3.167 \(\int \frac {\coth ^3(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\)

Optimal. Leaf size=152 \[ -\frac {b^4}{4 a^3 d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {b^3 (2 a+b)}{a^3 d (a+b)^3 \left (a \cosh ^2(c+d x)+b\right )}+\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \log \left (a \cosh ^2(c+d x)+b\right )}{2 a^3 d (a+b)^4}-\frac {\text {csch}^2(c+d x)}{2 d (a+b)^3}+\frac {(a+4 b) \log (\sinh (c+d x))}{d (a+b)^4} \]

[Out]

-1/4*b^4/a^3/(a+b)^2/d/(b+a*cosh(d*x+c)^2)^2+b^3*(2*a+b)/a^3/(a+b)^3/d/(b+a*cosh(d*x+c)^2)-1/2*csch(d*x+c)^2/d
/(a+b)^3+1/2*b^2*(6*a^2+4*a*b+b^2)*ln(b+a*cosh(d*x+c)^2)/a^3/(a+b)^4/d+(a+4*b)*ln(sinh(d*x+c))/d/(a+b)^4

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Rubi [A]  time = 0.24, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4138, 446, 88} \[ -\frac {b^4}{4 a^3 d (a+b)^2 \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {b^3 (2 a+b)}{a^3 d (a+b)^3 \left (a \cosh ^2(c+d x)+b\right )}+\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \log \left (a \cosh ^2(c+d x)+b\right )}{2 a^3 d (a+b)^4}-\frac {\text {csch}^2(c+d x)}{2 d (a+b)^3}+\frac {(a+4 b) \log (\sinh (c+d x))}{d (a+b)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b^4/(4*a^3*(a + b)^2*d*(b + a*Cosh[c + d*x]^2)^2) + (b^3*(2*a + b))/(a^3*(a + b)^3*d*(b + a*Cosh[c + d*x]^2))
- Csch[c + d*x]^2/(2*(a + b)^3*d) + (b^2*(6*a^2 + 4*a*b + b^2)*Log[b + a*Cosh[c + d*x]^2])/(2*a^3*(a + b)^4*d
) + ((a + 4*b)*Log[Sinh[c + d*x]])/((a + b)^4*d)

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]))

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 4138

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Module[{ff =
FreeFactors[Cos[e + f*x], x]}, -Dist[(f*ff^(m + n*p - 1))^(-1), Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*
(ff*x)^n)^p)/x^(m + n*p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\coth ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^9}{\left (1-x^2\right )^2 \left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{(1-x)^2 (b+a x)^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{(a+b)^3 (-1+x)^2}+\frac {a+4 b}{(a+b)^4 (-1+x)}+\frac {b^4}{a^2 (a+b)^2 (b+a x)^3}-\frac {2 b^3 (2 a+b)}{a^2 (a+b)^3 (b+a x)^2}+\frac {b^2 \left (6 a^2+4 a b+b^2\right )}{a^2 (a+b)^4 (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {b^4}{4 a^3 (a+b)^2 d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {b^3 (2 a+b)}{a^3 (a+b)^3 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\text {csch}^2(c+d x)}{2 (a+b)^3 d}+\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \log \left (b+a \cosh ^2(c+d x)\right )}{2 a^3 (a+b)^4 d}+\frac {(a+4 b) \log (\sinh (c+d x))}{(a+b)^4 d}\\ \end {align*}

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Mathematica [A]  time = 1.92, size = 172, normalized size = 1.13 \[ -\frac {\text {sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b)^3 \left (\frac {b^4 (a+b)^2}{a^3 \left (a \sinh ^2(c+d x)+a+b\right )^2}-\frac {4 b^3 (a+b) (2 a+b)}{a^3 \left (a \sinh ^2(c+d x)+a+b\right )}-\frac {2 b^2 \left (6 a^2+4 a b+b^2\right ) \log \left (a \sinh ^2(c+d x)+a+b\right )}{a^3}+2 (a+b) \text {csch}^2(c+d x)-4 (a+4 b) \log (\sinh (c+d x))\right )}{32 d (a+b)^4 \left (a+b \text {sech}^2(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/32*((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*(2*(a + b)*Csch[c + d*x]^2 - 4*(a + 4*b)*Log[Sinh[c +
d*x]] - (2*b^2*(6*a^2 + 4*a*b + b^2)*Log[a + b + a*Sinh[c + d*x]^2])/a^3 + (b^4*(a + b)^2)/(a^3*(a + b + a*Si
nh[c + d*x]^2)^2) - (4*b^3*(a + b)*(2*a + b))/(a^3*(a + b + a*Sinh[c + d*x]^2))))/((a + b)^4*d*(a + b*Sech[c +
d*x]^2)^3)

