Integrand size = 23, antiderivative size = 291 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=-\frac {a^3 x}{2}+\frac {105}{128} a b^2 x+\frac {3 a^2 b \cosh (c+d x)}{d}-\frac {b^3 \cosh (c+d x)}{d}-\frac {2 a^2 b \cosh ^3(c+d x)}{d}+\frac {5 b^3 \cosh ^3(c+d x)}{3 d}+\frac {3 a^2 b \cosh ^5(c+d x)}{5 d}-\frac {2 b^3 \cosh ^5(c+d x)}{d}+\frac {10 b^3 \cosh ^7(c+d x)}{7 d}-\frac {5 b^3 \cosh ^9(c+d x)}{9 d}+\frac {b^3 \cosh ^{11}(c+d x)}{11 d}+\frac {a^3 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {105 a b^2 \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {35 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{64 d}-\frac {7 a b^2 \cosh (c+d x) \sinh ^5(c+d x)}{16 d}+\frac {3 a b^2 \cosh (c+d x) \sinh ^7(c+d x)}{8 d} \]
-1/2*a^3*x+105/128*a*b^2*x+3*a^2*b*cosh(d*x+c)/d-b^3*cosh(d*x+c)/d-2*a^2*b *cosh(d*x+c)^3/d+5/3*b^3*cosh(d*x+c)^3/d+3/5*a^2*b*cosh(d*x+c)^5/d-2*b^3*c osh(d*x+c)^5/d+10/7*b^3*cosh(d*x+c)^7/d-5/9*b^3*cosh(d*x+c)^9/d+1/11*b^3*c osh(d*x+c)^11/d+1/2*a^3*cosh(d*x+c)*sinh(d*x+c)/d-105/128*a*b^2*cosh(d*x+c )*sinh(d*x+c)/d+35/64*a*b^2*cosh(d*x+c)*sinh(d*x+c)^3/d-7/16*a*b^2*cosh(d* x+c)*sinh(d*x+c)^5/d+3/8*a*b^2*cosh(d*x+c)*sinh(d*x+c)^7/d
Time = 6.77 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.67 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {-27720 a \left (64 a^2-105 b^2\right ) (c+d x)-20790 b \left (-320 a^2+77 b^2\right ) \cosh (c+d x)+34650 b \left (-32 a^2+11 b^2\right ) \cosh (3 (c+d x))-2079 b \left (-64 a^2+55 b^2\right ) \cosh (5 (c+d x))+27225 b^3 \cosh (7 (c+d x))-4235 b^3 \cosh (9 (c+d x))+315 b^3 \cosh (11 (c+d x))+110880 a \left (8 a^2-21 b^2\right ) \sinh (2 (c+d x))+582120 a b^2 \sinh (4 (c+d x))-110880 a b^2 \sinh (6 (c+d x))+10395 a b^2 \sinh (8 (c+d x))}{3548160 d} \]
(-27720*a*(64*a^2 - 105*b^2)*(c + d*x) - 20790*b*(-320*a^2 + 77*b^2)*Cosh[ c + d*x] + 34650*b*(-32*a^2 + 11*b^2)*Cosh[3*(c + d*x)] - 2079*b*(-64*a^2 + 55*b^2)*Cosh[5*(c + d*x)] + 27225*b^3*Cosh[7*(c + d*x)] - 4235*b^3*Cosh[ 9*(c + d*x)] + 315*b^3*Cosh[11*(c + d*x)] + 110880*a*(8*a^2 - 21*b^2)*Sinh [2*(c + d*x)] + 582120*a*b^2*Sinh[4*(c + d*x)] - 110880*a*b^2*Sinh[6*(c + d*x)] + 10395*a*b^2*Sinh[8*(c + d*x)])/(3548160*d)
Time = 0.51 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 25, 3699, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\sin (i c+i d x)^2 \left (a+i b \sin (i c+i d x)^3\right )^3dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \sin (i c+i d x)^2 \left (i b \sin (i c+i d x)^3+a\right )^3dx\) |
| \(\Big \downarrow \) 3699 |
| \(\displaystyle -\int \left (-b^3 \sinh ^{11}(c+d x)-3 a b^2 \sinh ^8(c+d x)-3 a^2 b \sinh ^5(c+d x)-a^3 \sinh ^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {a^3 x}{2}+\frac {3 a^2 b \cosh ^5(c+d x)}{5 d}-\frac {2 a^2 b \cosh ^3(c+d x)}{d}+\frac {3 a^2 b \cosh (c+d x)}{d}+\frac {3 a b^2 \sinh ^7(c+d x) \cosh (c+d x)}{8 d}-\frac {7 a b^2 \sinh ^5(c+d x) \cosh (c+d x)}{16 d}+\frac {35 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{64 d}-\frac {105 a b^2 \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac {105}{128} a b^2 x+\frac {b^3 \cosh ^{11}(c+d x)}{11 d}-\frac {5 b^3 \cosh ^9(c+d x)}{9 d}+\frac {10 b^3 \cosh ^7(c+d x)}{7 d}-\frac {2 b^3 \cosh ^5(c+d x)}{d}+\frac {5 b^3 \cosh ^3(c+d x)}{3 d}-\frac {b^3 \cosh (c+d x)}{d}\) |
