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

Optimal. Leaf size=178 \[ -\frac{\left (8 a^2+27 a b+15 b^2\right ) \coth (c+d x)}{8 a^3 d (a+b)^2}-\frac{b^{3/2} \left (35 a^2+42 a b+15 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{7/2} d (a+b)^3}+\frac{b (9 a+5 b) \coth (c+d x)}{8 a^2 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}+\frac{b \coth (c+d x)}{4 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{x}{(a+b)^3} \]

[Out]

x/(a + b)^3 - (b^(3/2)*(35*a^2 + 42*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(7/2)*(a + b)^
3*d) - ((8*a^2 + 27*a*b + 15*b^2)*Coth[c + d*x])/(8*a^3*(a + b)^2*d) + (b*Coth[c + d*x])/(4*a*(a + b)*d*(a + b
*Tanh[c + d*x]^2)^2) + (b*(9*a + 5*b)*Coth[c + d*x])/(8*a^2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))

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Rubi [A]  time = 0.290858, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3670, 472, 579, 583, 522, 206, 205} \[ -\frac{\left (8 a^2+27 a b+15 b^2\right ) \coth (c+d x)}{8 a^3 d (a+b)^2}-\frac{b^{3/2} \left (35 a^2+42 a b+15 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{7/2} d (a+b)^3}+\frac{b (9 a+5 b) \coth (c+d x)}{8 a^2 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}+\frac{b \coth (c+d x)}{4 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{x}{(a+b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(a + b)^3 - (b^(3/2)*(35*a^2 + 42*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(7/2)*(a + b)^
3*d) - ((8*a^2 + 27*a*b + 15*b^2)*Coth[c + d*x])/(8*a^3*(a + b)^2*d) + (b*Coth[c + d*x])/(4*a*(a + b)*d*(a + b
*Tanh[c + d*x]^2)^2) + (b*(9*a + 5*b)*Coth[c + d*x])/(8*a^2*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))

Rule 3670

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rule 472

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

Rule 579

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -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 206

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\coth ^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b \coth (c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-4 a-5 b+5 b x^2}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a (a+b) d}\\ &=\frac{b \coth (c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{b (9 a+5 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{8 a^2+27 a b+15 b^2-3 b (9 a+5 b) x^2}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 (a+b)^2 d}\\ &=-\frac{\left (8 a^2+27 a b+15 b^2\right ) \coth (c+d x)}{8 a^3 (a+b)^2 d}+\frac{b \coth (c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{b (9 a+5 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-8 a^3+8 a^2 b+27 a b^2+15 b^3-b \left (8 a^2+27 a b+15 b^2\right ) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^3 (a+b)^2 d}\\ &=-\frac{\left (8 a^2+27 a b+15 b^2\right ) \coth (c+d x)}{8 a^3 (a+b)^2 d}+\frac{b \coth (c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{b (9 a+5 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{(a+b)^3 d}-\frac{\left (b^2 \left (35 a^2+42 a b+15 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^3 (a+b)^3 d}\\ &=\frac{x}{(a+b)^3}-\frac{b^{3/2} \left (35 a^2+42 a b+15 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{7/2} (a+b)^3 d}-\frac{\left (8 a^2+27 a b+15 b^2\right ) \coth (c+d x)}{8 a^3 (a+b)^2 d}+\frac{b \coth (c+d x)}{4 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{b (9 a+5 b) \coth (c+d x)}{8 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 5.76014, size = 166, normalized size = 0.93 \[ -\frac{\frac{b^{3/2} \left (35 a^2+42 a b+15 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{7/2} (a+b)^3}+\frac{4 b^3 \sinh (2 (c+d x))}{a^2 (a+b)^2 ((a+b) \cosh (2 (c+d x))+a-b)^2}+\frac{b^2 (13 a+7 b) \sinh (2 (c+d x))}{a^3 (a+b)^2 ((a+b) \cosh (2 (c+d x))+a-b)}+\frac{8 \coth (c+d x)}{a^3}-\frac{8 (c+d x)}{(a+b)^3}}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-8*(c + d*x))/(a + b)^3 + (b^(3/2)*(35*a^2 + 42*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(
7/2)*(a + b)^3) + (8*Coth[c + d*x])/a^3 + (4*b^3*Sinh[2*(c + d*x)])/(a^2*(a + b)^2*(a - b + (a + b)*Cosh[2*(c
+ d*x)])^2) + (b^2*(13*a + 7*b)*Sinh[2*(c + d*x)])/(a^3*(a + b)^2*(a - b + (a + b)*Cosh[2*(c + d*x)])))/(8*d)

