3.199 \(\int \frac{\coth ^3(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\)
Optimal. Leaf size=171 \[ -\frac{b^2 (3 a+2 b)}{2 a^3 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac{b^2}{4 a^2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{b^2 \left (6 a^2+8 a b+3 b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^4 d (a+b)^3}+\frac{(a-3 b) \log (\tanh (c+d x))}{a^4 d}-\frac{\coth ^2(c+d x)}{2 a^3 d}+\frac{\log (\cosh (c+d x))}{d (a+b)^3} \]
[Out]
-Coth[c + d*x]^2/(2*a^3*d) + Log[Cosh[c + d*x]]/((a + b)^3*d) + ((a - 3*b)*Log[Tanh[c + d*x]])/(a^4*d) + (b^2*
(6*a^2 + 8*a*b + 3*b^2)*Log[a + b*Tanh[c + d*x]^2])/(2*a^4*(a + b)^3*d) - b^2/(4*a^2*(a + b)*d*(a + b*Tanh[c +
d*x]^2)^2) - (b^2*(3*a + 2*b))/(2*a^3*(a + b)^2*d*(a + b*Tanh[c + d*x]^2))
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Rubi [A] time = 0.261835, antiderivative size = 171, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used =
{3670, 446, 88} \[ -\frac{b^2 (3 a+2 b)}{2 a^3 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )}-\frac{b^2}{4 a^2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{b^2 \left (6 a^2+8 a b+3 b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^4 d (a+b)^3}+\frac{(a-3 b) \log (\tanh (c+d x))}{a^4 d}-\frac{\coth ^2(c+d x)}{2 a^3 d}+\frac{\log (\cosh (c+d x))}{d (a+b)^3} \]
Antiderivative was successfully verified.
[In]
Int[Coth[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3,x]
[Out]
-Coth[c + d*x]^2/(2*a^3*d) + Log[Cosh[c + d*x]]/((a + b)^3*d) + ((a - 3*b)*Log[Tanh[c + d*x]])/(a^4*d) + (b^2*
(6*a^2 + 8*a*b + 3*b^2)*Log[a + b*Tanh[c + d*x]^2])/(2*a^4*(a + b)^3*d) - b^2/(4*a^2*(a + b)*d*(a + b*Tanh[c +
d*x]^2)^2) - (b^2*(3*a + 2*b))/(2*a^3*(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 446
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rule 88
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))
Rubi steps
\begin{align*} \int \frac{\coth ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) x^2 (a+b x)^3} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{(a+b)^3 (-1+x)}+\frac{1}{a^3 x^2}+\frac{a-3 b}{a^4 x}+\frac{b^3}{a^2 (a+b) (a+b x)^3}+\frac{b^3 (3 a+2 b)}{a^3 (a+b)^2 (a+b x)^2}+\frac{b^3 \left (6 a^2+8 a b+3 b^2\right )}{a^4 (a+b)^3 (a+b x)}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=-\frac{\coth ^2(c+d x)}{2 a^3 d}+\frac{\log (\cosh (c+d x))}{(a+b)^3 d}+\frac{(a-3 b) \log (\tanh (c+d x))}{a^4 d}+\frac{b^2 \left (6 a^2+8 a b+3 b^2\right ) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^4 (a+b)^3 d}-\frac{b^2}{4 a^2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac{b^2 (3 a+2 b)}{2 a^3 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 1.71898, size = 138, normalized size = 0.81 \[ -\frac{\frac{b^4}{2 a^4 (a+b) \left (a \coth ^2(c+d x)+b\right )^2}-\frac{b^3 (4 a+3 b)}{a^4 (a+b)^2 \left (a \coth ^2(c+d x)+b\right )}-\frac{b^2 \left (6 a^2+8 a b+3 b^2\right ) \log \left (a \coth ^2(c+d x)+b\right )}{a^4 (a+b)^3}+\frac{\coth ^2(c+d x)}{a^3}-\frac{2 \log (\sinh (c+d x))}{(a+b)^3}}{2 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^3,x]
[Out]
-(Coth[c + d*x]^2/a^3 + b^4/(2*a^4*(a + b)*(b + a*Coth[c + d*x]^2)^2) - (b^3*(4*a + 3*b))/(a^4*(a + b)^2*(b +
a*Coth[c + d*x]^2)) - (b^2*(6*a^2 + 8*a*b + 3*b^2)*Log[b + a*Coth[c + d*x]^2])/(a^4*(a + b)^3) - (2*Log[Sinh[c
+ d*x]])/(a + b)^3)/(2*d)
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Maple [B] time = 0.128, size = 1020, normalized size = 6. \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)^3/(a+b*tanh(d*x+c)^2)^3,x)