-1/2*(a^3*x) + (105*a*b^2*x)/128 + (3*a^2*b*Cosh[c + d*x])/d - (b^3*Cosh[c + d*x])/d - (2*a^2*b*Cosh[c + d*x]^3)/d + (5*b^3*Cosh[c + d*x]^3)/(3*d) + (3*a^2*b*Cosh[c + d*x]^5)/(5*d) - (2*b^3*Cosh[c + d*x]^5)/d + (10*b^3*Cos h[c + d*x]^7)/(7*d) - (5*b^3*Cosh[c + d*x]^9)/(9*d) + (b^3*Cosh[c + d*x]^1 1)/(11*d) + (a^3*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) - (105*a*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + (35*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(64 *d) - (7*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^5)/(16*d) + (3*a*b^2*Cosh[c + d *x]*Sinh[c + d*x]^7)/(8*d)
3.2.61.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n) ^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4] || Gt Q[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Time = 3.96 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.65
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+3 a^{2} b \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )+3 a \,b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{7}}{8}-\frac {7 \sinh \left (d x +c \right )^{5}}{48}+\frac {35 \sinh \left (d x +c \right )^{3}}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )+b^{3} \left (-\frac {256}{693}+\frac {\sinh \left (d x +c \right )^{10}}{11}-\frac {10 \sinh \left (d x +c \right )^{8}}{99}+\frac {80 \sinh \left (d x +c \right )^{6}}{693}-\frac {32 \sinh \left (d x +c \right )^{4}}{231}+\frac {128 \sinh \left (d x +c \right )^{2}}{693}\right ) \cosh \left (d x +c \right )}{d}\) | \(188\) |
| default | \(\frac {a^{3} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+3 a^{2} b \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )+3 a \,b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{7}}{8}-\frac {7 \sinh \left (d x +c \right )^{5}}{48}+\frac {35 \sinh \left (d x +c \right )^{3}}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )+b^{3} \left (-\frac {256}{693}+\frac {\sinh \left (d x +c \right )^{10}}{11}-\frac {10 \sinh \left (d x +c \right )^{8}}{99}+\frac {80 \sinh \left (d x +c \right )^{6}}{693}-\frac {32 \sinh \left (d x +c \right )^{4}}{231}+\frac {128 \sinh \left (d x +c \right )^{2}}{693}\right ) \cosh \left (d x +c \right )}{d}\) | \(188\) |
| parts | \(\frac {a^{3} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d}+\frac {b^{3} \left (-\frac {256}{693}+\frac {\sinh \left (d x +c \right )^{10}}{11}-\frac {10 \sinh \left (d x +c \right )^{8}}{99}+\frac {80 \sinh \left (d x +c \right )^{6}}{693}-\frac {32 \sinh \left (d x +c \right )^{4}}{231}+\frac {128 \sinh \left (d x +c \right )^{2}}{693}\right ) \cosh \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{7}}{8}-\frac {7 \sinh \left (d x +c \right )^{5}}{48}+\frac {35 \sinh \left (d x +c \right )^{3}}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {3 a^{2} b \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )}{d}\) | \(196\) |