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Maple [B]  time = 0.125, size = 2045, normalized size = 11.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35/8/d*b^2/(a+b)^3/a/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-
2*b)*a)^(1/2))-21/4/d*b^3/(a+b)^3/a^2/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b
*(a+b))^(1/2)-a-2*b)*a)^(1/2))+35/8/d*b^2/(a+b)^3/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(
a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+35/8/d*b^2/(a+b)^3/a/((2*(b*(a+b))^(1/2)+a+2*b)*a)^
(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))+21/4/d*b^3/(a+b)^3/a^2/((2*(b*(a+b))^(
1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))+35/8/d*b^2/(a+b)^3/(b*(
a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/
2))+15/8/d*b^5/(a+b)^3/a^3/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/(
(2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+77/8/d*b^3/(a+b)^3/a/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*a
rctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+57/8/d*b^4/(a+b)^3/a^2/(b*(a+b))^(1/2)/((2*(
b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+77/8/d*b^3/(
a+b)^3/a/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+
a+2*b)*a)^(1/2))+57/8/d*b^4/(a+b)^3/a^2/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*
d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))+1/d/(a+b)^3*ln(tanh(1/2*d*x+1/2*c)+1)-1/d/(a+b)^3*ln(tanh(1/2*
d*x+1/2*c)-1)-7/d*b^5/(a+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*tanh(1/2*d*x+1/2*c)^3-15/8/d*b^4/(a+b)^3/a^3/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+
1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+15/8/d*b^4/(a+b)^3/a^3/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(
a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))-9/4/d*b^4/(a+b)^3/a^2/(tanh(1/2*d*x+1/2*c)^4*a+2*ta
nh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2*tanh(1/2*d*x+1/2*c)^7-39/4/d*b^2/(a+b)^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*tanh(1/2*d*x+1/2*c)^5-39/4/d*b^2/(a+b)^3/(ta
nh(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*tanh(1/2*d*x+1/2*c)^3-13/4/d*b^
2/(a+b)^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*tanh(1/2*d*x+1/2*c
)+15/8/d*b^5/(a+b)^3/a^3/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*
(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))-1/2/d/a^3*tanh(1/2*d*x+1/2*c)-1/2/d/a^3/tanh(1/2*d*x+1/2*c)-11/2/d*b^3/(a+b)^
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*tanh(1/2*d*x+1/2*c)^7-55
/2/d*b^3/(a+b)^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*tanh(1/2*
d*x+1/2*c)^5-99/4/d*b^4/(a+b)^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^2*tanh(1/2*d*x+1/2*c)^5-55/2/d*b^3/(a+b)^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*tanh(1/2*d*x+1/2*c)^3-99/4/d*b^4/(a+b)^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^2*tanh(1/2*d*x+1/2*c)^3-11/2/d*b^3/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+2*ta
nh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/a*tanh(1/2*d*x+1/2*c)-13/4/d*b^2/(a+b)^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*tanh(1/2*d*x+1/2*c)^7-9/4/d*b^4/(a+b)^3/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*tanh(1/2*d*x+1/2*c)-7/d*b^5/
(a+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*tanh(1/2*d*x+1/2
*c)^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^2/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.82515, size = 27774, normalized size = 156.03 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(16*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^10 + 160*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)*sin
h(d*x + c)^9 + 16*(a^5 + 2*a^4*b + a^3*b^2)*d*x*sinh(d*x + c)^10 - 4*(8*a^5 + 40*a^4*b + 67*a^3*b^2 + 77*a^2*b
^3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^8 - 4*(8*a^5 + 40*a^4*b + 67*a^3*b
^2 + 77*a^2*b^3 + 57*a*b^4 + 15*b^5 - 180*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^2 - 4*(3*a^5 - 2*a^4*b -
5*a^3*b^2)*d*x)*sinh(d*x + c)^8 + 32*(60*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^3 - (8*a^5 + 40*a^4*b +
67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c
)^7 - 8*(16*a^5 + 48*a^4*b + 19*a^3*b^2 - 28*a^2*b^3 - 69*a*b^4 - 30*b^5 - 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*
cosh(d*x + c)^6 + 8*(420*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^4 - 16*a^5 - 48*a^4*b - 19*a^3*b^2 + 28*a
^2*b^3 + 69*a*b^4 + 30*b^5 + 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x - 14*(8*a^5 + 40*a^4*b + 67*a^3*b^2 + 77*a^2*b^
3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 16*(252*(a^5 +
2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^5 - 14*(8*a^5 + 40*a^4*b + 67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 + 15*b^5 -