[Out]
-1/8/d*tanh(1/2*d*x+1/2*c)^2/a^3-1/d/(a+b)^3*ln(tanh(1/2*d*x+1/2*c)+1)+8/d*b^3/(a+b)^3/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*tanh(1/2*d*x+1/2*c)^6+14/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)^6+6/d*b^5/a^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*tanh(1/2*d*x+1/2*c)^
6+16/d*b^3/(a+b)^3/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*tanh(1/
2*d*x+1/2*c)^4+56/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)^4+60/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)^4+20/d*b^6/a^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*tanh(1/2*d*x+1/2*c)^4+8/d*b^3/(a+b)^3/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*tanh(1/2*d*x+1/2*c)^2+14/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)^2+6/d*b^5/a^3/(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)^2+3/d*b^2/(
a+b)^3/a^2*ln(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)+4/d*b^3/(a+b)^3/a
^3*ln(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)+3/2/d*b^4/a^4/(a+b)^3*ln(
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)-1/8/d/a^3/tanh(1/2*d*x+1/2*c)^2
+1/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-1/d/(a+b)^3*ln(tanh(1/2*d*x+1/2*c)-1)
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Maxima [B] time = 1.40136, size = 1040, normalized size = 6.08 \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)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
[Out]
1/2*(6*a^2*b^2 + 8*a*b^3 + 3*b^4)*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^7 + 3
*a^6*b + 3*a^5*b^2 + a^4*b^3)*d) + (d*x + c)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) - 2*((a^5 + 5*a^4*b + 10*a^3*
b^2 + 14*a^2*b^3 + 11*a*b^4 + 3*b^5)*e^(-2*d*x - 2*c) + 2*(2*a^5 + 6*a^4*b + 4*a^3*b^2 - 4*a^2*b^3 - 13*a*b^4
- 6*b^5)*e^(-4*d*x - 4*c) + 2*(3*a^5 + 7*a^4*b + 6*a^3*b^2 + 2*a^2*b^3 + 15*a*b^4 + 9*b^5)*e^(-6*d*x - 6*c) +
2*(2*a^5 + 6*a^4*b + 4*a^3*b^2 - 4*a^2*b^3 - 13*a*b^4 - 6*b^5)*e^(-8*d*x - 8*c) + (a^5 + 5*a^4*b + 10*a^3*b^2
+ 14*a^2*b^3 + 11*a*b^4 + 3*b^5)*e^(-10*d*x - 10*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 + 2*(a^8 + a^7*b - 6*a^6*b^2 - 14*a^5*b^3 - 11*a^4*b^4 - 3*a^3*b^5)*e^(-2*d*x - 2*c) - (a^8 + 5*a^7*b -
6*a^6*b^2 - 38*a^5*b^3 - 43*a^4*b^4 - 15*a^3*b^5)*e^(-4*d*x - 4*c) - 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)*e^(-6*d*x - 6*c) - (a^8 + 5*a^7*b - 6*a^6*b^2 - 38*a^5*b^3 - 43*a^4*b^4 - 15*a^3*b^5
)*e^(-8*d*x - 8*c) + 2*(a^8 + a^7*b - 6*a^6*b^2 - 14*a^5*b^3 - 11*a^4*b^4 - 3*a^3*b^5)*e^(-10*d*x - 10*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)*e^(-12*d*x - 12*c))*d) + (a - 3*b)*log(e^(-d*x -
c) + 1)/(a^4*d) + (a - 3*b)*log(e^(-d*x - c) - 1)/(a^4*d)
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Fricas [B] time = 7.57986, size = 24520, normalized size = 143.39 \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)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
[Out]
-1/2*(2*(a^6 + 2*a^5*b + a^4*b^2)*d*x*cosh(d*x + c)^12 + 24*(a^6 + 2*a^5*b + a^4*b^2)*d*x*cosh(d*x + c)*sinh(d
*x + c)^11 + 2*(a^6 + 2*a^5*b + a^4*b^2)*d*x*sinh(d*x + c)^12 + 4*(a^6 + 5*a^5*b + 10*a^4*b^2 + 14*a^3*b^3 + 1
1*a^2*b^4 + 3*a*b^5 + (a^6 - 2*a^5*b - 3*a^4*b^2)*d*x)*cosh(d*x + c)^10 + 4*(a^6 + 5*a^5*b + 10*a^4*b^2 + 14*a
^3*b^3 + 11*a^2*b^4 + 3*a*b^5 + 33*(a^6 + 2*a^5*b + a^4*b^2)*d*x*cosh(d*x + c)^2 + (a^6 - 2*a^5*b - 3*a^4*b^2)
