Integrand size = 23, antiderivative size = 152 \[ \int \text {csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {9}{8} a b^2 x+\frac {3 a^2 b \cosh (c+d x)}{d}-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{d}-\frac {3 b^3 \cosh ^5(c+d x)}{5 d}+\frac {b^3 \cosh ^7(c+d x)}{7 d}-\frac {a^3 \coth (c+d x)}{d}-\frac {9 a b^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {3 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d} \]
9/8*a*b^2*x+3*a^2*b*cosh(d*x+c)/d-b^3*cosh(d*x+c)/d+b^3*cosh(d*x+c)^3/d-3/ 5*b^3*cosh(d*x+c)^5/d+1/7*b^3*cosh(d*x+c)^7/d-a^3*coth(d*x+c)/d-9/8*a*b^2* cosh(d*x+c)*sinh(d*x+c)/d+3/4*a*b^2*cosh(d*x+c)*sinh(d*x+c)^3/d
Time = 4.59 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.92 \[ \int \text {csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {2520 a b^2 c+2520 a b^2 d x+35 b \left (192 a^2-35 b^2\right ) \cosh (c+d x)+245 b^3 \cosh (3 (c+d x))-49 b^3 \cosh (5 (c+d x))+5 b^3 \cosh (7 (c+d x))-1120 a^3 \coth \left (\frac {1}{2} (c+d x)\right )-1680 a b^2 \sinh (2 (c+d x))+210 a b^2 \sinh (4 (c+d x))-1120 a^3 \tanh \left (\frac {1}{2} (c+d x)\right )}{2240 d} \]
(2520*a*b^2*c + 2520*a*b^2*d*x + 35*b*(192*a^2 - 35*b^2)*Cosh[c + d*x] + 2 45*b^3*Cosh[3*(c + d*x)] - 49*b^3*Cosh[5*(c + d*x)] + 5*b^3*Cosh[7*(c + d* x)] - 1120*a^3*Coth[(c + d*x)/2] - 1680*a*b^2*Sinh[2*(c + d*x)] + 210*a*b^ 2*Sinh[4*(c + d*x)] - 1120*a^3*Tanh[(c + d*x)/2])/(2240*d)
Time = 0.38 (sec) , antiderivative size = 152, 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.
| \(\displaystyle \int \text {csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a+i b \sin (i c+i d x)^3\right )^3}{\sin (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (i b \sin (i c+i d x)^3+a\right )^3}{\sin (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 3699 |
| \(\displaystyle -\int \left (-b^3 \sinh ^7(c+d x)-3 a b^2 \sinh ^4(c+d x)-3 a^2 b \sinh (c+d x)-a^3 \text {csch}^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \coth (c+d x)}{d}+\frac {3 a^2 b \cosh (c+d x)}{d}+\frac {3 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {9 a b^2 \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {9}{8} a b^2 x+\frac {b^3 \cosh ^7(c+d x)}{7 d}-\frac {3 b^3 \cosh ^5(c+d x)}{5 d}+\frac {b^3 \cosh ^3(c+d x)}{d}-\frac {b^3 \cosh (c+d x)}{d}\) |
(9*a*b^2*x)/8 + (3*a^2*b*Cosh[c + d*x])/d - (b^3*Cosh[c + d*x])/d + (b^3*C osh[c + d*x]^3)/d - (3*b^3*Cosh[c + d*x]^5)/(5*d) + (b^3*Cosh[c + d*x]^7)/ (7*d) - (a^3*Coth[c + d*x])/d - (9*a*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(8*d ) + (3*a*b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)
3.2.65.3.1 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 = 0.92 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.73
| method | result | size |
| derivativedivides | \(\frac {-a^{3} \coth \left (d x +c \right )+3 a^{2} b \cosh \left (d x +c \right )+3 a \,b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b^{3} \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}\) | \(111\) |
| default | \(\frac {-a^{3} \coth \left (d x +c \right )+3 a^{2} b \cosh \left (d x +c \right )+3 a \,b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b^{3} \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}\) | \(111\) |
| parallelrisch | \(\frac {-2240 a^{3} \coth \left (\frac {d x}{2}+\frac {c}{2}\right )+1120 \,\operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3}+245 b^{3} \cosh \left (3 d x +3 c \right )-49 b^{3} \cosh \left (5 d x +5 c \right )+5 b^{3} \cosh \left (7 d x +7 c \right )+6720 b \left (-\frac {a b \sinh \left (2 d x +2 c \right )}{4}+\frac {a b \sinh \left (4 d x +4 c \right )}{32}+\left (a^{2}-\frac {35 b^{2}}{192}\right ) \cosh \left (d x +c \right )+\frac {3 a b d x}{8}+a^{2}-\frac {16 b^{2}}{105}\right )}{2240 d}\) | \(146\) |