4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^3 - 3*(16*a^5 + 48*a^4*b + 19*a^3*b^2 - 28*a^2*b^3 - 69*a*
b^4 - 30*b^5 - 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 32*a^5 - 160*a^4*b - 372*a^
3*b^2 - 452*a^2*b^3 - 268*a*b^4 - 60*b^5 - 8*(24*a^5 + 56*a^4*b + 48*a^3*b^2 + 33*a^2*b^3 + 86*a*b^4 + 45*b^5
+ 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c)^4 + 8*(420*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^6 -
24*a^5 - 56*a^4*b - 48*a^3*b^2 - 33*a^2*b^3 - 86*a*b^4 - 45*b^5 - 35*(8*a^5 + 40*a^4*b + 67*a^3*b^2 + 77*a^2*b
^3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^4 - 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*
d*x - 15*(16*a^5 + 48*a^4*b + 19*a^3*b^2 - 28*a^2*b^3 - 69*a*b^4 - 30*b^5 - 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)
*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 32*(60*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^7 - 7*(8*a^5 + 40*a^4*b
+ 67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^5 - 5*(16*
a^5 + 48*a^4*b + 19*a^3*b^2 - 28*a^2*b^3 - 69*a*b^4 - 30*b^5 - 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c
)^3 - (24*a^5 + 56*a^4*b + 48*a^3*b^2 + 33*a^2*b^3 + 86*a*b^4 + 45*b^5 + 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*co
sh(d*x + c))*sinh(d*x + c)^3 - 16*(a^5 + 2*a^4*b + a^3*b^2)*d*x - 8*(16*a^5 + 48*a^4*b + 45*a^3*b^2 - 36*a^2*b
^3 - 79*a*b^4 - 30*b^5 + 2*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^2 + 8*(90*(a^5 + 2*a^4*b + a^3*b^2
)*d*x*cosh(d*x + c)^8 - 14*(8*a^5 + 40*a^4*b + 67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*
b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^6 - 16*a^5 - 48*a^4*b - 45*a^3*b^2 + 36*a^2*b^3 + 79*a*b^4 + 30*b^5 - 15*(16
*a^5 + 48*a^4*b + 19*a^3*b^2 - 28*a^2*b^3 - 69*a*b^4 - 30*b^5 - 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x +
c)^4 - 2*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x - 6*(24*a^5 + 56*a^4*b + 48*a^3*b^2 + 33*a^2*b^3 + 86*a*b^4 + 45*b^
5 + 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((35*a^4*b + 112*a^3*b^2 + 134*a^2*b
^3 + 72*a*b^4 + 15*b^5)*cosh(d*x + c)^10 + 10*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(
d*x + c)*sinh(d*x + c)^9 + (35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*sinh(d*x + c)^10 + (105*
a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x + c)^8 + (105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3
- 240*a*b^4 - 75*b^5 + 45*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*x + c)^2)*sinh(d*
x + c)^8 + 8*(15*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*x + c)^3 + (105*a^4*b + 56*
a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(35*a^4*b - 28*a^3*b^2 + 106*a^
2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c)^6 + 2*(35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5 + 10
5*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*x + c)^4 + 14*(105*a^4*b + 56*a^3*b^2 - 21
4*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3
+ 72*a*b^4 + 15*b^5)*cosh(d*x + c)^5 + 14*(105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*
x + c)^3 + 3*(35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 - 35*a^
4*b - 112*a^3*b^2 - 134*a^2*b^3 - 72*a*b^4 - 15*b^5 - 2*(35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*
b^5)*cosh(d*x + c)^4 + 2*(105*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*x + c)^6 - 35*
a^4*b + 28*a^3*b^2 - 106*a^2*b^3 - 180*a*b^4 - 75*b^5 + 35*(105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 -
75*b^5)*cosh(d*x + c)^4 + 15*(35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c)^2)*sinh
(d*x + c)^4 + 8*(15*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*x + c)^7 + 7*(105*a^4*b
+ 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x + c)^5 + 5*(35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 + 18
0*a*b^4 + 75*b^5)*cosh(d*x + c)^3 - (35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c))*
sinh(d*x + c)^3 - (105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x + c)^2 + (45*(35*a^4*b
+ 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*x + c)^8 + 28*(105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 -
240*a*b^4 - 75*b^5)*cosh(d*x + c)^6 - 105*a^4*b - 56*a^3*b^2 + 214*a^2*b^3 + 240*a*b^4 + 75*b^5 + 30*(35*a^4*b
- 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c)^4 - 12*(35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 +
180*a*b^4 + 75*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 +
15*b^5)*cosh(d*x + c)^9 + 4*(105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x + c)^7 + 6*(
35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c)^5 - 4*(35*a^4*b - 28*a^3*b^2 + 106*a^2