*d*x)*sinh(d*x + c)^10 + 40*(11*(a^6 + 2*a^5*b + a^4*b^2)*d*x*cosh(d*x + c)^3 + (a^6 + 5*a^5*b + 10*a^4*b^2 +
14*a^3*b^3 + 11*a^2*b^4 + 3*a*b^5 + (a^6 - 2*a^5*b - 3*a^4*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^9 + 2*(8*a^6
+ 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^3 - 52*a^2*b^4 - 24*a*b^5 - (a^6 + 2*a^5*b - 15*a^4*b^2)*d*x)*cosh(d*x + c
)^8 + 2*(495*(a^6 + 2*a^5*b + a^4*b^2)*d*x*cosh(d*x + c)^4 + 8*a^6 + 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^3 - 52*a
^2*b^4 - 24*a*b^5 - (a^6 + 2*a^5*b - 15*a^4*b^2)*d*x + 90*(a^6 + 5*a^5*b + 10*a^4*b^2 + 14*a^3*b^3 + 11*a^2*b^
4 + 3*a*b^5 + (a^6 - 2*a^5*b - 3*a^4*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(99*(a^6 + 2*a^5*b + a^4*
b^2)*d*x*cosh(d*x + c)^5 + 30*(a^6 + 5*a^5*b + 10*a^4*b^2 + 14*a^3*b^3 + 11*a^2*b^4 + 3*a*b^5 + (a^6 - 2*a^5*b
- 3*a^4*b^2)*d*x)*cosh(d*x + c)^3 + (8*a^6 + 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^3 - 52*a^2*b^4 - 24*a*b^5 - (a^
6 + 2*a^5*b - 15*a^4*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^7 + 8*(3*a^6 + 7*a^5*b + 6*a^4*b^2 + 2*a^3*b^3 + 1
5*a^2*b^4 + 9*a*b^5 - (a^6 - 2*a^5*b + 5*a^4*b^2)*d*x)*cosh(d*x + c)^6 + 8*(231*(a^6 + 2*a^5*b + a^4*b^2)*d*x*
cosh(d*x + c)^6 + 3*a^6 + 7*a^5*b + 6*a^4*b^2 + 2*a^3*b^3 + 15*a^2*b^4 + 9*a*b^5 + 105*(a^6 + 5*a^5*b + 10*a^4
*b^2 + 14*a^3*b^3 + 11*a^2*b^4 + 3*a*b^5 + (a^6 - 2*a^5*b - 3*a^4*b^2)*d*x)*cosh(d*x + c)^4 - (a^6 - 2*a^5*b +
5*a^4*b^2)*d*x + 7*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^3 - 52*a^2*b^4 - 24*a*b^5 - (a^6 + 2*a^5*b - 15*
a^4*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 16*(99*(a^6 + 2*a^5*b + a^4*b^2)*d*x*cosh(d*x + c)^7 + 63*(a^
6 + 5*a^5*b + 10*a^4*b^2 + 14*a^3*b^3 + 11*a^2*b^4 + 3*a*b^5 + (a^6 - 2*a^5*b - 3*a^4*b^2)*d*x)*cosh(d*x + c)^
5 + 7*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^3 - 52*a^2*b^4 - 24*a*b^5 - (a^6 + 2*a^5*b - 15*a^4*b^2)*d*x)*
cosh(d*x + c)^3 + 3*(3*a^6 + 7*a^5*b + 6*a^4*b^2 + 2*a^3*b^3 + 15*a^2*b^4 + 9*a*b^5 - (a^6 - 2*a^5*b + 5*a^4*b
^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^3 - 52*a^2*b^4 - 24*a*b^
5 - (a^6 + 2*a^5*b - 15*a^4*b^2)*d*x)*cosh(d*x + c)^4 + 2*(495*(a^6 + 2*a^5*b + a^4*b^2)*d*x*cosh(d*x + c)^8 +
420*(a^6 + 5*a^5*b + 10*a^4*b^2 + 14*a^3*b^3 + 11*a^2*b^4 + 3*a*b^5 + (a^6 - 2*a^5*b - 3*a^4*b^2)*d*x)*cosh(d
*x + c)^6 + 8*a^6 + 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^3 - 52*a^2*b^4 - 24*a*b^5 + 70*(8*a^6 + 24*a^5*b + 16*a^4
*b^2 - 16*a^3*b^3 - 52*a^2*b^4 - 24*a*b^5 - (a^6 + 2*a^5*b - 15*a^4*b^2)*d*x)*cosh(d*x + c)^4 - (a^6 + 2*a^5*b
- 15*a^4*b^2)*d*x + 60*(3*a^6 + 7*a^5*b + 6*a^4*b^2 + 2*a^3*b^3 + 15*a^2*b^4 + 9*a*b^5 - (a^6 - 2*a^5*b + 5*a
^4*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(55*(a^6 + 2*a^5*b + a^4*b^2)*d*x*cosh(d*x + c)^9 + 60*(a^6
+ 5*a^5*b + 10*a^4*b^2 + 14*a^3*b^3 + 11*a^2*b^4 + 3*a*b^5 + (a^6 - 2*a^5*b - 3*a^4*b^2)*d*x)*cosh(d*x + c)^7
+ 14*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^3 - 52*a^2*b^4 - 24*a*b^5 - (a^6 + 2*a^5*b - 15*a^4*b^2)*d*x)*c
osh(d*x + c)^5 + 20*(3*a^6 + 7*a^5*b + 6*a^4*b^2 + 2*a^3*b^3 + 15*a^2*b^4 + 9*a*b^5 - (a^6 - 2*a^5*b + 5*a^4*b
^2)*d*x)*cosh(d*x + c)^3 + (8*a^6 + 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^3 - 52*a^2*b^4 - 24*a*b^5 - (a^6 + 2*a^5*
b - 15*a^4*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(a^6 + 2*a^5*b + a^4*b^2)*d*x + 4*(a^6 + 5*a^5*b + 10*
a^4*b^2 + 14*a^3*b^3 + 11*a^2*b^4 + 3*a*b^5 + (a^6 - 2*a^5*b - 3*a^4*b^2)*d*x)*cosh(d*x + c)^2 + 4*(33*(a^6 +