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fricas [B]  time = 1.34, size = 10255, normalized size = 67.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^12 + 24*(a^6 + 4*a^5*b + 6*a^4*b^2
+ 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)*sinh(d*x + c)^11 + 2*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^
4)*d*x*sinh(d*x + c)^12 + 4*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*
a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^10 + 4*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + 33*
(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^2 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b
^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*sinh(d*x + c)^10 + 40*(11*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*
x*cosh(d*x + c)^3 + (a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3
+ 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^9 + 2*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 -
40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x +
c)^8 + 2*(495*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^4 + 8*a^6 + 24*a^5*b + 16*a
^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 1
6*b^6)*d*x + 90*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3 + 17
*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(99*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^
2*b^4)*d*x*cosh(d*x + c)^5 + 30*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 +
28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^3 + (8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*
b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c))*
sinh(d*x + c)^7 + 8*(3*a^6 + 11*a^5*b + 16*a^4*b^2 + 12*a^3*b^3 + 20*a^2*b^4 + 22*a*b^5 + 6*b^6 - (a^6 + 8*a^5
*b + 30*a^4*b^2 + 60*a^3*b^3 + 65*a^2*b^4 + 36*a*b^5 + 8*b^6)*d*x)*cosh(d*x + c)^6 + 8*(231*(a^6 + 4*a^5*b + 6
*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^6 + 3*a^6 + 11*a^5*b + 16*a^4*b^2 + 12*a^3*b^3 + 20*a^2*b^4
+ 22*a*b^5 + 6*b^6 + 105*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3
*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^4 - (a^6 + 8*a^5*b + 30*a^4*b^2 + 60*a^3*b^3 + 65*a^2*b^4 + 36
*a*b^5 + 8*b^6)*d*x + 7*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*
a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 16*(99*(a^6 + 4
*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^7 + 63*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*
b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^5 + 7*(8*a^6 + 24*a^
5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64
*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^3 + 3*(3*a^6 + 11*a^5*b + 16*a^4*b^2 + 12*a^3*b^3 + 20*a^2*b^4 + 22*a*b^5
+ 6*b^6 - (a^6 + 8*a^5*b + 30*a^4*b^2 + 60*a^3*b^3 + 65*a^2*b^4 + 36*a*b^5 + 8*b^6)*d*x)*cosh(d*x + c))*sinh(d
*x + c)^5 + 2*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 -
60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^4 + 2*(495*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*
b^3 + a^2*b^4)*d*x*cosh(d*x + c)^8 + 420*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*
a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^6 + 8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4
- 40*a*b^5 - 12*b^6 + 70*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 1
0*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^4 - (a^6 + 4*a^5*b - 10*a^4*b^2 -
60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x + 60*(3*a^6 + 11*a^5*b + 16*a^4*b^2 + 12*a^3*b^3 + 20*a^2*b^4
+ 22*a*b^5 + 6*b^6 - (a^6 + 8*a^5*b + 30*a^4*b^2 + 60*a^3*b^3 + 65*a^2*b^4 + 36*a*b^5 + 8*b^6)*d*x)*cosh(d*x
+ c)^2)*sinh(d*x + c)^4 + 8*(55*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^9 + 60*(a^
6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)
*d*x)*cosh(d*x + c)^7 + 14*(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b -
10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^5 + 20*(3*a^6 + 11*a^5*b + 16*a^4
*b^2 + 12*a^3*b^3 + 20*a^2*b^4 + 22*a*b^5 + 6*b^6 - (a^6 + 8*a^5*b + 30*a^4*b^2 + 60*a^3*b^3 + 65*a^2*b^4 + 36
*a*b^5 + 8*b^6)*d*x)*cosh(d*x + c)^3 + (8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6
+ 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*
(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x + 4*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^
