sech}(c+d x)}{8 a^3 (a+b) d \left (a+b-b \text{sech}^2(c+d x)\right )}-\frac{(a+6 b) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\text{sech}(c+d x)\right )}{2 a^4 d}-\frac{\left (b \left (15 a^2+40 a b+24 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-b x^2} \, dx,x,\text{sech}(c+d x)\right )}{8 a^4 (a+b) d}\\ &=\frac{(a+6 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^4 d}-\frac{\sqrt{b} \left (15 a^2+40 a b+24 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{8 a^4 (a+b)^{3/2} d}-\frac{\coth (c+d x) \text{csch}(c+d x)}{2 a d \left (a+b-b \text{sech}^2(c+d x)\right )^2}-\frac{3 b \text{sech}(c+d x)}{4 a^2 d \left (a+b-b \text{sech}^2(c+d x)\right )^2}-\frac{b (11 a+12 b) \text{sech}(c+d x)}{8 a^3 (a+b) d \left (a+b-b \text{sech}^2(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 4.09531, size = 269, normalized size = 1.37 $-\frac{\frac{8 a^2 b^2 \cosh (c+d x)}{(a+b) ((a+b) \cosh (2 (c+d x))+a-b)^2}+\frac{i \sqrt{b} \left (15 a^2+40 a b+24 b^2\right ) \tan ^{-1}\left (\frac{-\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )}{(a+b)^{3/2}}+\frac{i \sqrt{b} \left (15 a^2+40 a b+24 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )}{(a+b)^{3/2}}+\frac{2 a b (9 a+8 b) \cosh (c+d x)}{(a+b) ((a+b) \cosh (2 (c+d x))+a-b)}+4 (a+6 b) \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+a \text{csch}^2\left (\frac{1}{2} (c+d x)\right )+a \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{8 a^4 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((I*Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b
)^(3/2) + (I*Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]
)/(a + b)^(3/2) + (8*a^2*b^2*Cosh[c + d*x])/((a + b)*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (2*a*b*(9*a + 8*
b)*Cosh[c + d*x])/((a + b)*(a - b + (a + b)*Cosh[2*(c + d*x)])) + a*Csch[(c + d*x)/2]^2 + 4*(a + 6*b)*Log[Tanh
[(c + d*x)/2]] + a*Sech[(c + d*x)/2]^2)/(8*a^4*d)