*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c)^3 - (105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(
d*x + c))*sinh(d*x + c))*sqrt(-b/a)*log(((a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x
+ c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b
+ b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 - 6*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c
)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c) - 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*
sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 - a*b)*sqrt(-b/a))/((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)) + 16*(
10*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^9 - 2*(8*a^5 + 40*a^4*b + 67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 +
15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^7 - 3*(16*a^5 + 48*a^4*b + 19*a^3*b^2 - 28*a^2*b^3
- 69*a*b^4 - 30*b^5 - 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c)^5 - 2*(24*a^5 + 56*a^4*b + 48*a^3*b^2
+ 33*a^2*b^3 + 86*a*b^4 + 45*b^5 + 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c)^3 - (16*a^5 + 48*a^4*b + 4
5*a^3*b^2 - 36*a^2*b^3 - 79*a*b^4 - 30*b^5 + 2*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)
)/((a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^10 + 10*(a^8 + 5*a^7*b + 10
*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^8 + 5*a^7*b + 10*a^6*b^2 + 1
0*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*sinh(d*x + c)^10 + (3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 -
5*a^3*b^5)*d*cosh(d*x + c)^8 + (45*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x
+ c)^2 + (3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d)*sinh(d*x + c)^8 + 2*(a^8 + a^
7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^6 + 8*(15*(a^8 + 5*a^7*b + 10*a^6*b^2 +
10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^3 + (3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4
- 5*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(105*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 +
a^3*b^5)*d*cosh(d*x + c)^4 + 14*(3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x
+ c)^2 + (a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d)*sinh(d*x + c)^6 - 2*(a^8 + a^7*b
+ 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^4 + 4*(63*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*
a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^5 + 14*(3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 -
5*a^3*b^5)*d*cosh(d*x + c)^3 + 3*(a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x +
c))*sinh(d*x + c)^5 + 2*(105*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^
6 + 35*(3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c)^4 + 15*(a^8 + a^7*b
+ 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^2 - (a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3
+ 13*a^4*b^4 + 5*a^3*b^5)*d)*sinh(d*x + c)^4 - (3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*
b^5)*d*cosh(d*x + c)^2 + 8*(15*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)
^7 + 7*(3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c)^5 + 5*(a^8 + a^7*b
+ 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^3 - (a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 +
13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (45*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a
^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^8 + 28*(3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d
*cosh(d*x + c)^6 + 30*(a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^4 - 12*(
a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^2 - (3*a^8 + 7*a^7*b - 2*a^6*b^
2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d)*sinh(d*x + c)^2 - (a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4
*b^4 + a^3*b^5)*d + 2*(5*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^9 + 4
*(3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c)^7 + 6*(a^8 + a^7*b + 2*a^
6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^5 - 4*(a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*
a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^3 - (3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d
*cosh(d*x + c))*sinh(d*x + c)), 1/8*(8*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^10 + 80*(a^5 + 2*a^4*b + a^
3*b^2)*d*x*cosh(d*x + c)*sinh(d*x + c)^9 + 8*(a^5 + 2*a^4*b + a^3*b^2)*d*x*sinh(d*x + c)^10 - 2*(8*a^5 + 40*a^
4*b + 67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^8 - 2*(
8*a^5 + 40*a^4*b + 67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 + 15*b^5 - 180*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x +