6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^2 + 4*(33*(a^6 + 4*a^5*b + 6*
a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^10 + 45*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^6
+ 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^8 + 14*(8*a^6 + 24*a^5*b + 16*
a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 -
16*b^6)*d*x)*cosh(d*x + c)^6 + a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + 30*(3*a^6 + 11*a^5*b + 16*a^4*b
^2 + 12*a^3*b^3 + 20*a^2*b^4 + 22*a*b^5 + 6*b^6 - (a^6 + 8*a^5*b + 30*a^4*b^2 + 60*a^3*b^3 + 65*a^2*b^4 + 36*a
*b^5 + 8*b^6)*d*x)*cosh(d*x + c)^4 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x + 3*
(8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 9
5*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cosh
(d*x + c)^12 + 12*(6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cosh(d*x + c)*sinh(d*x + c)^11 + (6*a^4*b^2 + 4*a^3*b^3 +
a^2*b^4)*sinh(d*x + c)^12 + 2*(6*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*cosh(d*x + c)^10 + 2*(6*a^4*b^2
+ 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5 + 33*(6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^10 +
20*(11*(6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cosh(d*x + c)^3 + (6*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*co
sh(d*x + c))*sinh(d*x + c)^9 - (6*a^4*b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*cosh(d*x + c)^8 - (6*a
^4*b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6 - 495*(6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cosh(d*x + c)^4 -
90*(6*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(99*(6*a^4*b^2 + 4*a^3
*b^3 + a^2*b^4)*cosh(d*x + c)^5 + 30*(6*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*cosh(d*x + c)^3 - (6*a^4*
b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(6*a^4*b^2 + 28*a^3*b^3 +
65*a^2*b^4 + 36*a*b^5 + 8*b^6)*cosh(d*x + c)^6 + 4*(231*(6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cosh(d*x + c)^6 - 6
*a^4*b^2 - 28*a^3*b^3 - 65*a^2*b^4 - 36*a*b^5 - 8*b^6 + 105*(6*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*co
sh(d*x + c)^4 - 7*(6*a^4*b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^6 +
6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4 + 8*(99*(6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cosh(d*x + c)^7 + 63*(6*a^4*b^2 + 28
*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*cosh(d*x + c)^5 - 7*(6*a^4*b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*
cosh(d*x + c)^3 - 3*(6*a^4*b^2 + 28*a^3*b^3 + 65*a^2*b^4 + 36*a*b^5 + 8*b^6)*cosh(d*x + c))*sinh(d*x + c)^5 -
(6*a^4*b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*cosh(d*x + c)^4 + (495*(6*a^4*b^2 + 4*a^3*b^3 + a^2*b
^4)*cosh(d*x + c)^8 + 420*(6*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*cosh(d*x + c)^6 - 6*a^4*b^2 - 4*a^3*
b^3 + 95*a^2*b^4 + 64*a*b^5 + 16*b^6 - 70*(6*a^4*b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*cosh(d*x +
c)^4 - 60*(6*a^4*b^2 + 28*a^3*b^3 + 65*a^2*b^4 + 36*a*b^5 + 8*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(55*(6
*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cosh(d*x + c)^9 + 60*(6*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*cosh(d*x
+ c)^7 - 14*(6*a^4*b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*cosh(d*x + c)^5 - 20*(6*a^4*b^2 + 28*a^3*
b^3 + 65*a^2*b^4 + 36*a*b^5 + 8*b^6)*cosh(d*x + c)^3 - (6*a^4*b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6
)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(6*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*cosh(d*x + c)^2 + 2*(33*(
6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cosh(d*x + c)^10 + 45*(6*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*cosh(d*
x + c)^8 - 14*(6*a^4*b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*cosh(d*x + c)^6 + 6*a^4*b^2 + 28*a^3*b^
3 + 17*a^2*b^4 + 4*a*b^5 - 30*(6*a^4*b^2 + 28*a^3*b^3 + 65*a^2*b^4 + 36*a*b^5 + 8*b^6)*cosh(d*x + c)^4 - 3*(6*
a^4*b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(3*(6*a^4*b^2 + 4*a
^3*b^3 + a^2*b^4)*cosh(d*x + c)^11 + 5*(6*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*cosh(d*x + c)^9 - 2*(6*
a^4*b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*cosh(d*x + c)^7 - 6*(6*a^4*b^2 + 28*a^3*b^3 + 65*a^2*b^4
+ 36*a*b^5 + 8*b^6)*cosh(d*x + c)^5 - (6*a^4*b^2 + 4*a^3*b^3 - 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*cosh(d*x + c)^