2*a^5*b + a^4*b^2)*d*x*cosh(d*x + c)^10 + 45*(a^6 + 5*a^5*b + 10*a^4*b^2 + 14*a^3*b^3 + 11*a^2*b^4 + 3*a*b^5 +
(a^6 - 2*a^5*b - 3*a^4*b^2)*d*x)*cosh(d*x + c)^8 + 14*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^3 - 52*a^2*b^
4 - 24*a*b^5 - (a^6 + 2*a^5*b - 15*a^4*b^2)*d*x)*cosh(d*x + c)^6 + a^6 + 5*a^5*b + 10*a^4*b^2 + 14*a^3*b^3 + 1
1*a^2*b^4 + 3*a*b^5 + 30*(3*a^6 + 7*a^5*b + 6*a^4*b^2 + 2*a^3*b^3 + 15*a^2*b^4 + 9*a*b^5 - (a^6 - 2*a^5*b + 5*
a^4*b^2)*d*x)*cosh(d*x + c)^4 + (a^6 - 2*a^5*b - 3*a^4*b^2)*d*x + 3*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^
3 - 52*a^2*b^4 - 24*a*b^5 - (a^6 + 2*a^5*b - 15*a^4*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((6*a^4*b^2 +
20*a^3*b^3 + 25*a^2*b^4 + 14*a*b^5 + 3*b^6)*cosh(d*x + c)^12 + 12*(6*a^4*b^2 + 20*a^3*b^3 + 25*a^2*b^4 + 14*a
*b^5 + 3*b^6)*cosh(d*x + c)*sinh(d*x + c)^11 + (6*a^4*b^2 + 20*a^3*b^3 + 25*a^2*b^4 + 14*a*b^5 + 3*b^6)*sinh(d
*x + c)^12 + 2*(6*a^4*b^2 - 4*a^3*b^3 - 31*a^2*b^4 - 30*a*b^5 - 9*b^6)*cosh(d*x + c)^10 + 2*(6*a^4*b^2 - 4*a^3
*b^3 - 31*a^2*b^4 - 30*a*b^5 - 9*b^6 + 33*(6*a^4*b^2 + 20*a^3*b^3 + 25*a^2*b^4 + 14*a*b^5 + 3*b^6)*cosh(d*x +
c)^2)*sinh(d*x + c)^10 + 20*(11*(6*a^4*b^2 + 20*a^3*b^3 + 25*a^2*b^4 + 14*a*b^5 + 3*b^6)*cosh(d*x + c)^3 + (6*
a^4*b^2 - 4*a^3*b^3 - 31*a^2*b^4 - 30*a*b^5 - 9*b^6)*cosh(d*x + c))*sinh(d*x + c)^9 - (6*a^4*b^2 + 20*a^3*b^3
- 71*a^2*b^4 - 114*a*b^5 - 45*b^6)*cosh(d*x + c)^8 - (6*a^4*b^2 + 20*a^3*b^3 - 71*a^2*b^4 - 114*a*b^5 - 45*b^6
- 495*(6*a^4*b^2 + 20*a^3*b^3 + 25*a^2*b^4 + 14*a*b^5 + 3*b^6)*cosh(d*x + c)^4 - 90*(6*a^4*b^2 - 4*a^3*b^3 -
31*a^2*b^4 - 30*a*b^5 - 9*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(99*(6*a^4*b^2 + 20*a^3*b^3 + 25*a^2*b^4 +
14*a*b^5 + 3*b^6)*cosh(d*x + c)^5 + 30*(6*a^4*b^2 - 4*a^3*b^3 - 31*a^2*b^4 - 30*a*b^5 - 9*b^6)*cosh(d*x + c)^
3 - (6*a^4*b^2 + 20*a^3*b^3 - 71*a^2*b^4 - 114*a*b^5 - 45*b^6)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(6*a^4*b^2 -
4*a^3*b^3 + 17*a^2*b^4 + 34*a*b^5 + 15*b^6)*cosh(d*x + c)^6 + 4*(231*(6*a^4*b^2 + 20*a^3*b^3 + 25*a^2*b^4 + 1
4*a*b^5 + 3*b^6)*cosh(d*x + c)^6 - 6*a^4*b^2 + 4*a^3*b^3 - 17*a^2*b^4 - 34*a*b^5 - 15*b^6 + 105*(6*a^4*b^2 - 4
*a^3*b^3 - 31*a^2*b^4 - 30*a*b^5 - 9*b^6)*cosh(d*x + c)^4 - 7*(6*a^4*b^2 + 20*a^3*b^3 - 71*a^2*b^4 - 114*a*b^5
- 45*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 6*a^4*b^2 + 20*a^3*b^3 + 25*a^2*b^4 + 14*a*b^5 + 3*b^6 + 8*(99*(
6*a^4*b^2 + 20*a^3*b^3 + 25*a^2*b^4 + 14*a*b^5 + 3*b^6)*cosh(d*x + c)^7 + 63*(6*a^4*b^2 - 4*a^3*b^3 - 31*a^2*b
^4 - 30*a*b^5 - 9*b^6)*cosh(d*x + c)^5 - 7*(6*a^4*b^2 + 20*a^3*b^3 - 71*a^2*b^4 - 114*a*b^5 - 45*b^6)*cosh(d*x
+ c)^3 - 3*(6*a^4*b^2 - 4*a^3*b^3 + 17*a^2*b^4 + 34*a*b^5 + 15*b^6)*cosh(d*x + c))*sinh(d*x + c)^5 - (6*a^4*b
^2 + 20*a^3*b^3 - 71*a^2*b^4 - 114*a*b^5 - 45*b^6)*cosh(d*x + c)^4 + (495*(6*a^4*b^2 + 20*a^3*b^3 + 25*a^2*b^4
+ 14*a*b^5 + 3*b^6)*cosh(d*x + c)^8 + 420*(6*a^4*b^2 - 4*a^3*b^3 - 31*a^2*b^4 - 30*a*b^5 - 9*b^6)*cosh(d*x +
c)^6 - 6*a^4*b^2 - 20*a^3*b^3 + 71*a^2*b^4 + 114*a*b^5 + 45*b^6 - 70*(6*a^4*b^2 + 20*a^3*b^3 - 71*a^2*b^4 - 11
4*a*b^5 - 45*b^6)*cosh(d*x + c)^4 - 60*(6*a^4*b^2 - 4*a^3*b^3 + 17*a^2*b^4 + 34*a*b^5 + 15*b^6)*cosh(d*x + c)^
2)*sinh(d*x + c)^4 + 4*(55*(6*a^4*b^2 + 20*a^3*b^3 + 25*a^2*b^4 + 14*a*b^5 + 3*b^6)*cosh(d*x + c)^9 + 60*(6*a^
4*b^2 - 4*a^3*b^3 - 31*a^2*b^4 - 30*a*b^5 - 9*b^6)*cosh(d*x + c)^7 - 14*(6*a^4*b^2 + 20*a^3*b^3 - 71*a^2*b^4 -
114*a*b^5 - 45*b^6)*cosh(d*x + c)^5 - 20*(6*a^4*b^2 - 4*a^3*b^3 + 17*a^2*b^4 + 34*a*b^5 + 15*b^6)*cosh(d*x +