| parallelrisch | \(\frac {\left (-1108800 a^{2} b +381150 b^{3}\right ) \cosh \left (3 d x +3 c \right )+\left (133056 a^{2} b -114345 b^{3}\right ) \cosh \left (5 d x +5 c \right )+\left (887040 a^{3}-2328480 a \,b^{2}\right ) \sinh \left (2 d x +2 c \right )+315 b^{3} \cosh \left (11 d x +11 c \right )+27225 b^{3} \cosh \left (7 d x +7 c \right )-4235 b^{3} \cosh \left (9 d x +9 c \right )+582120 a \,b^{2} \sinh \left (4 d x +4 c \right )-110880 a \,b^{2} \sinh \left (6 d x +6 c \right )+10395 a \,b^{2} \sinh \left (8 d x +8 c \right )+\left (6652800 a^{2} b -1600830 b^{3}\right ) \cosh \left (d x +c \right )-1774080 x \,a^{3} d +2910600 x a \,b^{2} d +5677056 a^{2} b -1310720 b^{3}}{3548160 d}\) | \(205\) |
| risch | \(\frac {3 b \,{\mathrm e}^{-5 d x -5 c} a^{2}}{160 d}+\frac {3 b \,{\mathrm e}^{5 d x +5 c} a^{2}}{160 d}+\frac {105 a \,b^{2} x}{128}-\frac {a^{3} x}{2}+\frac {{\mathrm e}^{2 d x +2 c} a^{3}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{3}}{8 d}-\frac {3 b^{2} {\mathrm e}^{-8 d x -8 c} a}{2048 d}+\frac {21 \,{\mathrm e}^{4 d x +4 c} a \,b^{2}}{256 d}-\frac {21 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{64 d}+\frac {21 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{64 d}-\frac {21 \,{\mathrm e}^{-4 d x -4 c} a \,b^{2}}{256 d}+\frac {b^{2} {\mathrm e}^{-6 d x -6 c} a}{64 d}+\frac {3 b^{2} {\mathrm e}^{8 d x +8 c} a}{2048 d}-\frac {b^{2} {\mathrm e}^{6 d x +6 c} a}{64 d}+\frac {b^{3} {\mathrm e}^{11 d x +11 c}}{22528 d}-\frac {11 b^{3} {\mathrm e}^{9 d x +9 c}}{18432 d}-\frac {11 b^{3} {\mathrm e}^{-9 d x -9 c}}{18432 d}+\frac {b^{3} {\mathrm e}^{-11 d x -11 c}}{22528 d}+\frac {55 b^{3} {\mathrm e}^{7 d x +7 c}}{14336 d}-\frac {33 b^{3} {\mathrm e}^{5 d x +5 c}}{2048 d}+\frac {55 \,{\mathrm e}^{3 d x +3 c} b^{3}}{1024 d}-\frac {231 \,{\mathrm e}^{d x +c} b^{3}}{1024 d}-\frac {231 \,{\mathrm e}^{-d x -c} b^{3}}{1024 d}+\frac {55 \,{\mathrm e}^{-3 d x -3 c} b^{3}}{1024 d}-\frac {33 b^{3} {\mathrm e}^{-5 d x -5 c}}{2048 d}+\frac {55 b^{3} {\mathrm e}^{-7 d x -7 c}}{14336 d}+\frac {15 \,{\mathrm e}^{-d x -c} a^{2} b}{16 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} a^{2} b}{32 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} a^{2} b}{32 d}+\frac {15 \,{\mathrm e}^{d x +c} a^{2} b}{16 d}\) | \(473\) |
1/d*(a^3*(1/2*sinh(d*x+c)*cosh(d*x+c)-1/2*d*x-1/2*c)+3*a^2*b*(8/15+1/5*sin h(d*x+c)^4-4/15*sinh(d*x+c)^2)*cosh(d*x+c)+3*a*b^2*((1/8*sinh(d*x+c)^7-7/4 8*sinh(d*x+c)^5+35/192*sinh(d*x+c)^3-35/128*sinh(d*x+c))*cosh(d*x+c)+35/12 8*d*x+35/128*c)+b^3*(-256/693+1/11*sinh(d*x+c)^10-10/99*sinh(d*x+c)^8+80/6 93*sinh(d*x+c)^6-32/231*sinh(d*x+c)^4+128/693*sinh(d*x+c)^2)*cosh(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (267) = 534\).