| risch | \(\frac {9 a \,b^{2} x}{8}+\frac {b^{3} {\mathrm e}^{7 d x +7 c}}{896 d}-\frac {7 b^{3} {\mathrm e}^{5 d x +5 c}}{640 d}+\frac {3 \,{\mathrm e}^{4 d x +4 c} a \,b^{2}}{64 d}+\frac {7 \,{\mathrm e}^{3 d x +3 c} b^{3}}{128 d}-\frac {3 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{8 d}+\frac {3 \,{\mathrm e}^{d x +c} a^{2} b}{2 d}-\frac {35 \,{\mathrm e}^{d x +c} b^{3}}{128 d}+\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{2 d}-\frac {35 \,{\mathrm e}^{-d x -c} b^{3}}{128 d}+\frac {3 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{8 d}+\frac {7 \,{\mathrm e}^{-3 d x -3 c} b^{3}}{128 d}-\frac {3 \,{\mathrm e}^{-4 d x -4 c} a \,b^{2}}{64 d}-\frac {7 b^{3} {\mathrm e}^{-5 d x -5 c}}{640 d}+\frac {b^{3} {\mathrm e}^{-7 d x -7 c}}{896 d}-\frac {2 a^{3}}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )}\) | \(268\) |
1/d*(-a^3*coth(d*x+c)+3*a^2*b*cosh(d*x+c)+3*a*b^2*((1/4*sinh(d*x+c)^3-3/8* sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c)+b^3*(-16/35+1/7*sinh(d*x+c)^6-6/35 *sinh(d*x+c)^4+8/35*sinh(d*x+c)^2)*cosh(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (142) = 284\).
Time = 0.28 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.99 \[ \int \text {csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {20 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 105 \, a b^{2} \cosh \left (d x + c\right )^{5} + 525 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - 945 \, a b^{2} \cosh \left (d x + c\right )^{3} + 2 \, {\left (70 \, b^{3} \cosh \left (d x + c\right )^{3} - 81 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 4 \, {\left (35 \, b^{3} \cosh \left (d x + c\right )^{5} - 135 \, b^{3} \cosh \left (d x + c\right )^{3} + 147 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 105 \, {\left (10 \, a b^{2} \cosh \left (d x + c\right )^{3} - 27 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 280 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cosh \left (d x + c\right ) + 2 \, {\left (10 \, b^{3} \cosh \left (d x + c\right )^{7} - 81 \, b^{3} \cosh \left (d x + c\right )^{5} + 294 \, b^{3} \cosh \left (d x + c\right )^{3} + 1260 \, a b^{2} d x + 1120 \, a^{3} + 105 \, {\left (32 \, a^{2} b - 7 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2240 \, d \sinh \left (d x + c\right )} \]
1/2240*(20*b^3*cosh(d*x + c)*sinh(d*x + c)^7 + 105*a*b^2*cosh(d*x + c)^5 + 525*a*b^2*cosh(d*x + c)*sinh(d*x + c)^4 - 945*a*b^2*cosh(d*x + c)^3 + 2*( 70*b^3*cosh(d*x + c)^3 - 81*b^3*cosh(d*x + c))*sinh(d*x + c)^5 + 4*(35*b^3 *cosh(d*x + c)^5 - 135*b^3*cosh(d*x + c)^3 + 147*b^3*cosh(d*x + c))*sinh(d *x + c)^3 + 105*(10*a*b^2*cosh(d*x + c)^3 - 27*a*b^2*cosh(d*x + c))*sinh(d *x + c)^2 - 280*(8*a^3 - 3*a*b^2)*cosh(d*x + c) + 2*(10*b^3*cosh(d*x + c)^ 7 - 81*b^3*cosh(d*x + c)^5 + 294*b^3*cosh(d*x + c)^3 + 1260*a*b^2*d*x + 11 20*a^3 + 105*(32*a^2*b - 7*b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*sinh(d*x + c))
Timed out. \[ \int \text {csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.45 \[ \int \text {csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {3}{64} \, a b^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{4480} \, b^{3} {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {2 \, a^{3}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