3 + (6*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*cosh(d*x + c))*sinh(d*x + c))*log(2*(a*cosh(d*x + c)^2 + a
*sinh(d*x + c)^2 + a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 2*((a^6 + 4
*a^5*b)*cosh(d*x + c)^12 + 12*(a^6 + 4*a^5*b)*cosh(d*x + c)*sinh(d*x + c)^11 + (a^6 + 4*a^5*b)*sinh(d*x + c)^1
2 + 2*(a^6 + 8*a^5*b + 16*a^4*b^2)*cosh(d*x + c)^10 + 2*(a^6 + 8*a^5*b + 16*a^4*b^2 + 33*(a^6 + 4*a^5*b)*cosh(
d*x + c)^2)*sinh(d*x + c)^10 + 20*(11*(a^6 + 4*a^5*b)*cosh(d*x + c)^3 + (a^6 + 8*a^5*b + 16*a^4*b^2)*cosh(d*x
+ c))*sinh(d*x + c)^9 - (a^6 + 4*a^5*b - 16*a^4*b^2 - 64*a^3*b^3)*cosh(d*x + c)^8 - (a^6 + 4*a^5*b - 16*a^4*b^
2 - 64*a^3*b^3 - 495*(a^6 + 4*a^5*b)*cosh(d*x + c)^4 - 90*(a^6 + 8*a^5*b + 16*a^4*b^2)*cosh(d*x + c)^2)*sinh(d
*x + c)^8 + 8*(99*(a^6 + 4*a^5*b)*cosh(d*x + c)^5 + 30*(a^6 + 8*a^5*b + 16*a^4*b^2)*cosh(d*x + c)^3 - (a^6 + 4
*a^5*b - 16*a^4*b^2 - 64*a^3*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(a^6 + 8*a^5*b + 24*a^4*b^2 + 32*a^3*b^3)
*cosh(d*x + c)^6 + 4*(231*(a^6 + 4*a^5*b)*cosh(d*x + c)^6 - a^6 - 8*a^5*b - 24*a^4*b^2 - 32*a^3*b^3 + 105*(a^6
+ 8*a^5*b + 16*a^4*b^2)*cosh(d*x + c)^4 - 7*(a^6 + 4*a^5*b - 16*a^4*b^2 - 64*a^3*b^3)*cosh(d*x + c)^2)*sinh(d
*x + c)^6 + a^6 + 4*a^5*b + 8*(99*(a^6 + 4*a^5*b)*cosh(d*x + c)^7 + 63*(a^6 + 8*a^5*b + 16*a^4*b^2)*cosh(d*x +
c)^5 - 7*(a^6 + 4*a^5*b - 16*a^4*b^2 - 64*a^3*b^3)*cosh(d*x + c)^3 - 3*(a^6 + 8*a^5*b + 24*a^4*b^2 + 32*a^3*b
^3)*cosh(d*x + c))*sinh(d*x + c)^5 - (a^6 + 4*a^5*b - 16*a^4*b^2 - 64*a^3*b^3)*cosh(d*x + c)^4 + (495*(a^6 + 4
*a^5*b)*cosh(d*x + c)^8 + 420*(a^6 + 8*a^5*b + 16*a^4*b^2)*cosh(d*x + c)^6 - a^6 - 4*a^5*b + 16*a^4*b^2 + 64*a
^3*b^3 - 70*(a^6 + 4*a^5*b - 16*a^4*b^2 - 64*a^3*b^3)*cosh(d*x + c)^4 - 60*(a^6 + 8*a^5*b + 24*a^4*b^2 + 32*a^
3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(55*(a^6 + 4*a^5*b)*cosh(d*x + c)^9 + 60*(a^6 + 8*a^5*b + 16*a^4*b
^2)*cosh(d*x + c)^7 - 14*(a^6 + 4*a^5*b - 16*a^4*b^2 - 64*a^3*b^3)*cosh(d*x + c)^5 - 20*(a^6 + 8*a^5*b + 24*a^
4*b^2 + 32*a^3*b^3)*cosh(d*x + c)^3 - (a^6 + 4*a^5*b - 16*a^4*b^2 - 64*a^3*b^3)*cosh(d*x + c))*sinh(d*x + c)^3
+ 2*(a^6 + 8*a^5*b + 16*a^4*b^2)*cosh(d*x + c)^2 + 2*(33*(a^6 + 4*a^5*b)*cosh(d*x + c)^10 + 45*(a^6 + 8*a^5*b
+ 16*a^4*b^2)*cosh(d*x + c)^8 - 14*(a^6 + 4*a^5*b - 16*a^4*b^2 - 64*a^3*b^3)*cosh(d*x + c)^6 + a^6 + 8*a^5*b
+ 16*a^4*b^2 - 30*(a^6 + 8*a^5*b + 24*a^4*b^2 + 32*a^3*b^3)*cosh(d*x + c)^4 - 3*(a^6 + 4*a^5*b - 16*a^4*b^2 -
64*a^3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(3*(a^6 + 4*a^5*b)*cosh(d*x + c)^11 + 5*(a^6 + 8*a^5*b + 16*a
^4*b^2)*cosh(d*x + c)^9 - 2*(a^6 + 4*a^5*b - 16*a^4*b^2 - 64*a^3*b^3)*cosh(d*x + c)^7 - 6*(a^6 + 8*a^5*b + 24*
a^4*b^2 + 32*a^3*b^3)*cosh(d*x + c)^5 - (a^6 + 4*a^5*b - 16*a^4*b^2 - 64*a^3*b^3)*cosh(d*x + c)^3 + (a^6 + 8*a
^5*b + 16*a^4*b^2)*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 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cosh(d*x + c)^11 + 5*(a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4
- 2*a*b^5 + (a^6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c)^9 + 2*(8*a^6
+ 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3 - 95*a^2*b
^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^7 + 6*(3*a^6 + 11*a^5*b + 16*a^4*b^2 + 12*a^3*b^3 + 20*a^2*b^4 + 22
*a*b^5 + 6*b^6 - (a^6 + 8*a^5*b + 30*a^4*b^2 + 60*a^3*b^3 + 65*a^2*b^4 + 36*a*b^5 + 8*b^6)*d*x)*cosh(d*x + c)^
5 + (8*a^6 + 24*a^5*b + 16*a^4*b^2 - 28*a^2*b^4 - 40*a*b^5 - 12*b^6 - (a^6 + 4*a^5*b - 10*a^4*b^2 - 60*a^3*b^3
- 95*a^2*b^4 - 64*a*b^5 - 16*b^6)*d*x)*cosh(d*x + c)^3 + (a^6 + a^5*b - 4*a^3*b^3 - 6*a^2*b^4 - 2*a*b^5 + (a^
6 + 8*a^5*b + 22*a^4*b^2 + 28*a^3*b^3 + 17*a^2*b^4 + 4*a*b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c))/((a^9 + 4*a^8
*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*d*cosh(d*x + c)^12 + 12*(a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4
)*d*cosh(d*x + c)*sinh(d*x + c)^11 + (a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*d*sinh(d*x + c)^12 + 2*
(a^9 + 8*a^8*b + 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5*b^4 + 4*a^4*b^5)*d*cosh(d*x + c)^10 + 2*(33*(a^9 + 4*a^8*b +
6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*d*cosh(d*x + c)^2 + (a^9 + 8*a^8*b + 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5*b^4 +
4*a^4*b^5)*d)*sinh(d*x + c)^10 - (a^9 + 4*a^8*b - 10*a^7*b^2 - 60*a^6*b^3 - 95*a^5*b^4 - 64*a^4*b^5 - 16*a^3*b