Time = 0.27 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.95 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {315 \, b^{3} \cosh \left (d x + c\right )^{11} + 3465 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{10} - 4235 \, b^{3} \cosh \left (d x + c\right )^{9} + 83160 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 27225 \, b^{3} \cosh \left (d x + c\right )^{7} + 3465 \, {\left (15 \, b^{3} \cosh \left (d x + c\right )^{3} - 11 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} + 1155 \, {\left (126 \, b^{3} \cosh \left (d x + c\right )^{5} - 308 \, b^{3} \cosh \left (d x + c\right )^{3} + 165 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 2079 \, {\left (64 \, a^{2} b - 55 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 83160 \, {\left (7 \, a b^{2} \cosh \left (d x + c\right )^{3} - 8 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 3465 \, {\left (30 \, b^{3} \cosh \left (d x + c\right )^{7} - 154 \, b^{3} \cosh \left (d x + c\right )^{5} + 275 \, b^{3} \cosh \left (d x + c\right )^{3} + 3 \, {\left (64 \, a^{2} b - 55 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} - 34650 \, {\left (32 \, a^{2} b - 11 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 27720 \, {\left (21 \, a b^{2} \cosh \left (d x + c\right )^{5} - 80 \, a b^{2} \cosh \left (d x + c\right )^{3} + 84 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 27720 \, {\left (64 \, a^{3} - 105 \, a b^{2}\right )} d x + 3465 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{9} - 44 \, b^{3} \cosh \left (d x + c\right )^{7} + 165 \, b^{3} \cosh \left (d x + c\right )^{5} + 6 \, {\left (64 \, a^{2} b - 55 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 30 \, {\left (32 \, a^{2} b - 11 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 20790 \, {\left (320 \, a^{2} b - 77 \, b^{3}\right )} \cosh \left (d x + c\right ) + 27720 \, {\left (3 \, a b^{2} \cosh \left (d x + c\right )^{7} - 24 \, a b^{2} \cosh \left (d x + c\right )^{5} + 84 \, a b^{2} \cosh \left (d x + c\right )^{3} + 8 \, {\left (8 \, a^{3} - 21 \, a b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{3548160 \, d} \]
1/3548160*(315*b^3*cosh(d*x + c)^11 + 3465*b^3*cosh(d*x + c)*sinh(d*x + c) ^10 - 4235*b^3*cosh(d*x + c)^9 + 83160*a*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + 27225*b^3*cosh(d*x + c)^7 + 3465*(15*b^3*cosh(d*x + c)^3 - 11*b^3*cosh( d*x + c))*sinh(d*x + c)^8 + 1155*(126*b^3*cosh(d*x + c)^5 - 308*b^3*cosh(d *x + c)^3 + 165*b^3*cosh(d*x + c))*sinh(d*x + c)^6 + 2079*(64*a^2*b - 55*b ^3)*cosh(d*x + c)^5 + 83160*(7*a*b^2*cosh(d*x + c)^3 - 8*a*b^2*cosh(d*x + c))*sinh(d*x + c)^5 + 3465*(30*b^3*cosh(d*x + c)^7 - 154*b^3*cosh(d*x + c) ^5 + 275*b^3*cosh(d*x + c)^3 + 3*(64*a^2*b - 55*b^3)*cosh(d*x + c))*sinh(d *x + c)^4 - 34650*(32*a^2*b - 11*b^3)*cosh(d*x + c)^3 + 27720*(21*a*b^2*co sh(d*x + c)^5 - 80*a*b^2*cosh(d*x + c)^3 + 84*a*b^2*cosh(d*x + c))*sinh(d* x + c)^3 - 27720*(64*a^3 - 105*a*b^2)*d*x + 3465*(5*b^3*cosh(d*x + c)^9 - 44*b^3*cosh(d*x + c)^7 + 165*b^3*cosh(d*x + c)^5 + 6*(64*a^2*b - 55*b^3)*c osh(d*x + c)^3 - 30*(32*a^2*b - 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 2 0790*(320*a^2*b - 77*b^3)*cosh(d*x + c) + 27720*(3*a*b^2*cosh(d*x + c)^7 - 24*a*b^2*cosh(d*x + c)^5 + 84*a*b^2*cosh(d*x + c)^3 + 8*(8*a^3 - 21*a*b^2 )*cosh(d*x + c))*sinh(d*x + c))/d
