3.224 \(\int \text{csch}^{12}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)
Optimal. Leaf size=147 \[ -\frac{a \left (5 a^2+9 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}+\frac{a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac{a^2 (10 a+3 b) \coth ^7(c+d x)}{7 d}+\frac{a^2 (10 a+9 b) \coth ^5(c+d x)}{5 d}-\frac{a^3 \coth ^{11}(c+d x)}{11 d}+\frac{5 a^3 \coth ^9(c+d x)}{9 d}+b^3 x \]
[Out]
b^3*x + (a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x])/d - (a*(5*a^2 + 9*a*b + 3*b^2)*Coth[c + d*x]^3)/(3*d) + (a^2*(
10*a + 9*b)*Coth[c + d*x]^5)/(5*d) - (a^2*(10*a + 3*b)*Coth[c + d*x]^7)/(7*d) + (5*a^3*Coth[c + d*x]^9)/(9*d)
- (a^3*Coth[c + d*x]^11)/(11*d)
________________________________________________________________________________________
Rubi [A] time = 0.149097, antiderivative size = 147, 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 =
{3217, 1261, 207} \[ -\frac{a \left (5 a^2+9 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}+\frac{a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac{a^2 (10 a+3 b) \coth ^7(c+d x)}{7 d}+\frac{a^2 (10 a+9 b) \coth ^5(c+d x)}{5 d}-\frac{a^3 \coth ^{11}(c+d x)}{11 d}+\frac{5 a^3 \coth ^9(c+d x)}{9 d}+b^3 x \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^12*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
b^3*x + (a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x])/d - (a*(5*a^2 + 9*a*b + 3*b^2)*Coth[c + d*x]^3)/(3*d) + (a^2*(
10*a + 9*b)*Coth[c + d*x]^5)/(5*d) - (a^2*(10*a + 3*b)*Coth[c + d*x]^7)/(7*d) + (5*a^3*Coth[c + d*x]^9)/(9*d)
- (a^3*Coth[c + d*x]^11)/(11*d)
Rule 3217
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p)/(1 + ff^2
*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Rule 1261
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Rule 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \int \text{csch}^{12}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-2 a x^2+(a+b) x^4\right )^3}{x^{12} \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^3}{x^{12}}-\frac{5 a^3}{x^{10}}+\frac{a^2 (10 a+3 b)}{x^8}-\frac{a^2 (10 a+9 b)}{x^6}+\frac{a \left (5 a^2+9 a b+3 b^2\right )}{x^4}-\frac{a \left (a^2+3 a b+3 b^2\right )}{x^2}-\frac{b^3}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac{a \left (5 a^2+9 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}+\frac{a^2 (10 a+9 b) \coth ^5(c+d x)}{5 d}-\frac{a^2 (10 a+3 b) \coth ^7(c+d x)}{7 d}+\frac{5 a^3 \coth ^9(c+d x)}{9 d}-\frac{a^3 \coth ^{11}(c+d x)}{11 d}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=b^3 x+\frac{a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac{a \left (5 a^2+9 a b+3 b^2\right ) \coth ^3(c+d x)}{3 d}+\frac{a^2 (10 a+9 b) \coth ^5(c+d x)}{5 d}-\frac{a^2 (10 a+3 b) \coth ^7(c+d x)}{7 d}+\frac{5 a^3 \coth ^9(c+d x)}{9 d}-\frac{a^3 \coth ^{11}(c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 6.10498, size = 239, normalized size = 1.63 \[ \frac{\text{csch}^3(c+d x) \left (-2376 a^2 b \cosh (c+d x)-640 a^3 \cosh (c+d x)-3465 a b^2 \cosh (c+d x)\right )}{3465 d}+\frac{2 \text{csch}(c+d x) \left (2376 a^2 b \cosh (c+d x)+640 a^3 \cosh (c+d x)+3465 a b^2 \cosh (c+d x)\right )}{3465 d}+\frac{\text{csch}^7(c+d x) \left (-297 a^2 b \cosh (c+d x)-80 a^3 \cosh (c+d x)\right )}{693 d}+\frac{2 \text{csch}^5(c+d x) \left (297 a^2 b \cosh (c+d x)+80 a^3 \cosh (c+d x)\right )}{1155 d}-\frac{a^3 \coth (c+d x) \text{csch}^{10}(c+d x)}{11 d}+\frac{10 a^3 \coth (c+d x) \text{csch}^8(c+d x)}{99 d}+\frac{b^3 (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^12*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
(b^3*(c + d*x))/d + (2*(640*a^3*Cosh[c + d*x] + 2376*a^2*b*Cosh[c + d*x] + 3465*a*b^2*Cosh[c + d*x])*Csch[c +