^6)*d*cosh(d*x + c)^8 + 20*(11*(a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*d*cosh(d*x + c)^3 + (a^9 + 8*
a^8*b + 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5*b^4 + 4*a^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^9 + (495*(a^9 + 4*a^8
*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*d*cosh(d*x + c)^4 + 90*(a^9 + 8*a^8*b + 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5
*b^4 + 4*a^4*b^5)*d*cosh(d*x + c)^2 - (a^9 + 4*a^8*b - 10*a^7*b^2 - 60*a^6*b^3 - 95*a^5*b^4 - 64*a^4*b^5 - 16*
a^3*b^6)*d)*sinh(d*x + c)^8 - 4*(a^9 + 8*a^8*b + 30*a^7*b^2 + 60*a^6*b^3 + 65*a^5*b^4 + 36*a^4*b^5 + 8*a^3*b^6
)*d*cosh(d*x + c)^6 + 8*(99*(a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*d*cosh(d*x + c)^5 + 30*(a^9 + 8*
a^8*b + 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5*b^4 + 4*a^4*b^5)*d*cosh(d*x + c)^3 - (a^9 + 4*a^8*b - 10*a^7*b^2 - 60
*a^6*b^3 - 95*a^5*b^4 - 64*a^4*b^5 - 16*a^3*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^7 + 4*(231*(a^9 + 4*a^8*b + 6*
a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*d*cosh(d*x + c)^6 + 105*(a^9 + 8*a^8*b + 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5*b^4 +
4*a^4*b^5)*d*cosh(d*x + c)^4 - 7*(a^9 + 4*a^8*b - 10*a^7*b^2 - 60*a^6*b^3 - 95*a^5*b^4 - 64*a^4*b^5 - 16*a^3*
b^6)*d*cosh(d*x + c)^2 - (a^9 + 8*a^8*b + 30*a^7*b^2 + 60*a^6*b^3 + 65*a^5*b^4 + 36*a^4*b^5 + 8*a^3*b^6)*d)*si
nh(d*x + c)^6 - (a^9 + 4*a^8*b - 10*a^7*b^2 - 60*a^6*b^3 - 95*a^5*b^4 - 64*a^4*b^5 - 16*a^3*b^6)*d*cosh(d*x +
c)^4 + 8*(99*(a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*d*cosh(d*x + c)^7 + 63*(a^9 + 8*a^8*b + 22*a^7*
b^2 + 28*a^6*b^3 + 17*a^5*b^4 + 4*a^4*b^5)*d*cosh(d*x + c)^5 - 7*(a^9 + 4*a^8*b - 10*a^7*b^2 - 60*a^6*b^3 - 95
*a^5*b^4 - 64*a^4*b^5 - 16*a^3*b^6)*d*cosh(d*x + c)^3 - 3*(a^9 + 8*a^8*b + 30*a^7*b^2 + 60*a^6*b^3 + 65*a^5*b^
4 + 36*a^4*b^5 + 8*a^3*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^5 + (495*(a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a
^5*b^4)*d*cosh(d*x + c)^8 + 420*(a^9 + 8*a^8*b + 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5*b^4 + 4*a^4*b^5)*d*cosh(d*x
+ c)^6 - 70*(a^9 + 4*a^8*b - 10*a^7*b^2 - 60*a^6*b^3 - 95*a^5*b^4 - 64*a^4*b^5 - 16*a^3*b^6)*d*cosh(d*x + c)^4
- 60*(a^9 + 8*a^8*b + 30*a^7*b^2 + 60*a^6*b^3 + 65*a^5*b^4 + 36*a^4*b^5 + 8*a^3*b^6)*d*cosh(d*x + c)^2 - (a^9
+ 4*a^8*b - 10*a^7*b^2 - 60*a^6*b^3 - 95*a^5*b^4 - 64*a^4*b^5 - 16*a^3*b^6)*d)*sinh(d*x + c)^4 + 2*(a^9 + 8*a
^8*b + 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5*b^4 + 4*a^4*b^5)*d*cosh(d*x + c)^2 + 4*(55*(a^9 + 4*a^8*b + 6*a^7*b^2
+ 4*a^6*b^3 + a^5*b^4)*d*cosh(d*x + c)^9 + 60*(a^9 + 8*a^8*b + 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5*b^4 + 4*a^4*b^
5)*d*cosh(d*x + c)^7 - 14*(a^9 + 4*a^8*b - 10*a^7*b^2 - 60*a^6*b^3 - 95*a^5*b^4 - 64*a^4*b^5 - 16*a^3*b^6)*d*c
osh(d*x + c)^5 - 20*(a^9 + 8*a^8*b + 30*a^7*b^2 + 60*a^6*b^3 + 65*a^5*b^4 + 36*a^4*b^5 + 8*a^3*b^6)*d*cosh(d*x
+ c)^3 - (a^9 + 4*a^8*b - 10*a^7*b^2 - 60*a^6*b^3 - 95*a^5*b^4 - 64*a^4*b^5 - 16*a^3*b^6)*d*cosh(d*x + c))*si
nh(d*x + c)^3 + 2*(33*(a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*d*cosh(d*x + c)^10 + 45*(a^9 + 8*a^8*b
+ 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5*b^4 + 4*a^4*b^5)*d*cosh(d*x + c)^8 - 14*(a^9 + 4*a^8*b - 10*a^7*b^2 - 60*a
^6*b^3 - 95*a^5*b^4 - 64*a^4*b^5 - 16*a^3*b^6)*d*cosh(d*x + c)^6 - 30*(a^9 + 8*a^8*b + 30*a^7*b^2 + 60*a^6*b^3
+ 65*a^5*b^4 + 36*a^4*b^5 + 8*a^3*b^6)*d*cosh(d*x + c)^4 - 3*(a^9 + 4*a^8*b - 10*a^7*b^2 - 60*a^6*b^3 - 95*a^
5*b^4 - 64*a^4*b^5 - 16*a^3*b^6)*d*cosh(d*x + c)^2 + (a^9 + 8*a^8*b + 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5*b^4 + 4
*a^4*b^5)*d)*sinh(d*x + c)^2 + (a^9 + 4*a^8*b + 6*a^7*b^2 + 4*a^6*b^3 + a^5*b^4)*d + 4*(3*(a^9 + 4*a^8*b + 6*a
^7*b^2 + 4*a^6*b^3 + a^5*b^4)*d*cosh(d*x + c)^11 + 5*(a^9 + 8*a^8*b + 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5*b^4 + 4
*a^4*b^5)*d*cosh(d*x + c)^9 - 2*(a^9 + 4*a^8*b - 10*a^7*b^2 - 60*a^6*b^3 - 95*a^5*b^4 - 64*a^4*b^5 - 16*a^3*b^
6)*d*cosh(d*x + c)^7 - 6*(a^9 + 8*a^8*b + 30*a^7*b^2 + 60*a^6*b^3 + 65*a^5*b^4 + 36*a^4*b^5 + 8*a^3*b^6)*d*cos
h(d*x + c)^5 - (a^9 + 4*a^8*b - 10*a^7*b^2 - 60*a^6*b^3 - 95*a^5*b^4 - 64*a^4*b^5 - 16*a^3*b^6)*d*cosh(d*x + c
)^3 + (a^9 + 8*a^8*b + 22*a^7*b^2 + 28*a^6*b^3 + 17*a^5*b^4 + 4*a^4*b^5)*d*cosh(d*x + c))*sinh(d*x + c))