Time = 1.89 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.71 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\begin {cases} \frac {a^{3} x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {a^{3} x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {a^{3} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} + \frac {3 a^{2} b \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {8 a^{2} b \cosh ^{5}{\left (c + d x \right )}}{5 d} + \frac {105 a b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac {105 a b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac {315 a b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac {105 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac {105 a b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac {279 a b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} - \frac {511 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac {385 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{128 d} - \frac {105 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} + \frac {b^{3} \sinh ^{10}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {10 b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {16 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{3 d} - \frac {32 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{7 d} + \frac {128 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{63 d} - \frac {256 b^{3} \cosh ^{11}{\left (c + d x \right )}}{693 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right )^{3} \sinh ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a**3*x*sinh(c + d*x)**2/2 - a**3*x*cosh(c + d*x)**2/2 + a**3*si nh(c + d*x)*cosh(c + d*x)/(2*d) + 3*a**2*b*sinh(c + d*x)**4*cosh(c + d*x)/ d - 4*a**2*b*sinh(c + d*x)**2*cosh(c + d*x)**3/d + 8*a**2*b*cosh(c + d*x)* *5/(5*d) + 105*a*b**2*x*sinh(c + d*x)**8/128 - 105*a*b**2*x*sinh(c + d*x)* *6*cosh(c + d*x)**2/32 + 315*a*b**2*x*sinh(c + d*x)**4*cosh(c + d*x)**4/64 - 105*a*b**2*x*sinh(c + d*x)**2*cosh(c + d*x)**6/32 + 105*a*b**2*x*cosh(c + d*x)**8/128 + 279*a*b**2*sinh(c + d*x)**7*cosh(c + d*x)/(128*d) - 511*a *b**2*sinh(c + d*x)**5*cosh(c + d*x)**3/(128*d) + 385*a*b**2*sinh(c + d*x) **3*cosh(c + d*x)**5/(128*d) - 105*a*b**2*sinh(c + d*x)*cosh(c + d*x)**7/( 128*d) + b**3*sinh(c + d*x)**10*cosh(c + d*x)/d - 10*b**3*sinh(c + d*x)**8 *cosh(c + d*x)**3/(3*d) + 16*b**3*sinh(c + d*x)**6*cosh(c + d*x)**5/(3*d) - 32*b**3*sinh(c + d*x)**4*cosh(c + d*x)**7/(7*d) + 128*b**3*sinh(c + d*x) **2*cosh(c + d*x)**9/(63*d) - 256*b**3*cosh(c + d*x)**11/(693*d), Ne(d, 0) ), (x*(a + b*sinh(c)**3)**3*sinh(c)**2, True))