c)^3 - (6*a^4*b^2 + 20*a^3*b^3 - 71*a^2*b^4 - 114*a*b^5 - 45*b^6)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(6*a^4*b^
2 - 4*a^3*b^3 - 31*a^2*b^4 - 30*a*b^5 - 9*b^6)*cosh(d*x + c)^2 + 2*(33*(6*a^4*b^2 + 20*a^3*b^3 + 25*a^2*b^4 +
14*a*b^5 + 3*b^6)*cosh(d*x + c)^10 + 45*(6*a^4*b^2 - 4*a^3*b^3 - 31*a^2*b^4 - 30*a*b^5 - 9*b^6)*cosh(d*x + c)^
8 - 14*(6*a^4*b^2 + 20*a^3*b^3 - 71*a^2*b^4 - 114*a*b^5 - 45*b^6)*cosh(d*x + c)^6 + 6*a^4*b^2 - 4*a^3*b^3 - 31
*a^2*b^4 - 30*a*b^5 - 9*b^6 - 30*(6*a^4*b^2 - 4*a^3*b^3 + 17*a^2*b^4 + 34*a*b^5 + 15*b^6)*cosh(d*x + c)^4 - 3*
(6*a^4*b^2 + 20*a^3*b^3 - 71*a^2*b^4 - 114*a*b^5 - 45*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(3*(6*a^4*b^2
+ 20*a^3*b^3 + 25*a^2*b^4 + 14*a*b^5 + 3*b^6)*cosh(d*x + c)^11 + 5*(6*a^4*b^2 - 4*a^3*b^3 - 31*a^2*b^4 - 30*a*
b^5 - 9*b^6)*cosh(d*x + c)^9 - 2*(6*a^4*b^2 + 20*a^3*b^3 - 71*a^2*b^4 - 114*a*b^5 - 45*b^6)*cosh(d*x + c)^7 -
6*(6*a^4*b^2 - 4*a^3*b^3 + 17*a^2*b^4 + 34*a*b^5 + 15*b^6)*cosh(d*x + c)^5 - (6*a^4*b^2 + 20*a^3*b^3 - 71*a^2*
b^4 - 114*a*b^5 - 45*b^6)*cosh(d*x + c)^3 + (6*a^4*b^2 - 4*a^3*b^3 - 31*a^2*b^4 - 30*a*b^5 - 9*b^6)*cosh(d*x +
c))*sinh(d*x + c))*log(2*((a + b)*cosh(d*x + c)^2 + (a + b)*sinh(d*x + c)^2 + a - b)/(cosh(d*x + c)^2 - 2*cos
h(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 2*((a^6 + 2*a^5*b - 5*a^4*b^2 - 20*a^3*b^3 - 25*a^2*b^4 - 14*a*
b^5 - 3*b^6)*cosh(d*x + c)^12 + 12*(a^6 + 2*a^5*b - 5*a^4*b^2 - 20*a^3*b^3 - 25*a^2*b^4 - 14*a*b^5 - 3*b^6)*co
sh(d*x + c)*sinh(d*x + c)^11 + (a^6 + 2*a^5*b - 5*a^4*b^2 - 20*a^3*b^3 - 25*a^2*b^4 - 14*a*b^5 - 3*b^6)*sinh(d
*x + c)^12 + 2*(a^6 - 2*a^5*b - 9*a^4*b^2 + 4*a^3*b^3 + 31*a^2*b^4 + 30*a*b^5 + 9*b^6)*cosh(d*x + c)^10 + 2*(a
^6 - 2*a^5*b - 9*a^4*b^2 + 4*a^3*b^3 + 31*a^2*b^4 + 30*a*b^5 + 9*b^6 + 33*(a^6 + 2*a^5*b - 5*a^4*b^2 - 20*a^3*
b^3 - 25*a^2*b^4 - 14*a*b^5 - 3*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 20*(11*(a^6 + 2*a^5*b - 5*a^4*b^2 - 2
0*a^3*b^3 - 25*a^2*b^4 - 14*a*b^5 - 3*b^6)*cosh(d*x + c)^3 + (a^6 - 2*a^5*b - 9*a^4*b^2 + 4*a^3*b^3 + 31*a^2*b
^4 + 30*a*b^5 + 9*b^6)*cosh(d*x + c))*sinh(d*x + c)^9 - (a^6 + 2*a^5*b - 21*a^4*b^2 - 20*a^3*b^3 + 71*a^2*b^4
+ 114*a*b^5 + 45*b^6)*cosh(d*x + c)^8 - (a^6 + 2*a^5*b - 21*a^4*b^2 - 20*a^3*b^3 + 71*a^2*b^4 + 114*a*b^5 + 45
*b^6 - 495*(a^6 + 2*a^5*b - 5*a^4*b^2 - 20*a^3*b^3 - 25*a^2*b^4 - 14*a*b^5 - 3*b^6)*cosh(d*x + c)^4 - 90*(a^6
- 2*a^5*b - 9*a^4*b^2 + 4*a^3*b^3 + 31*a^2*b^4 + 30*a*b^5 + 9*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(99*(a
^6 + 2*a^5*b - 5*a^4*b^2 - 20*a^3*b^3 - 25*a^2*b^4 - 14*a*b^5 - 3*b^6)*cosh(d*x + c)^5 + 30*(a^6 - 2*a^5*b - 9
*a^4*b^2 + 4*a^3*b^3 + 31*a^2*b^4 + 30*a*b^5 + 9*b^6)*cosh(d*x + c)^3 - (a^6 + 2*a^5*b - 21*a^4*b^2 - 20*a^3*b
^3 + 71*a^2*b^4 + 114*a*b^5 + 45*b^6)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3
- 17*a^2*b^4 - 34*a*b^5 - 15*b^6)*cosh(d*x + c)^6 + 4*(231*(a^6 + 2*a^5*b - 5*a^4*b^2 - 20*a^3*b^3 - 25*a^2*b^
4 - 14*a*b^5 - 3*b^6)*cosh(d*x + c)^6 - a^6 + 2*a^5*b + a^4*b^2 - 4*a^3*b^3 + 17*a^2*b^4 + 34*a*b^5 + 15*b^6 +
105*(a^6 - 2*a^5*b - 9*a^4*b^2 + 4*a^3*b^3 + 31*a^2*b^4 + 30*a*b^5 + 9*b^6)*cosh(d*x + c)^4 - 7*(a^6 + 2*a^5*
b - 21*a^4*b^2 - 20*a^3*b^3 + 71*a^2*b^4 + 114*a*b^5 + 45*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + a^6 + 2*a^5*
b - 5*a^4*b^2 - 20*a^3*b^3 - 25*a^2*b^4 - 14*a*b^5 - 3*b^6 + 8*(99*(a^6 + 2*a^5*b - 5*a^4*b^2 - 20*a^3*b^3 - 2