d*x])/(3465*d) + ((-640*a^3*Cosh[c + d*x] - 2376*a^2*b*Cosh[c + d*x] - 3465*a*b^2*Cosh[c + d*x])*Csch[c + d*x]
^3)/(3465*d) + (2*(80*a^3*Cosh[c + d*x] + 297*a^2*b*Cosh[c + d*x])*Csch[c + d*x]^5)/(1155*d) + ((-80*a^3*Cosh[
c + d*x] - 297*a^2*b*Cosh[c + d*x])*Csch[c + d*x]^7)/(693*d) + (10*a^3*Coth[c + d*x]*Csch[c + d*x]^8)/(99*d) -
(a^3*Coth[c + d*x]*Csch[c + d*x]^10)/(11*d)
________________________________________________________________________________________
Maple [A] time = 0.086, size = 145, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{256}{693}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{10}}{11}}+{\frac{10\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{8}}{99}}-{\frac{80\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{693}}+{\frac{32\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{231}}-{\frac{128\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{693}} \right ){\rm coth} \left (dx+c\right )+3\,{a}^{2}b \left ({\frac{16}{35}}-1/7\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}+{\frac{6\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{35}}-{\frac{8\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{35}} \right ){\rm coth} \left (dx+c\right )+3\,a{b}^{2} \left ( 2/3-1/3\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2} \right ){\rm coth} \left (dx+c\right )+{b}^{3} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^12*(a+b*sinh(d*x+c)^4)^3,x)
[Out]
1/d*(a^3*(256/693-1/11*csch(d*x+c)^10+10/99*csch(d*x+c)^8-80/693*csch(d*x+c)^6+32/231*csch(d*x+c)^4-128/693*cs
ch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(16/35-1/7*csch(d*x+c)^6+6/35*csch(d*x+c)^4-8/35*csch(d*x+c)^2)*coth(d*x+c)+3
*a*b^2*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)+b^3*(d*x+c))
________________________________________________________________________________________
Maxima [B] time = 1.24004, size = 1743, normalized size = 11.86 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^12*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")
[Out]
b^3*x + 512/693*a^3*(11*e^(-2*d*x - 2*c)/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(-6*d*x - 6*c)
- 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x - 10*c) - 462*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-16
*d*x - 16*c) + 55*e^(-18*d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) - 55*e^(-4*d*x - 4*c)/
(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(-6*d*x - 6*c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x -
10*c) - 462*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-16*d*x - 16*c) + 55*e^(-18*d*x - 18*c) - 11
*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) + 165*e^(-6*d*x - 6*c)/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x -
4*c) + 165*e^(-6*d*x - 6*c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x - 10*c) - 462*e^(-12*d*x - 12*c) + 330*e^
(-14*d*x - 14*c) - 165*e^(-16*d*x - 16*c) + 55*e^(-18*d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c)
- 1)) - 330*e^(-8*d*x - 8*c)/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(-6*d*x - 6*c) - 330*e^(-8
*d*x - 8*c) + 462*e^(-10*d*x - 10*c) - 462*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-16*d*x - 16*c
) + 55*e^(-18*d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) + 462*e^(-10*d*x - 10*c)/(d*(11*e
^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(-6*d*x - 6*c) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x - 10*c) -
462*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-16*d*x - 16*c) + 55*e^(-18*d*x - 18*c) - 11*e^(-20*
d*x - 20*c) + e^(-22*d*x - 22*c) - 1)) - 1/(d*(11*e^(-2*d*x - 2*c) - 55*e^(-4*d*x - 4*c) + 165*e^(-6*d*x - 6*c