Time = 0.19 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.33 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=-\frac {1}{8} \, a^{3} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{1419264} \, b^{3} {\left (\frac {{\left (847 \, e^{\left (-2 \, d x - 2 \, c\right )} - 5445 \, e^{\left (-4 \, d x - 4 \, c\right )} + 22869 \, e^{\left (-6 \, d x - 6 \, c\right )} - 76230 \, e^{\left (-8 \, d x - 8 \, c\right )} + 320166 \, e^{\left (-10 \, d x - 10 \, c\right )} - 63\right )} e^{\left (11 \, d x + 11 \, c\right )}}{d} + \frac {320166 \, e^{\left (-d x - c\right )} - 76230 \, e^{\left (-3 \, d x - 3 \, c\right )} + 22869 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5445 \, e^{\left (-7 \, d x - 7 \, c\right )} + 847 \, e^{\left (-9 \, d x - 9 \, c\right )} - 63 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d}\right )} - \frac {1}{2048} \, a b^{2} {\left (\frac {{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac {1680 \, {\left (d x + c\right )}}{d} - \frac {672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} + \frac {1}{160} \, a^{2} b {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} \]
-1/8*a^3*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/1419264*b^3*(( 847*e^(-2*d*x - 2*c) - 5445*e^(-4*d*x - 4*c) + 22869*e^(-6*d*x - 6*c) - 76 230*e^(-8*d*x - 8*c) + 320166*e^(-10*d*x - 10*c) - 63)*e^(11*d*x + 11*c)/d + (320166*e^(-d*x - c) - 76230*e^(-3*d*x - 3*c) + 22869*e^(-5*d*x - 5*c) - 5445*e^(-7*d*x - 7*c) + 847*e^(-9*d*x - 9*c) - 63*e^(-11*d*x - 11*c))/d) - 1/2048*a*b^2*((32*e^(-2*d*x - 2*c) - 168*e^(-4*d*x - 4*c) + 672*e^(-6*d *x - 6*c) - 3)*e^(8*d*x + 8*c)/d - 1680*(d*x + c)/d - (672*e^(-2*d*x - 2*c ) - 168*e^(-4*d*x - 4*c) + 32*e^(-6*d*x - 6*c) - 3*e^(-8*d*x - 8*c))/d) + 1/160*a^2*b*(3*e^(5*d*x + 5*c)/d - 25*e^(3*d*x + 3*c)/d + 150*e^(d*x + c)/ d + 150*e^(-d*x - c)/d - 25*e^(-3*d*x - 3*c)/d + 3*e^(-5*d*x - 5*c)/d)
Time = 0.40 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.48 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {b^{3} e^{\left (11 \, d x + 11 \, c\right )}}{22528 \, d} - \frac {11 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )}}{18432 \, d} + \frac {3 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )}}{2048 \, d} + \frac {55 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )}}{14336 \, d} - \frac {a b^{2} e^{\left (6 \, d x + 6 \, c\right )}}{64 \, d} + \frac {21 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )}}{256 \, d} - \frac {21 \, a b^{2} e^{\left (-4 \, d x - 4 \, c\right )}}{256 \, d} + \frac {a b^{2} e^{\left (-6 \, d x - 6 \, c\right )}}{64 \, d} + \frac {55 \, b^{3} e^{\left (-7 \, d x - 7 \, c\right )}}{14336 \, d} - \frac {3 \, a b^{2} e^{\left (-8 \, d x - 8 \, c\right )}}{2048 \, d} - \frac {11 \, b^{3} e^{\left (-9 \, d x - 9 \, c\right )}}{18432 \, d} + \frac {b^{3} e^{\left (-11 \, d x - 11 \, c\right )}}{22528 \, d} - \frac {1}{128} \, {\left (64 \, a^{3} - 105 \, a b^{2}\right )} x + \frac {3 \, {\left (64 \, a^{2} b - 55 \, b^{3}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{10240 \, d} - \frac {5 \, {\left (32 \, a^{2} b - 11 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{1024 \, d} + \frac {{\left (8 \, a^{3} - 21 \, a b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{64 \, d} + \frac {3 \, {\left (320 \, a^{2} b - 77 \, b^{3}\right )} e^{\left (d x + c\right )}}{1024 \, d} + \frac {3 \, {\left (320 \, a^{2} b - 77 \, b^{3}\right )} e^{\left (-d x - c\right )}}{1024 \, d} - \frac {{\left (8 \, a^{3} - 21 \, a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{64 \, d} - \frac {5 \, {\left (32 \, a^{2} b - 11 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{1024 \, d} + \frac {3 \, {\left (64 \, a^{2} b - 55 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{10240 \, d} \]
1/22528*b^3*e^(11*d*x + 11*c)/d - 11/18432*b^3*e^(9*d*x + 9*c)/d + 3/2048* a*b^2*e^(8*d*x + 8*c)/d + 55/14336*b^3*e^(7*d*x + 7*c)/d - 1/64*a*b^2*e^(6 *d*x + 6*c)/d + 21/256*a*b^2*e^(4*d*x + 4*c)/d - 21/256*a*b^2*e^(-4*d*x - 4*c)/d + 1/64*a*b^2*e^(-6*d*x - 6*c)/d + 55/14336*b^3*e^(-7*d*x - 7*c)/d - 3/2048*a*b^2*e^(-8*d*x - 8*c)/d - 11/18432*b^3*e^(-9*d*x - 9*c)/d + 1/225 28*b^3*e^(-11*d*x - 11*c)/d - 1/128*(64*a^3 - 105*a*b^2)*x + 3/10240*(64*a ^2*b - 55*b^3)*e^(5*d*x + 5*c)/d - 5/1024*(32*a^2*b - 11*b^3)*e^(3*d*x + 3 *c)/d + 1/64*(8*a^3 - 21*a*b^2)*e^(2*d*x + 2*c)/d + 3/1024*(320*a^2*b - 77 *b^3)*e^(d*x + c)/d + 3/1024*(320*a^2*b - 77*b^3)*e^(-d*x - c)/d - 1/64*(8 *a^3 - 21*a*b^2)*e^(-2*d*x - 2*c)/d - 5/1024*(32*a^2*b - 11*b^3)*e^(-3*d*x - 3*c)/d + 3/10240*(64*a^2*b - 55*b^3)*e^(-5*d*x - 5*c)/d
Time = 1.88 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.79 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {\frac {\mathrm {sinh}\left (c+d\,x\right )\,a^3\,\mathrm {cosh}\left (c+d\,x\right )}{2}-\frac {d\,x\,a^3}{2}+\frac {3\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-2\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,a^2\,b\,\mathrm {cosh}\left (c+d\,x\right )+\frac {3\,\mathrm {sinh}\left (c+d\,x\right )\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{8}-\frac {25\,\mathrm {sinh}\left (c+d\,x\right )\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{16}+\frac {163\,\mathrm {sinh}\left (c+d\,x\right )\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{64}-\frac {279\,\mathrm {sinh}\left (c+d\,x\right )\,a\,b^2\,\mathrm {cosh}\left (c+d\,x\right )}{128}+\frac {105\,d\,x\,a\,b^2}{128}+\frac {b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}-\frac {5\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{9}+\frac {10\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}-2\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^5+\frac {5\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}-b^3\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]
((5*b^3*cosh(c + d*x)^3)/3 - b^3*cosh(c + d*x) - 2*b^3*cosh(c + d*x)^5 + ( 10*b^3*cosh(c + d*x)^7)/7 - (5*b^3*cosh(c + d*x)^9)/9 + (b^3*cosh(c + d*x) ^11)/11 - 2*a^2*b*cosh(c + d*x)^3 + (3*a^2*b*cosh(c + d*x)^5)/5 + 3*a^2*b* cosh(c + d*x) + (a^3*cosh(c + d*x)*sinh(c + d*x))/2 - (a^3*d*x)/2 - (279*a *b^2*cosh(c + d*x)*sinh(c + d*x))/128 + (105*a*b^2*d*x)/128 + (163*a*b^2*c osh(c + d*x)^3*sinh(c + d*x))/64 - (25*a*b^2*cosh(c + d*x)^5*sinh(c + d*x) )/16 + (3*a*b^2*cosh(c + d*x)^7*sinh(c + d*x))/8)/d