5*a^2*b^4 - 14*a*b^5 - 3*b^6)*cosh(d*x + c)^7 + 63*(a^6 - 2*a^5*b - 9*a^4*b^2 + 4*a^3*b^3 + 31*a^2*b^4 + 30*a*
b^5 + 9*b^6)*cosh(d*x + c)^5 - 7*(a^6 + 2*a^5*b - 21*a^4*b^2 - 20*a^3*b^3 + 71*a^2*b^4 + 114*a*b^5 + 45*b^6)*c
osh(d*x + c)^3 - 3*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - 17*a^2*b^4 - 34*a*b^5 - 15*b^6)*cosh(d*x + c))*sinh(
d*x + c)^5 - (a^6 + 2*a^5*b - 21*a^4*b^2 - 20*a^3*b^3 + 71*a^2*b^4 + 114*a*b^5 + 45*b^6)*cosh(d*x + c)^4 + (49
5*(a^6 + 2*a^5*b - 5*a^4*b^2 - 20*a^3*b^3 - 25*a^2*b^4 - 14*a*b^5 - 3*b^6)*cosh(d*x + c)^8 + 420*(a^6 - 2*a^5*
b - 9*a^4*b^2 + 4*a^3*b^3 + 31*a^2*b^4 + 30*a*b^5 + 9*b^6)*cosh(d*x + c)^6 - a^6 - 2*a^5*b + 21*a^4*b^2 + 20*a
^3*b^3 - 71*a^2*b^4 - 114*a*b^5 - 45*b^6 - 70*(a^6 + 2*a^5*b - 21*a^4*b^2 - 20*a^3*b^3 + 71*a^2*b^4 + 114*a*b^
5 + 45*b^6)*cosh(d*x + c)^4 - 60*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - 17*a^2*b^4 - 34*a*b^5 - 15*b^6)*cosh(d
*x + c)^2)*sinh(d*x + c)^4 + 4*(55*(a^6 + 2*a^5*b - 5*a^4*b^2 - 20*a^3*b^3 - 25*a^2*b^4 - 14*a*b^5 - 3*b^6)*co
sh(d*x + c)^9 + 60*(a^6 - 2*a^5*b - 9*a^4*b^2 + 4*a^3*b^3 + 31*a^2*b^4 + 30*a*b^5 + 9*b^6)*cosh(d*x + c)^7 - 1
4*(a^6 + 2*a^5*b - 21*a^4*b^2 - 20*a^3*b^3 + 71*a^2*b^4 + 114*a*b^5 + 45*b^6)*cosh(d*x + c)^5 - 20*(a^6 - 2*a^
5*b - a^4*b^2 + 4*a^3*b^3 - 17*a^2*b^4 - 34*a*b^5 - 15*b^6)*cosh(d*x + c)^3 - (a^6 + 2*a^5*b - 21*a^4*b^2 - 20
*a^3*b^3 + 71*a^2*b^4 + 114*a*b^5 + 45*b^6)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(a^6 - 2*a^5*b - 9*a^4*b^2 + 4*
a^3*b^3 + 31*a^2*b^4 + 30*a*b^5 + 9*b^6)*cosh(d*x + c)^2 + 2*(33*(a^6 + 2*a^5*b - 5*a^4*b^2 - 20*a^3*b^3 - 25*
a^2*b^4 - 14*a*b^5 - 3*b^6)*cosh(d*x + c)^10 + 45*(a^6 - 2*a^5*b - 9*a^4*b^2 + 4*a^3*b^3 + 31*a^2*b^4 + 30*a*b
^5 + 9*b^6)*cosh(d*x + c)^8 - 14*(a^6 + 2*a^5*b - 21*a^4*b^2 - 20*a^3*b^3 + 71*a^2*b^4 + 114*a*b^5 + 45*b^6)*c
osh(d*x + c)^6 + a^6 - 2*a^5*b - 9*a^4*b^2 + 4*a^3*b^3 + 31*a^2*b^4 + 30*a*b^5 + 9*b^6 - 30*(a^6 - 2*a^5*b - a
^4*b^2 + 4*a^3*b^3 - 17*a^2*b^4 - 34*a*b^5 - 15*b^6)*cosh(d*x + c)^4 - 3*(a^6 + 2*a^5*b - 21*a^4*b^2 - 20*a^3*
b^3 + 71*a^2*b^4 + 114*a*b^5 + 45*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(3*(a^6 + 2*a^5*b - 5*a^4*b^2 - 20
*a^3*b^3 - 25*a^2*b^4 - 14*a*b^5 - 3*b^6)*cosh(d*x + c)^11 + 5*(a^6 - 2*a^5*b - 9*a^4*b^2 + 4*a^3*b^3 + 31*a^2
*b^4 + 30*a*b^5 + 9*b^6)*cosh(d*x + c)^9 - 2*(a^6 + 2*a^5*b - 21*a^4*b^2 - 20*a^3*b^3 + 71*a^2*b^4 + 114*a*b^5
+ 45*b^6)*cosh(d*x + c)^7 - 6*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - 17*a^2*b^4 - 34*a*b^5 - 15*b^6)*cosh(d*x
+ c)^5 - (a^6 + 2*a^5*b - 21*a^4*b^2 - 20*a^3*b^3 + 71*a^2*b^4 + 114*a*b^5 + 45*b^6)*cosh(d*x + c)^3 + (a^6 -
2*a^5*b - 9*a^4*b^2 + 4*a^3*b^3 + 31*a^2*b^4 + 30*a*b^5 + 9*b^6)*cosh(d*x + c))*sinh(d*x + c))*log(2*sinh(d*x
+ c)/(cosh(d*x + c) - sinh(d*x + c))) + 8*(3*(a^6 + 2*a^5*b + a^4*b^2)*d*x*cosh(d*x + c)^11 + 5*(a^6 + 5*a^5*
b + 10*a^4*b^2 + 14*a^3*b^3 + 11*a^2*b^4 + 3*a*b^5 + (a^6 - 2*a^5*b - 3*a^4*b^2)*d*x)*cosh(d*x + c)^9 + 2*(8*a
^6 + 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^3 - 52*a^2*b^4 - 24*a*b^5 - (a^6 + 2*a^5*b - 15*a^4*b^2)*d*x)*cosh(d*x +
c)^7 + 6*(3*a^6 + 7*a^5*b + 6*a^4*b^2 + 2*a^3*b^3 + 15*a^2*b^4 + 9*a*b^5 - (a^6 - 2*a^5*b + 5*a^4*b^2)*d*x)*c
osh(d*x + c)^5 + (8*a^6 + 24*a^5*b + 16*a^4*b^2 - 16*a^3*b^3 - 52*a^2*b^4 - 24*a*b^5 - (a^6 + 2*a^5*b - 15*a^4
*b^2)*d*x)*cosh(d*x + c)^3 + (a^6 + 5*a^5*b + 10*a^4*b^2 + 14*a^3*b^3 + 11*a^2*b^4 + 3*a*b^5 + (a^6 - 2*a^5*b
- 3*a^4*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^