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giac [B]  time = 2.47, size = 766, normalized size = 5.04 \[ \frac {\frac {{\left (6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}} + \frac {2 \, {\left (a e^{\left (2 \, c\right )} + 4 \, b e^{\left (2 \, c\right )}\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{4} e^{\left (2 \, c\right )} + 4 \, a^{3} b e^{\left (2 \, c\right )} + 6 \, a^{2} b^{2} e^{\left (2 \, c\right )} + 4 \, a b^{3} e^{\left (2 \, c\right )} + b^{4} e^{\left (2 \, c\right )}} - \frac {2 \, d x}{a^{3}} - \frac {a^{5} e^{\left (12 \, d x + 12 \, c\right )} + 3 \, a^{4} b e^{\left (12 \, d x + 12 \, c\right )} + 3 \, a^{3} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + a^{2} b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 6 \, a^{5} e^{\left (10 \, d x + 10 \, c\right )} + 14 \, a^{4} b e^{\left (10 \, d x + 10 \, c\right )} + 30 \, a^{3} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 10 \, a^{2} b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 15 \, a^{5} e^{\left (8 \, d x + 8 \, c\right )} + 29 \, a^{4} b e^{\left (8 \, d x + 8 \, c\right )} + 13 \, a^{3} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 47 \, a^{2} b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 8 \, a b^{4} e^{\left (8 \, d x + 8 \, c\right )} - 8 \, b^{5} e^{\left (8 \, d x + 8 \, c\right )} + 20 \, a^{5} e^{\left (6 \, d x + 6 \, c\right )} + 36 \, a^{4} b e^{\left (6 \, d x + 6 \, c\right )} - 28 \, a^{3} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 116 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 15 \, a^{5} e^{\left (4 \, d x + 4 \, c\right )} + 29 \, a^{4} b e^{\left (4 \, d x + 4 \, c\right )} + 13 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 47 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a^{5} e^{\left (2 \, d x + 2 \, c\right )} + 14 \, a^{4} b e^{\left (2 \, d x + 2 \, c\right )} + 30 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 10 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}}{{\left (a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right )} {\left (a e^{\left (6 \, d x + 6 \, c\right )} + a e^{\left (4 \, d x + 4 \, c\right )} + 4 \, b e^{\left (4 \, d x + 4 \, c\right )} - a e^{\left (2 \, d x + 2 \, c\right )} - 4 \, b e^{\left (2 \, d x + 2 \, c\right )} - a\right )}^{2}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((6*a^2*b^2 + 4*a*b^3 + b^4)*log(a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)/(a^7 +
4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4) + 2*(a*e^(2*c) + 4*b*e^(2*c))*log(abs(e^(2*d*x + 2*c) - 1))/(a^4*e
^(2*c) + 4*a^3*b*e^(2*c) + 6*a^2*b^2*e^(2*c) + 4*a*b^3*e^(2*c) + b^4*e^(2*c)) - 2*d*x/a^3 - (a^5*e^(12*d*x + 1
2*c) + 3*a^4*b*e^(12*d*x + 12*c) + 3*a^3*b^2*e^(12*d*x + 12*c) + a^2*b^3*e^(12*d*x + 12*c) + 6*a^5*e^(10*d*x +
10*c) + 14*a^4*b*e^(10*d*x + 10*c) + 30*a^3*b^2*e^(10*d*x + 10*c) + 10*a^2*b^3*e^(10*d*x + 10*c) + 15*a^5*e^(
8*d*x + 8*c) + 29*a^4*b*e^(8*d*x + 8*c) + 13*a^3*b^2*e^(8*d*x + 8*c) + 47*a^2*b^3*e^(8*d*x + 8*c) - 8*a*b^4*e^
(8*d*x + 8*c) - 8*b^5*e^(8*d*x + 8*c) + 20*a^5*e^(6*d*x + 6*c) + 36*a^4*b*e^(6*d*x + 6*c) - 28*a^3*b^2*e^(6*d*
x + 6*c) - 116*a^2*b^3*e^(6*d*x + 6*c) + 16*a*b^4*e^(6*d*x + 6*c) + 16*b^5*e^(6*d*x + 6*c) + 15*a^5*e^(4*d*x +
4*c) + 29*a^4*b*e^(4*d*x + 4*c) + 13*a^3*b^2*e^(4*d*x + 4*c) + 47*a^2*b^3*e^(4*d*x + 4*c) - 8*a*b^4*e^(4*d*x
+ 4*c) - 8*b^5*e^(4*d*x + 4*c) + 6*a^5*e^(2*d*x + 2*c) + 14*a^4*b*e^(2*d*x + 2*c) + 30*a^3*b^2*e^(2*d*x + 2*c)
+ 10*a^2*b^3*e^(2*d*x + 2*c) + a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)/((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*(a
*e^(6*d*x + 6*c) + a*e^(4*d*x + 4*c) + 4*b*e^(4*d*x + 4*c) - a*e^(2*d*x + 2*c) - 4*b*e^(2*d*x + 2*c) - a)^2))/
d