3/64*a*b^2*(24*x + e^(4*d*x + 4*c)/d - 8*e^(2*d*x + 2*c)/d + 8*e^(-2*d*x - 2*c)/d - e^(-4*d*x - 4*c)/d) - 1/4480*b^3*((49*e^(-2*d*x - 2*c) - 245*e^( -4*d*x - 4*c) + 1225*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (1225*e^(-d *x - c) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c)) /d) + 3/2*a^2*b*(e^(d*x + c)/d + e^(-d*x - c)/d) + 2*a^3/(d*(e^(-2*d*x - 2 *c) - 1))
Time = 0.40 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.82 \[ \int \text {csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {5040 \, {\left (d x + c\right )} a b^{2} + 5 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 49 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 210 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 245 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 1680 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6720 \, a^{2} b e^{\left (d x + c\right )} - 1225 \, b^{3} e^{\left (d x + c\right )} - \frac {{\left (1890 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 294 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 210 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 54 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3} - 35 \, {\left (192 \, a^{2} b - 35 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )} + 560 \, {\left (16 \, a^{3} - 3 \, a b^{2}\right )} e^{\left (7 \, d x + 7 \, c\right )} + 210 \, {\left (32 \, a^{2} b - 7 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )} {\left (e^{\left (d x + c\right )} - 1\right )}}}{4480 \, d} \]
1/4480*(5040*(d*x + c)*a*b^2 + 5*b^3*e^(7*d*x + 7*c) - 49*b^3*e^(5*d*x + 5 *c) + 210*a*b^2*e^(4*d*x + 4*c) + 245*b^3*e^(3*d*x + 3*c) - 1680*a*b^2*e^( 2*d*x + 2*c) + 6720*a^2*b*e^(d*x + c) - 1225*b^3*e^(d*x + c) - (1890*a*b^2 *e^(5*d*x + 5*c) + 294*b^3*e^(4*d*x + 4*c) - 210*a*b^2*e^(3*d*x + 3*c) - 5 4*b^3*e^(2*d*x + 2*c) + 5*b^3 - 35*(192*a^2*b - 35*b^3)*e^(8*d*x + 8*c) + 560*(16*a^3 - 3*a*b^2)*e^(7*d*x + 7*c) + 210*(32*a^2*b - 7*b^3)*e^(6*d*x + 6*c))*e^(-7*d*x - 7*c)/((e^(d*x + c) + 1)*(e^(d*x + c) - 1)))/d
Time = 0.42 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.66 \[ \int \text {csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {{\mathrm {e}}^{c+d\,x}\,\left (192\,a^2\,b-35\,b^3\right )}{128\,d}-\frac {2\,a^3}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {7\,b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{128\,d}+\frac {7\,b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{128\,d}-\frac {7\,b^3\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{640\,d}-\frac {7\,b^3\,{\mathrm {e}}^{5\,c+5\,d\,x}}{640\,d}+\frac {b^3\,{\mathrm {e}}^{-7\,c-7\,d\,x}}{896\,d}+\frac {b^3\,{\mathrm {e}}^{7\,c+7\,d\,x}}{896\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (192\,a^2\,b-35\,b^3\right )}{128\,d}+\frac {9\,a\,b^2\,x}{8}+\frac {3\,a\,b^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}-\frac {3\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {3\,a\,b^2\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,d}+\frac {3\,a\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,d} \]
(exp(c + d*x)*(192*a^2*b - 35*b^3))/(128*d) - (2*a^3)/(d*(exp(2*c + 2*d*x) - 1)) + (7*b^3*exp(- 3*c - 3*d*x))/(128*d) + (7*b^3*exp(3*c + 3*d*x))/(12 8*d) - (7*b^3*exp(- 5*c - 5*d*x))/(640*d) - (7*b^3*exp(5*c + 5*d*x))/(640* d) + (b^3*exp(- 7*c - 7*d*x))/(896*d) + (b^3*exp(7*c + 7*d*x))/(896*d) + ( exp(- c - d*x)*(192*a^2*b - 35*b^3))/(128*d) + (9*a*b^2*x)/8 + (3*a*b^2*ex p(- 2*c - 2*d*x))/(8*d) - (3*a*b^2*exp(2*c + 2*d*x))/(8*d) - (3*a*b^2*exp( - 4*c - 4*d*x))/(64*d) + (3*a*b^2*exp(4*c + 4*d*x))/(64*d)