5)*d*cosh(d*x + c)^12 + 12*(a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*d*cosh(d*x + c)*sin
h(d*x + c)^11 + (a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*d*sinh(d*x + c)^12 + 2*(a^9 +
a^8*b - 6*a^7*b^2 - 14*a^6*b^3 - 11*a^5*b^4 - 3*a^4*b^5)*d*cosh(d*x + c)^10 + 2*(33*(a^9 + 5*a^8*b + 10*a^7*b^
2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*d*cosh(d*x + c)^2 + (a^9 + a^8*b - 6*a^7*b^2 - 14*a^6*b^3 - 11*a^5*b^4 -
3*a^4*b^5)*d)*sinh(d*x + c)^10 - (a^9 + 5*a^8*b - 6*a^7*b^2 - 38*a^6*b^3 - 43*a^5*b^4 - 15*a^4*b^5)*d*cosh(d*
x + c)^8 + 20*(11*(a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*d*cosh(d*x + c)^3 + (a^9 + a
^8*b - 6*a^7*b^2 - 14*a^6*b^3 - 11*a^5*b^4 - 3*a^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^9 + (495*(a^9 + 5*a^8*b
+ 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*d*cosh(d*x + c)^4 + 90*(a^9 + a^8*b - 6*a^7*b^2 - 14*a^6*b^3
- 11*a^5*b^4 - 3*a^4*b^5)*d*cosh(d*x + c)^2 - (a^9 + 5*a^8*b - 6*a^7*b^2 - 38*a^6*b^3 - 43*a^5*b^4 - 15*a^4*b
^5)*d)*sinh(d*x + c)^8 - 4*(a^9 + a^8*b + 2*a^7*b^2 + 10*a^6*b^3 + 13*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^6 +
8*(99*(a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*d*cosh(d*x + c)^5 + 30*(a^9 + a^8*b - 6
*a^7*b^2 - 14*a^6*b^3 - 11*a^5*b^4 - 3*a^4*b^5)*d*cosh(d*x + c)^3 - (a^9 + 5*a^8*b - 6*a^7*b^2 - 38*a^6*b^3 -
43*a^5*b^4 - 15*a^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 4*(231*(a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 +
5*a^5*b^4 + a^4*b^5)*d*cosh(d*x + c)^6 + 105*(a^9 + a^8*b - 6*a^7*b^2 - 14*a^6*b^3 - 11*a^5*b^4 - 3*a^4*b^5)*d
*cosh(d*x + c)^4 - 7*(a^9 + 5*a^8*b - 6*a^7*b^2 - 38*a^6*b^3 - 43*a^5*b^4 - 15*a^4*b^5)*d*cosh(d*x + c)^2 - (a
^9 + a^8*b + 2*a^7*b^2 + 10*a^6*b^3 + 13*a^5*b^4 + 5*a^4*b^5)*d)*sinh(d*x + c)^6 - (a^9 + 5*a^8*b - 6*a^7*b^2
- 38*a^6*b^3 - 43*a^5*b^4 - 15*a^4*b^5)*d*cosh(d*x + c)^4 + 8*(99*(a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5
*a^5*b^4 + a^4*b^5)*d*cosh(d*x + c)^7 + 63*(a^9 + a^8*b - 6*a^7*b^2 - 14*a^6*b^3 - 11*a^5*b^4 - 3*a^4*b^5)*d*c
osh(d*x + c)^5 - 7*(a^9 + 5*a^8*b - 6*a^7*b^2 - 38*a^6*b^3 - 43*a^5*b^4 - 15*a^4*b^5)*d*cosh(d*x + c)^3 - 3*(a
^9 + a^8*b + 2*a^7*b^2 + 10*a^6*b^3 + 13*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + (495*(a^9 + 5
*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*d*cosh(d*x + c)^8 + 420*(a^9 + a^8*b - 6*a^7*b^2 - 14*
a^6*b^3 - 11*a^5*b^4 - 3*a^4*b^5)*d*cosh(d*x + c)^6 - 70*(a^9 + 5*a^8*b - 6*a^7*b^2 - 38*a^6*b^3 - 43*a^5*b^4
- 15*a^4*b^5)*d*cosh(d*x + c)^4 - 60*(a^9 + a^8*b + 2*a^7*b^2 + 10*a^6*b^3 + 13*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*
x + c)^2 - (a^9 + 5*a^8*b - 6*a^7*b^2 - 38*a^6*b^3 - 43*a^5*b^4 - 15*a^4*b^5)*d)*sinh(d*x + c)^4 + 2*(a^9 + a^
8*b - 6*a^7*b^2 - 14*a^6*b^3 - 11*a^5*b^4 - 3*a^4*b^5)*d*cosh(d*x + c)^2 + 4*(55*(a^9 + 5*a^8*b + 10*a^7*b^2 +
10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*d*cosh(d*x + c)^9 + 60*(a^9 + a^8*b - 6*a^7*b^2 - 14*a^6*b^3 - 11*a^5*b^4 -
3*a^4*b^5)*d*cosh(d*x + c)^7 - 14*(a^9 + 5*a^8*b - 6*a^7*b^2 - 38*a^6*b^3 - 43*a^5*b^4 - 15*a^4*b^5)*d*cosh(d
*x + c)^5 - 20*(a^9 + a^8*b + 2*a^7*b^2 + 10*a^6*b^3 + 13*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^3 - (a^9 + 5*a^
8*b - 6*a^7*b^2 - 38*a^6*b^3 - 43*a^5*b^4 - 15*a^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(33*(a^9 + 5*a^8*
b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*d*cosh(d*x + c)^10 + 45*(a^9 + a^8*b - 6*a^7*b^2 - 14*a^6*b