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maple [B]  time = 0.57, size = 1128, normalized size = 7.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/d*tanh(1/2*d*x+1/2*c)^2/(a^3+3*a^2*b+3*a*b^2+b^3)-1/d/a^3*ln(tanh(1/2*d*x+1/2*c)-1)-1/d/a^3*ln(tanh(1/2*d
*x+1/2*c)+1)-8/d*b^3/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh
(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^6-10/d*b^4/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*
c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/a*tanh(1/2*d*x+1/2*c)^6-2/d*b^5/(a+b)^4/a^2/(t
anh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh
(1/2*d*x+1/2*c)^6-16/d*b^3/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-
2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^4+8/d*b^4/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x
+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/a*tanh(1/2*d*x+1/2*c)^4+4/d*b^5/(a+b)^4/a
^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2
*tanh(1/2*d*x+1/2*c)^4-8/d*b^3/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^
2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d*x+1/2*c)^2-10/d*b^4/(a+b)^4/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/
2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/a*tanh(1/2*d*x+1/2*c)^2-2/d*b^5/(a+b
)^4/a^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a
+b)^2*tanh(1/2*d*x+1/2*c)^2+3/d*b^2/(a+b)^4/a*ln(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*
x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)+2/d*b^3/(a+b)^4/a^2*ln(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*
c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)+1/2/d*b^4/(a+b)^4/a^3*ln(tanh(1/2*d*x+1/2*c)^4*a
+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)-1/8/d/(a+b)^3/tanh(1/2*d*x+1
/2*c)^2+1/d/(a+b)^4*ln(tanh(1/2*d*x+1/2*c))*a+4/d/(a+b)^4*ln(tanh(1/2*d*x+1/2*c))*b