c)^2 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*sinh(d*x + c)^8 + 16*(60*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x +
c)^3 - (8*a^5 + 40*a^4*b + 67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*
cosh(d*x + c))*sinh(d*x + c)^7 - 4*(16*a^5 + 48*a^4*b + 19*a^3*b^2 - 28*a^2*b^3 - 69*a*b^4 - 30*b^5 - 4*(a^5 -
2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c)^6 + 4*(420*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^4 - 16*a^5 - 4
8*a^4*b - 19*a^3*b^2 + 28*a^2*b^3 + 69*a*b^4 + 30*b^5 + 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x - 14*(8*a^5 + 40*a^4
*b + 67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^2)*sinh(
d*x + c)^6 + 8*(252*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^5 - 14*(8*a^5 + 40*a^4*b + 67*a^3*b^2 + 77*a^2
*b^3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^3 - 3*(16*a^5 + 48*a^4*b + 19*a^
3*b^2 - 28*a^2*b^3 - 69*a*b^4 - 30*b^5 - 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 - 1
6*a^5 - 80*a^4*b - 186*a^3*b^2 - 226*a^2*b^3 - 134*a*b^4 - 30*b^5 - 4*(24*a^5 + 56*a^4*b + 48*a^3*b^2 + 33*a^2
*b^3 + 86*a*b^4 + 45*b^5 + 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c)^4 + 4*(420*(a^5 + 2*a^4*b + a^3*b^
2)*d*x*cosh(d*x + c)^6 - 24*a^5 - 56*a^4*b - 48*a^3*b^2 - 33*a^2*b^3 - 86*a*b^4 - 45*b^5 - 35*(8*a^5 + 40*a^4*
b + 67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^4 - 4*(a^
5 - 2*a^4*b + 5*a^3*b^2)*d*x - 15*(16*a^5 + 48*a^4*b + 19*a^3*b^2 - 28*a^2*b^3 - 69*a*b^4 - 30*b^5 - 4*(a^5 -
2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(60*(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c
)^7 - 7*(8*a^5 + 40*a^4*b + 67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)
*cosh(d*x + c)^5 - 5*(16*a^5 + 48*a^4*b + 19*a^3*b^2 - 28*a^2*b^3 - 69*a*b^4 - 30*b^5 - 4*(a^5 - 2*a^4*b + 5*a
^3*b^2)*d*x)*cosh(d*x + c)^3 - (24*a^5 + 56*a^4*b + 48*a^3*b^2 + 33*a^2*b^3 + 86*a*b^4 + 45*b^5 + 4*(a^5 - 2*a
^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 8*(a^5 + 2*a^4*b + a^3*b^2)*d*x - 4*(16*a^5 + 48*a^4*b
+ 45*a^3*b^2 - 36*a^2*b^3 - 79*a*b^4 - 30*b^5 + 2*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^2 + 4*(90*
(a^5 + 2*a^4*b + a^3*b^2)*d*x*cosh(d*x + c)^8 - 14*(8*a^5 + 40*a^4*b + 67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 + 15
*b^5 - 4*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c)^6 - 16*a^5 - 48*a^4*b - 45*a^3*b^2 + 36*a^2*b^3 + 79
*a*b^4 + 30*b^5 - 15*(16*a^5 + 48*a^4*b + 19*a^3*b^2 - 28*a^2*b^3 - 69*a*b^4 - 30*b^5 - 4*(a^5 - 2*a^4*b + 5*a
^3*b^2)*d*x)*cosh(d*x + c)^4 - 2*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x - 6*(24*a^5 + 56*a^4*b + 48*a^3*b^2 + 33*a^
2*b^3 + 86*a*b^4 + 45*b^5 + 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((35*a^4*b +
112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*x + c)^10 + 10*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 +
72*a*b^4 + 15*b^5)*cosh(d*x + c)*sinh(d*x + c)^9 + (35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*
sinh(d*x + c)^10 + (105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x + c)^8 + (105*a^4*b +
56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5 + 45*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*
cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(15*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*x +
c)^3 + (105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(35*a^4
*b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c)^6 + 2*(35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 +
180*a*b^4 + 75*b^5 + 105*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*x + c)^4 + 14*(105
*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(35*a^4*b + 1
12*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*x + c)^5 + 14*(105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240
*a*b^4 - 75*b^5)*cosh(d*x + c)^3 + 3*(35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c))
*sinh(d*x + c)^5 - 35*a^4*b - 112*a^3*b^2 - 134*a^2*b^3 - 72*a*b^4 - 15*b^5 - 2*(35*a^4*b - 28*a^3*b^2 + 106*a
^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c)^4 + 2*(105*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^
5)*cosh(d*x + c)^6 - 35*a^4*b + 28*a^3*b^2 - 106*a^2*b^3 - 180*a*b^4 - 75*b^5 + 35*(105*a^4*b + 56*a^3*b^2 - 2
14*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x + c)^4 + 15*(35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^
5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(15*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*
x + c)^7 + 7*(105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x + c)^5 + 5*(35*a^4*b - 28*a^
3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c)^3 - (35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 +