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Maple [B]  time = 0.118, size = 1083, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/d*tanh(1/2*d*x+1/2*c)^2/a^3-9/4/d*b/a/(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/(a+b)*tanh(1/2*d*x+1/2*c)^6-8/d*b^2/a^2/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tan
h(1/2*d*x+1/2*c)^2*b+a)^2/(a+b)*tanh(1/2*d*x+1/2*c)^6-6/d*b^3/a^3/(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/(a+b)*tanh(1/2*d*x+1/2*c)^6-27/4/d*b/a/(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/(a+b)*tanh(1/2*d*x+1/2*c)^4-51/2/d*b^2/a^2/(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/(a+b)*tanh(1/2*d*x+1/2*c)^4-38/d*b^3/a^3/(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/(a+b)*tanh(1/2*d*x+1/2*c)^4-20/d*
b^4/a^4/(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/(a+b)*tanh(1/2*d*x+1
/2*c)^4-27/4/d*b/a/(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/(a+b)*tan
h(1/2*d*x+1/2*c)^2-20/d*b^2/a^2/(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/(a+b)*tanh(1/2*d*x+1/2*c)^2-14/d*b^3/a^3/(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/(a+b)*tanh(1/2*d*x+1/2*c)^2-9/4/d*b/a/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*ta
nh(1/2*d*x+1/2*c)^2*b+a)^2/(a+b)-5/2/d*b^2/a^2/(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/(a+b)-15/8/d*b/a^2/(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))-5/d*b^2/a^3/(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))-3/d*b^3/a^4/(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))-1/8/d/a
^3/tanh(1/2*d*x+1/2*c)^2-1/2/d/a^3*ln(tanh(1/2*d*x+1/2*c))-3/d/a^4*ln(tanh(1/2*d*x+1/2*c))*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \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)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-1/4*((4*a^3*e^(11*c) + 21*a^2*b*e^(11*c) + 29*a*b^2*e^(11*c) + 12*b^3*e^(11*c))*e^(11*d*x) + (20*a^3*e^(9*c)
+ 37*a^2*b*e^(9*c) - 15*a*b^2*e^(9*c) - 36*b^3*e^(9*c))*e^(9*d*x) + 2*(20*a^3*e^(7*c) + 3*a^2*b*e^(7*c) - 7*a*
b^2*e^(7*c) + 12*b^3*e^(7*c))*e^(7*d*x) + 2*(20*a^3*e^(5*c) + 3*a^2*b*e^(5*c) - 7*a*b^2*e^(5*c) + 12*b^3*e^(5*
c))*e^(5*d*x) + (20*a^3*e^(3*c) + 37*a^2*b*e^(3*c) - 15*a*b^2*e^(3*c) - 36*b^3*e^(3*c))*e^(3*d*x) + (4*a^3*e^c
+ 21*a^2*b*e^c + 29*a*b^2*e^c + 12*b^3*e^c)*e^(d*x))/(a^6*d + 3*a^5*b*d + 3*a^4*b^2*d + a^3*b^3*d + (a^6*d*e^
(12*c) + 3*a^5*b*d*e^(12*c) + 3*a^4*b^2*d*e^(12*c) + a^3*b^3*d*e^(12*c))*e^(12*d*x) + 2*(a^6*d*e^(10*c) - a^5*
b*d*e^(10*c) - 5*a^4*b^2*d*e^(10*c) - 3*a^3*b^3*d*e^(10*c))*e^(10*d*x) - (a^6*d*e^(8*c) + 3*a^5*b*d*e^(8*c) -
13*a^4*b^2*d*e^(8*c) - 15*a^3*b^3*d*e^(8*c))*e^(8*d*x) - 4*(a^6*d*e^(6*c) - a^5*b*d*e^(6*c) + 3*a^4*b^2*d*e^(6
*c) + 5*a^3*b^3*d*e^(6*c))*e^(6*d*x) - (a^6*d*e^(4*c) + 3*a^5*b*d*e^(4*c) - 13*a^4*b^2*d*e^(4*c) - 15*a^3*b^3*
d*e^(4*c))*e^(4*d*x) + 2*(a^6*d*e^(2*c) - a^5*b*d*e^(2*c) - 5*a^4*b^2*d*e^(2*c) - 3*a^3*b^3*d*e^(2*c))*e^(2*d*
x)) + 1/2*(a + 6*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^4*d) - 1/2*(a + 6*b)*log((e^(d*x + c) - 1)*e^(-c))/(a^4*d
) + 8*integrate(1/32*((15*a^2*b*e^(3*c) + 40*a*b^2*e^(3*c) + 24*b^3*e^(3*c))*e^(3*d*x) - (15*a^2*b*e^c + 40*a*
b^2*e^c + 24*b^3*e^c)*e^(d*x))/(a^6 + 2*a^5*b + a^4*b^2 + (a^6*e^(4*c) + 2*a^5*b*e^(4*c) + a^4*b^2*e^(4*c))*e^
(4*d*x) + 2*(a^6*e^(2*c) - a^4*b^2*e^(2*c))*e^(2*d*x)), x)

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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(csch(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**3/(a+b*tanh(d*x+c)**2)**3,x)

[Out]