^3 - 11*a^5*b^4 - 3*a^4*b^5)*d*cosh(d*x + c)^8 - 14*(a^9 + 5*a^8*b - 6*a^7*b^2 - 38*a^6*b^3 - 43*a^5*b^4 - 15*
a^4*b^5)*d*cosh(d*x + c)^6 - 30*(a^9 + a^8*b + 2*a^7*b^2 + 10*a^6*b^3 + 13*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c
)^4 - 3*(a^9 + 5*a^8*b - 6*a^7*b^2 - 38*a^6*b^3 - 43*a^5*b^4 - 15*a^4*b^5)*d*cosh(d*x + c)^2 + (a^9 + a^8*b -
6*a^7*b^2 - 14*a^6*b^3 - 11*a^5*b^4 - 3*a^4*b^5)*d)*sinh(d*x + c)^2 + (a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3
+ 5*a^5*b^4 + a^4*b^5)*d + 4*(3*(a^9 + 5*a^8*b + 10*a^7*b^2 + 10*a^6*b^3 + 5*a^5*b^4 + a^4*b^5)*d*cosh(d*x +
c)^11 + 5*(a^9 + a^8*b - 6*a^7*b^2 - 14*a^6*b^3 - 11*a^5*b^4 - 3*a^4*b^5)*d*cosh(d*x + c)^9 - 2*(a^9 + 5*a^8*b
- 6*a^7*b^2 - 38*a^6*b^3 - 43*a^5*b^4 - 15*a^4*b^5)*d*cosh(d*x + c)^7 - 6*(a^9 + a^8*b + 2*a^7*b^2 + 10*a^6*b
^3 + 13*a^5*b^4 + 5*a^4*b^5)*d*cosh(d*x + c)^5 - (a^9 + 5*a^8*b - 6*a^7*b^2 - 38*a^6*b^3 - 43*a^5*b^4 - 15*a^4
*b^5)*d*cosh(d*x + c)^3 + (a^9 + a^8*b - 6*a^7*b^2 - 14*a^6*b^3 - 11*a^5*b^4 - 3*a^4*b^5)*d*cosh(d*x + c))*sin
h(d*x + c))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{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(coth(d*x+c)**3/(a+b*tanh(d*x+c)**2)**3,x)
[Out]
Integral(coth(c + d*x)**3/(a + b*tanh(c + d*x)**2)**3, x)
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Giac [B] time = 1.41218, size = 656, normalized size = 3.84 \begin{align*} \frac{{\left (6 \, a^{2} b^{2} + 8 \, a b^{3} + 3 \, b^{4}\right )} \log \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 \,{\left (a^{7} d + 3 \, a^{6} b d + 3 \, a^{5} b^{2} d + a^{4} b^{3} d\right )}} - \frac{d x + c}{a^{3} d + 3 \, a^{2} b d + 3 \, a b^{2} d + b^{3} d} + \frac{{\left (a - 3 \, b\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{4} d} - \frac{2 \,{\left ({\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 14 \, a^{2} b^{3} + 11 \, a b^{4} + 3 \, b^{5}\right )} e^{\left (10 \, d x + 10 \, c\right )} + 2 \,{\left (2 \, a^{5} + 6 \, a^{4} b + 4 \, a^{3} b^{2} - 4 \, a^{2} b^{3} - 13 \, a b^{4} - 6 \, b^{5}\right )} e^{\left (8 \, d x + 8 \, c\right )} + 2 \,{\left (3 \, a^{5} + 7 \, a^{4} b + 6 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + 15 \, a b^{4} + 9 \, b^{5}\right )} e^{\left (6 \, d x + 6 \, c\right )} + 2 \,{\left (2 \, a^{5} + 6 \, a^{4} b + 4 \, a^{3} b^{2} - 4 \, a^{2} b^{3} - 13 \, a b^{4} - 6 \, b^{5}\right )} e^{\left (4 \, d x + 4 \, c\right )} +{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 14 \, a^{2} b^{3} + 11 \, a b^{4} + 3 \, b^{5}\right )} e^{\left (2 \, d x + 2 \, c\right )}\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}{\left (a + b\right )}^{3} a^{3} d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
[Out]
1/2*(6*a^2*b^2 + 8*a*b^3 + 3*b^4)*log(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)/(a^7*d + 3*a^6*b*d + 3*a^5*b^2*d + a^4*b^3*d) - (d*x + c)/(a^3*d + 3*a^2*b*d + 3*a*b^2*d +
b^3*d) + (a - 3*b)*log(abs(e^(2*d*x + 2*c) - 1))/(a^4*d) - 2*((a^5 + 5*a^4*b + 10*a^3*b^2 + 14*a^2*b^3 + 11*a*
b^4 + 3*b^5)*e^(10*d*x + 10*c) + 2*(2*a^5 + 6*a^4*b + 4*a^3*b^2 - 4*a^2*b^3 - 13*a*b^4 - 6*b^5)*e^(8*d*x + 8*c
) + 2*(3*a^5 + 7*a^4*b + 6*a^3*b^2 + 2*a^2*b^3 + 15*a*b^4 + 9*b^5)*e^(6*d*x + 6*c) + 2*(2*a^5 + 6*a^4*b + 4*a^
3*b^2 - 4*a^2*b^3 - 13*a*b^4 - 6*b^5)*e^(4*d*x + 4*c) + (a^5 + 5*a^4*b + 10*a^3*b^2 + 14*a^2*b^3 + 11*a*b^4 +
3*b^5)*e^(2*d*x + 2*c))/((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*(a + b)^3*a^3*d*(e^(2*d*x + 2*c) - 1)^2)