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maxima [B]  time = 0.54, size = 692, normalized size = 4.55 \[ \frac {{\left (6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, {\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} d} + \frac {{\left (a + 4 \, b\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d} + \frac {{\left (a + 4 \, b\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} d} - \frac {2 \, {\left ({\left (a^{5} - 4 \, a^{2} b^{3} - 2 \, a b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (2 \, a^{5} + 4 \, a^{4} b - 7 \, a b^{4} - 3 \, b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (3 \, a^{5} + 8 \, a^{4} b + 8 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + 16 \, a b^{4} + 6 \, b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 2 \, {\left (2 \, a^{5} + 4 \, a^{4} b - 7 \, a b^{4} - 3 \, b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + {\left (a^{5} - 4 \, a^{2} b^{3} - 2 \, a b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )}\right )}}{{\left (a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3} + 2 \, {\left (a^{8} + 7 \, a^{7} b + 15 \, a^{6} b^{2} + 13 \, a^{5} b^{3} + 4 \, a^{4} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (a^{8} + 3 \, a^{7} b - 13 \, a^{6} b^{2} - 47 \, a^{5} b^{3} - 48 \, a^{4} b^{4} - 16 \, a^{3} b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, {\left (a^{8} + 7 \, a^{7} b + 23 \, a^{6} b^{2} + 37 \, a^{5} b^{3} + 28 \, a^{4} b^{4} + 8 \, a^{3} b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - {\left (a^{8} + 3 \, a^{7} b - 13 \, a^{6} b^{2} - 47 \, a^{5} b^{3} - 48 \, a^{4} b^{4} - 16 \, a^{3} b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + 2 \, {\left (a^{8} + 7 \, a^{7} b + 15 \, a^{6} b^{2} + 13 \, a^{5} b^{3} + 4 \, a^{4} b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )} + {\left (a^{8} + 3 \, a^{7} b + 3 \, a^{6} b^{2} + a^{5} b^{3}\right )} e^{\left (-12 \, d x - 12 \, c\right )}\right )} d} + \frac {d x + c}{a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(6*a^2*b^2 + 4*a*b^3 + b^4)*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/((a^7 + 4*a^6*b + 6
*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d) + (a + 4*b)*log(e^(-d*x - c) + 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b
^4)*d) + (a + 4*b)*log(e^(-d*x - c) - 1)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) - 2*((a^5 - 4*a^2*b^3
- 2*a*b^4)*e^(-2*d*x - 2*c) + 2*(2*a^5 + 4*a^4*b - 7*a*b^4 - 3*b^5)*e^(-4*d*x - 4*c) + 2*(3*a^5 + 8*a^4*b + 8
*a^3*b^2 + 4*a^2*b^3 + 16*a*b^4 + 6*b^5)*e^(-6*d*x - 6*c) + 2*(2*a^5 + 4*a^4*b - 7*a*b^4 - 3*b^5)*e^(-8*d*x -
8*c) + (a^5 - 4*a^2*b^3 - 2*a*b^4)*e^(-10*d*x - 10*c))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3 + 2*(a^8 + 7*a^7*
b + 15*a^6*b^2 + 13*a^5*b^3 + 4*a^4*b^4)*e^(-2*d*x - 2*c) - (a^8 + 3*a^7*b - 13*a^6*b^2 - 47*a^5*b^3 - 48*a^4*
b^4 - 16*a^3*b^5)*e^(-4*d*x - 4*c) - 4*(a^8 + 7*a^7*b + 23*a^6*b^2 + 37*a^5*b^3 + 28*a^4*b^4 + 8*a^3*b^5)*e^(-
6*d*x - 6*c) - (a^8 + 3*a^7*b - 13*a^6*b^2 - 47*a^5*b^3 - 48*a^4*b^4 - 16*a^3*b^5)*e^(-8*d*x - 8*c) + 2*(a^8 +
7*a^7*b + 15*a^6*b^2 + 13*a^5*b^3 + 4*a^4*b^4)*e^(-10*d*x - 10*c) + (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*e^(
-12*d*x - 12*c))*d) + (d*x + c)/(a^3*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\mathrm {coth}\left (c+d\,x\right )}^3}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((cosh(c + d*x)^6*coth(c + d*x)^3)/(b + a*cosh(c + d*x)^2)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{3}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)**3/(a+b*sech(d*x+c)**2)**3,x)

[Out]

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

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