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

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Giac [C]  time = 2.44519, size = 8479, normalized size = 43.26 \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)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/32*(2*(3*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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)))) - (15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^
(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c)
+ 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^
4*e^(4*c) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^
7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c
) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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))))*si
nh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(
4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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 - (15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) +
88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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)))) + (15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*
e^(4*c) + 24*a^4*b^5*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^6 + 2*a^5*b + a^4*b^2)/(a^6*e^(4*c) + 2*a^5*b*e^(4*c) + a^4*b^2*e^(4*c)))^
(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) + e^(d*x))/(((a^6 + 2*a^5*b + a^4*b^2)/(a^6*e^(4*c) + 2*a^5*b*e^(4*c)
+ a^4*b^2*e^(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(2*(a^5*e^(2*c) + a^4*b*e^(2*c))^2*a*b + (a^6*e^
(2*c) - a^4*b^2*e^(2*c))*sqrt(-a*b)*abs(-a^5*e^(2*c) - a^4*b*e^(2*c))) + 2*(3*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e
^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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)))) - (15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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*si
n(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(15*a
^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*e^(4*c))*cosh(1/2*im
ag_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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^
5*b^4*e^(4*c) + 24*a^4*b^5*e^(4*c))*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c
) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^
(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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 + (15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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 - (15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*e^(4*c))*co
sh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) + (15*a^8
*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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^6 + 2*
a^5*b + a^4*b^2)/(a^6*e^(4*c) + 2*a^5*b*e^(4*c) + a^4*b^2*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b))) -
e^(d*x))/(((a^6 + 2*a^5*b + a^4*b^2)/(a^6*e^(4*c) + 2*a^5*b*e^(4*c) + a^4*b^2*e^(4*c)))^(1/4)*sin(1/2*arccos(-
(a - b)/(a + b)))))/(2*(a^5*e^(2*c) + a^4*b*e^(2*c))^2*a*b + (a^6*e^(2*c) - a^4*b^2*e^(2*c))*sqrt(-a*b)*abs(-a
^5*e^(2*c) - a^4*b*e^(2*c))) + ((15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4
*c) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) +
24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c
) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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*s
in(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*(1
5*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88
*a^5*b^4*e^(4*c) + 24*a^4*b^5*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*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))))^2*sinh(1/2*imag_part(arccos
(-a/(a + b) + b/(a + b))))^2 - (15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*
c) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) +
24*a^4*b^5*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 - (15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) +
119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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)))) + (15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^
3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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^6 + 2*a^5*b + a^4*b^2)/(a^6*e^(4*c) + 2*a^5*b*e^(4*c)
+ a^4*b^2*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a^6 + 2*a^5*b + a^4*b^2)/(a^6*e^(
4*c) + 2*a^5*b*e^(4*c) + a^4*b^2*e^(4*c))) + e^(2*d*x))/(2*(a^5*e^(2*c) + a^4*b*e^(2*c))^2*a*b + (a^6*e^(2*c)
- a^4*b^2*e^(2*c))*sqrt(-a*b)*abs(-a^5*e^(2*c) - a^4*b*e^(2*c))) - ((15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 1
19*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^
6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*e^(4*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*(
15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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*i
mag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 8
8*a^5*b^4*e^(4*c) + 24*a^4*b^5*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(arc
cos(-a/(a + b) + b/(a + b)))) + 3*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^
(4*c) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^
2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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 - (15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 11
9*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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*(15*a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6
*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*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 - (15*
a^8*b*e^(4*c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*e^(4*c))*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)))) + (15*a^8*b*e^(4*
c) + 70*a^7*b^2*e^(4*c) + 119*a^6*b^3*e^(4*c) + 88*a^5*b^4*e^(4*c) + 24*a^4*b^5*e^(4*c))*cos(1/2*real_part(arc
cos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))))*log(-2*((a^6 + 2*a^5*b + a^
4*b^2)/(a^6*e^(4*c) + 2*a^5*b*e^(4*c) + a^4*b^2*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sq
rt((a^6 + 2*a^5*b + a^4*b^2)/(a^6*e^(4*c) + 2*a^5*b*e^(4*c) + a^4*b^2*e^(4*c))) + e^(2*d*x))/(2*(a^5*e^(2*c) +
a^4*b*e^(2*c))^2*a*b + (a^6*e^(2*c) - a^4*b^2*e^(2*c))*sqrt(-a*b)*abs(-a^5*e^(2*c) - a^4*b*e^(2*c))) + 16*(a*
e^c + 6*b*e^c)*e^(-c)*log(e^(d*x + c) + 1)/a^4 - 16*(a*e^c + 6*b*e^c)*e^(-c)*log(abs(-e^(d*x + c) + 1))/a^4 -
8*(4*a^3*e^(11*d*x + 11*c) + 21*a^2*b*e^(11*d*x + 11*c) + 29*a*b^2*e^(11*d*x + 11*c) + 12*b^3*e^(11*d*x + 11*c
) + 20*a^3*e^(9*d*x + 9*c) + 37*a^2*b*e^(9*d*x + 9*c) - 15*a*b^2*e^(9*d*x + 9*c) - 36*b^3*e^(9*d*x + 9*c) + 40
*a^3*e^(7*d*x + 7*c) + 6*a^2*b*e^(7*d*x + 7*c) - 14*a*b^2*e^(7*d*x + 7*c) + 24*b^3*e^(7*d*x + 7*c) + 40*a^3*e^
(5*d*x + 5*c) + 6*a^2*b*e^(5*d*x + 5*c) - 14*a*b^2*e^(5*d*x + 5*c) + 24*b^3*e^(5*d*x + 5*c) + 20*a^3*e^(3*d*x
+ 3*c) + 37*a^2*b*e^(3*d*x + 3*c) - 15*a*b^2*e^(3*d*x + 3*c) - 36*b^3*e^(3*d*x + 3*c) + 4*a^3*e^(d*x + c) + 21
*a^2*b*e^(d*x + c) + 29*a*b^2*e^(d*x + c) + 12*b^3*e^(d*x + c))/((a^4 + a^3*b)*(a*e^(6*d*x + 6*c) + b*e^(6*d*x
+ 6*c) + a*e^(4*d*x + 4*c) - 3*b*e^(4*d*x + 4*c) - a*e^(2*d*x + 2*c) + 3*b*e^(2*d*x + 2*c) - a - b)^2))/d