75*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 - (105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x
+ c)^2 + (45*(35*a^4*b + 112*a^3*b^2 + 134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*x + c)^8 + 28*(105*a^4*b + 56*
a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^5)*cosh(d*x + c)^6 - 105*a^4*b - 56*a^3*b^2 + 214*a^2*b^3 + 240*a*b^4
+ 75*b^5 + 30*(35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c)^4 - 12*(35*a^4*b - 28*
a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(5*(35*a^4*b + 112*a^3*b^2 +
134*a^2*b^3 + 72*a*b^4 + 15*b^5)*cosh(d*x + c)^9 + 4*(105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 240*a*b^4 - 75*b^
5)*cosh(d*x + c)^7 + 6*(35*a^4*b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c)^5 - 4*(35*a^4*
b - 28*a^3*b^2 + 106*a^2*b^3 + 180*a*b^4 + 75*b^5)*cosh(d*x + c)^3 - (105*a^4*b + 56*a^3*b^2 - 214*a^2*b^3 - 2
40*a*b^4 - 75*b^5)*cosh(d*x + c))*sinh(d*x + c))*sqrt(b/a)*arctan(1/2*((a + b)*cosh(d*x + c)^2 + 2*(a + b)*cos
h(d*x + c)*sinh(d*x + c) + (a + b)*sinh(d*x + c)^2 + a - b)*sqrt(b/a)/b) + 8*(10*(a^5 + 2*a^4*b + a^3*b^2)*d*x
*cosh(d*x + c)^9 - 2*(8*a^5 + 40*a^4*b + 67*a^3*b^2 + 77*a^2*b^3 + 57*a*b^4 + 15*b^5 - 4*(3*a^5 - 2*a^4*b - 5*
a^3*b^2)*d*x)*cosh(d*x + c)^7 - 3*(16*a^5 + 48*a^4*b + 19*a^3*b^2 - 28*a^2*b^3 - 69*a*b^4 - 30*b^5 - 4*(a^5 -
2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c)^5 - 2*(24*a^5 + 56*a^4*b + 48*a^3*b^2 + 33*a^2*b^3 + 86*a*b^4 + 45*b^5
+ 4*(a^5 - 2*a^4*b + 5*a^3*b^2)*d*x)*cosh(d*x + c)^3 - (16*a^5 + 48*a^4*b + 45*a^3*b^2 - 36*a^2*b^3 - 79*a*b^
4 - 30*b^5 + 2*(3*a^5 - 2*a^4*b - 5*a^3*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^8 + 5*a^7*b + 10*a^6*b^2 +
10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^10 + 10*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^
4 + a^3*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^9 + (a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)
*d*sinh(d*x + c)^10 + (3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c)^8 +
(45*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^2 + (3*a^8 + 7*a^7*b - 2*a
^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d)*sinh(d*x + c)^8 + 2*(a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 1
3*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^6 + 8*(15*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^
5)*d*cosh(d*x + c)^3 + (3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c))*si
nh(d*x + c)^7 + 2*(105*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^4 + 14*
(3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c)^2 + (a^8 + a^7*b + 2*a^6*b
^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d)*sinh(d*x + c)^6 - 2*(a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^
4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^4 + 4*(63*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d
*cosh(d*x + c)^5 + 14*(3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c)^3 +
3*(a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(105*(a
^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^6 + 35*(3*a^8 + 7*a^7*b - 2*a^6*
b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c)^4 + 15*(a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a
^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^2 - (a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d)*sin
h(d*x + c)^4 - (3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c)^2 + 8*(15*(
a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^7 + 7*(3*a^8 + 7*a^7*b - 2*a^6*
b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c)^5 + 5*(a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^
4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^3 - (a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(
d*x + c))*sinh(d*x + c)^3 + (45*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c
)^8 + 28*(3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c)^6 + 30*(a^8 + a^7
*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^4 - 12*(a^8 + a^7*b + 2*a^6*b^2 + 10*a^5
*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x + c)^2 - (3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*
a^3*b^5)*d)*sinh(d*x + c)^2 - (a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d + 2*(5*(a^8 +
5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^9 + 4*(3*a^8 + 7*a^7*b - 2*a^6*b^2 -
18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c)^7 + 6*(a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4
+ 5*a^3*b^5)*d*cosh(d*x + c)^5 - 4*(a^8 + a^7*b + 2*a^6*b^2 + 10*a^5*b^3 + 13*a^4*b^4 + 5*a^3*b^5)*d*cosh(d*x
+ c)^3 - (3*a^8 + 7*a^7*b - 2*a^6*b^2 - 18*a^5*b^3 - 17*a^4*b^4 - 5*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c))]