) - 330*e^(-8*d*x - 8*c) + 462*e^(-10*d*x - 10*c) - 462*e^(-12*d*x - 12*c) + 330*e^(-14*d*x - 14*c) - 165*e^(-
16*d*x - 16*c) + 55*e^(-18*d*x - 18*c) - 11*e^(-20*d*x - 20*c) + e^(-22*d*x - 22*c) - 1))) + 96/35*a^2*b*(7*e^
(-2*d*x - 2*c)/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e
^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1)) - 21*e^(-4*d*x - 4*c)/(d*(7*e^(-2*d*x - 2*
c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x -
12*c) + e^(-14*d*x - 14*c) - 1)) + 35*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6
*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1)) -
1/(d*(7*e^(-2*d*x - 2*c) - 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) - 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 1
0*c) - 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) - 1))) + 4*a*b^2*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) -
3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c
) - 1)))
________________________________________________________________________________________
Fricas [B] time = 1.72119, size = 4510, normalized size = 30.68 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^12*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")
[Out]
1/3465*(2*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)^11 + 22*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d
*x + c)*sinh(d*x + c)^10 + (3465*b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930*a*b^2)*sinh(d*x + c)^11 - 22*(640*a^3
+ 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)^9 - 11*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930*a*b^2 - 5*(3465*b
^3*d*x - 1280*a^3 - 4752*a^2*b - 6930*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 66*(5*(640*a^3 + 2376*a^2*b +
3465*a*b^2)*cosh(d*x + c)^3 - 3*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 + 110*(640*
a^3 + 2376*a^2*b + 3087*a*b^2)*cosh(d*x + c)^7 + 11*(17325*b^3*d*x + 30*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*b
- 6930*a*b^2)*cosh(d*x + c)^4 - 6400*a^3 - 23760*a^2*b - 34650*a*b^2 - 36*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*
b - 6930*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 154*(6*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)^5
- 12*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)^3 + 5*(640*a^3 + 2376*a^2*b + 3087*a*b^2)*cosh(d*x + c)
)*sinh(d*x + c)^6 - 330*(640*a^3 + 2376*a^2*b + 2415*a*b^2)*cosh(d*x + c)^5 + 33*(14*(3465*b^3*d*x - 1280*a^3
- 4752*a^2*b - 6930*a*b^2)*cosh(d*x + c)^6 - 17325*b^3*d*x - 42*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930*a
*b^2)*cosh(d*x + c)^4 + 6400*a^3 + 23760*a^2*b + 34650*a*b^2 + 35*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930
*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 22*(30*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)^7 - 126*(6
40*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)^5 + 175*(640*a^3 + 2376*a^2*b + 3087*a*b^2)*cosh(d*x + c)^3 -
75*(640*a^3 + 2376*a^2*b + 2415*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 660*(640*a^3 + 1872*a^2*b + 1533*a*b^2
)*cosh(d*x + c)^3 + 11*(15*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930*a*b^2)*cosh(d*x + c)^8 - 84*(3465*b^3*
d*x - 1280*a^3 - 4752*a^2*b - 6930*a*b^2)*cosh(d*x + c)^6 + 103950*b^3*d*x + 175*(3465*b^3*d*x - 1280*a^3 - 47
52*a^2*b - 6930*a*b^2)*cosh(d*x + c)^4 - 38400*a^3 - 142560*a^2*b - 207900*a*b^2 - 150*(3465*b^3*d*x - 1280*a^
3 - 4752*a^2*b - 6930*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 22*(5*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh
(d*x + c)^9 - 36*(640*a^3 + 2376*a^2*b + 3465*a*b^2)*cosh(d*x + c)^7 + 105*(640*a^3 + 2376*a^2*b + 3087*a*b^2)