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

[Out]

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

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Giac [B]  time = 1.33385, size = 609, normalized size = 3.42 \begin{align*} -\frac{{\left (35 \, a^{2} b^{2} + 42 \, a b^{3} + 15 \, b^{4}\right )} \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt{a b}}\right )}{8 \,{\left (a^{6} d + 3 \, a^{5} b d + 3 \, a^{4} b^{2} d + a^{3} b^{3} d\right )} \sqrt{a b}} + \frac{d x + c}{a^{3} d + 3 \, a^{2} b d + 3 \, a b^{2} d + b^{3} d} + \frac{13 \, a^{3} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 17 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} - 7 \, b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 39 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 5 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 25 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 21 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 39 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 25 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 35 \, a b^{4} e^{\left (2 \, d x + 2 \, c\right )} - 21 \, b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 13 \, a^{3} b^{2} + 33 \, a^{2} b^{3} + 27 \, a b^{4} + 7 \, b^{5}}{4 \,{\left (a^{6} d + 3 \, a^{5} b d + 3 \, a^{4} b^{2} d + a^{3} b^{3} d\right )}{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2}} - \frac{2}{a^{3} d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^2/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-1/8*(35*a^2*b^2 + 42*a*b^3 + 15*b^4)*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((
a^6*d + 3*a^5*b*d + 3*a^4*b^2*d + a^3*b^3*d)*sqrt(a*b)) + (d*x + c)/(a^3*d + 3*a^2*b*d + 3*a*b^2*d + b^3*d) +
1/4*(13*a^3*b^2*e^(6*d*x + 6*c) + 3*a^2*b^3*e^(6*d*x + 6*c) - 17*a*b^4*e^(6*d*x + 6*c) - 7*b^5*e^(6*d*x + 6*c)
+ 39*a^3*b^2*e^(4*d*x + 4*c) - 5*a^2*b^3*e^(4*d*x + 4*c) + 25*a*b^4*e^(4*d*x + 4*c) + 21*b^5*e^(4*d*x + 4*c)
+ 39*a^3*b^2*e^(2*d*x + 2*c) + 25*a^2*b^3*e^(2*d*x + 2*c) - 35*a*b^4*e^(2*d*x + 2*c) - 21*b^5*e^(2*d*x + 2*c)
+ 13*a^3*b^2 + 33*a^2*b^3 + 27*a*b^4 + 7*b^5)/((a^6*d + 3*a^5*b*d + 3*a^4*b^2*d + a^3*b^3*d)*(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) - 2/(a^3*d*(e^(2*d*x + 2*c) - 1)
)