*cosh(d*x + c)^5 - 150*(640*a^3 + 2376*a^2*b + 2415*a*b^2)*cosh(d*x + c)^3 + 90*(640*a^3 + 1872*a^2*b + 1533*a
*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 4620*(128*a^3 + 144*a^2*b + 105*a*b^2)*cosh(d*x + c) + 11*((3465*b^3*d*
x - 1280*a^3 - 4752*a^2*b - 6930*a*b^2)*cosh(d*x + c)^10 - 9*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930*a*b^
2)*cosh(d*x + c)^8 + 35*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930*a*b^2)*cosh(d*x + c)^6 - 145530*b^3*d*x -
75*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930*a*b^2)*cosh(d*x + c)^4 + 53760*a^3 + 199584*a^2*b + 291060*a*
b^2 + 90*(3465*b^3*d*x - 1280*a^3 - 4752*a^2*b - 6930*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*sinh(d*x + c)^
11 + 11*(5*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^9 + 11*(30*d*cosh(d*x + c)^4 - 36*d*cosh(d*x + c)^2 + 5*d)*sin
h(d*x + c)^7 + 33*(14*d*cosh(d*x + c)^6 - 42*d*cosh(d*x + c)^4 + 35*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x + c)^5 +
11*(15*d*cosh(d*x + c)^8 - 84*d*cosh(d*x + c)^6 + 175*d*cosh(d*x + c)^4 - 150*d*cosh(d*x + c)^2 + 30*d)*sinh(
d*x + c)^3 + 11*(d*cosh(d*x + c)^10 - 9*d*cosh(d*x + c)^8 + 35*d*cosh(d*x + c)^6 - 75*d*cosh(d*x + c)^4 + 90*d
*cosh(d*x + c)^2 - 42*d)*sinh(d*x + c))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)**12*(a+b*sinh(d*x+c)**4)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.6987, size = 485, normalized size = 3.3 \begin{align*} \frac{{\left (d x + c\right )} b^{3}}{d} - \frac{4 \,{\left (10395 \, a b^{2} e^{\left (18 \, d x + 18 \, c\right )} - 86625 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} + 83160 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 318780 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} - 382536 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} - 679140 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 295680 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 715176 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 921690 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 211200 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 700920 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 824670 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 105600 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 392040 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 485100 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 35200 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 130680 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 180180 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 7040 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 26136 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 38115 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 640 \, a^{3} - 2376 \, a^{2} b - 3465 \, a b^{2}\right )}}{3465 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^12*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")
[Out]
(d*x + c)*b^3/d - 4/3465*(10395*a*b^2*e^(18*d*x + 18*c) - 86625*a*b^2*e^(16*d*x + 16*c) + 83160*a^2*b*e^(14*d*
x + 14*c) + 318780*a*b^2*e^(14*d*x + 14*c) - 382536*a^2*b*e^(12*d*x + 12*c) - 679140*a*b^2*e^(12*d*x + 12*c) +
295680*a^3*e^(10*d*x + 10*c) + 715176*a^2*b*e^(10*d*x + 10*c) + 921690*a*b^2*e^(10*d*x + 10*c) - 211200*a^3*e
^(8*d*x + 8*c) - 700920*a^2*b*e^(8*d*x + 8*c) - 824670*a*b^2*e^(8*d*x + 8*c) + 105600*a^3*e^(6*d*x + 6*c) + 39
2040*a^2*b*e^(6*d*x + 6*c) + 485100*a*b^2*e^(6*d*x + 6*c) - 35200*a^3*e^(4*d*x + 4*c) - 130680*a^2*b*e^(4*d*x
+ 4*c) - 180180*a*b^2*e^(4*d*x + 4*c) + 7040*a^3*e^(2*d*x + 2*c) + 26136*a^2*b*e^(2*d*x + 2*c) + 38115*a*b^2*e
^(2*d*x + 2*c) - 640*a^3 - 2376*a^2*b - 3465*a*b^2)/(d*(e^(2*d*x + 2*c) - 1)^11)