Integrand size = 23, antiderivative size = 180 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=-\frac {a^2 x}{2}+\frac {35 b^2 x}{128}+\frac {2 a b \cosh (c+d x)}{d}-\frac {4 a b \cosh ^3(c+d x)}{3 d}+\frac {2 a b \cosh ^5(c+d x)}{5 d}+\frac {a^2 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {35 b^2 \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {35 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{192 d}-\frac {7 b^2 \cosh (c+d x) \sinh ^5(c+d x)}{48 d}+\frac {b^2 \cosh (c+d x) \sinh ^7(c+d x)}{8 d} \]
-1/2*a^2*x+35/128*b^2*x+2*a*b*cosh(d*x+c)/d-4/3*a*b*cosh(d*x+c)^3/d+2/5*a* b*cosh(d*x+c)^5/d+1/2*a^2*cosh(d*x+c)*sinh(d*x+c)/d-35/128*b^2*cosh(d*x+c) *sinh(d*x+c)/d+35/192*b^2*cosh(d*x+c)*sinh(d*x+c)^3/d-7/48*b^2*cosh(d*x+c) *sinh(d*x+c)^5/d+1/8*b^2*cosh(d*x+c)*sinh(d*x+c)^7/d
Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.74 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {-7680 a^2 c+4200 b^2 c-7680 a^2 d x+4200 b^2 d x+19200 a b \cosh (c+d x)-3200 a b \cosh (3 (c+d x))+384 a b \cosh (5 (c+d x))+3840 a^2 \sinh (2 (c+d x))-3360 b^2 \sinh (2 (c+d x))+840 b^2 \sinh (4 (c+d x))-160 b^2 \sinh (6 (c+d x))+15 b^2 \sinh (8 (c+d x))}{15360 d} \]
(-7680*a^2*c + 4200*b^2*c - 7680*a^2*d*x + 4200*b^2*d*x + 19200*a*b*Cosh[c + d*x] - 3200*a*b*Cosh[3*(c + d*x)] + 384*a*b*Cosh[5*(c + d*x)] + 3840*a^ 2*Sinh[2*(c + d*x)] - 3360*b^2*Sinh[2*(c + d*x)] + 840*b^2*Sinh[4*(c + d*x )] - 160*b^2*Sinh[6*(c + d*x)] + 15*b^2*Sinh[8*(c + d*x)])/(15360*d)
Time = 0.42 (sec) , antiderivative size = 180, 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.
| \(\displaystyle \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\sin (i c+i d x)^2 \left (a+i b \sin (i c+i d x)^3\right )^2dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \sin (i c+i d x)^2 \left (i b \sin (i c+i d x)^3+a\right )^2dx\) |
| \(\Big \downarrow \) 3699 |
| \(\displaystyle -\int \left (-b^2 \sinh ^8(c+d x)-2 a b \sinh ^5(c+d x)-a^2 \sinh ^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {a^2 x}{2}+\frac {2 a b \cosh ^5(c+d x)}{5 d}-\frac {4 a b \cosh ^3(c+d x)}{3 d}+\frac {2 a b \cosh (c+d x)}{d}+\frac {b^2 \sinh ^7(c+d x) \cosh (c+d x)}{8 d}-\frac {7 b^2 \sinh ^5(c+d x) \cosh (c+d x)}{48 d}+\frac {35 b^2 \sinh ^3(c+d x) \cosh (c+d x)}{192 d}-\frac {35 b^2 \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac {35 b^2 x}{128}\) |
-1/2*(a^2*x) + (35*b^2*x)/128 + (2*a*b*Cosh[c + d*x])/d - (4*a*b*Cosh[c + d*x]^3)/(3*d) + (2*a*b*Cosh[c + d*x]^5)/(5*d) + (a^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) - (35*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + (35*b^2*Cos h[c + d*x]*Sinh[c + d*x]^3)/(192*d) - (7*b^2*Cosh[c + d*x]*Sinh[c + d*x]^5 )/(48*d) + (b^2*Cosh[c + d*x]*Sinh[c + d*x]^7)/(8*d)
3.2.51.3.1 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 = 0.94 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.68
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+2 a b \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )+b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{7}}{8}-\frac {7 \sinh \left (d x +c \right )^{5}}{48}+\frac {35 \sinh \left (d x +c \right )^{3}}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(122\) |
| default | \(\frac {a^{2} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+2 a b \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )+b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{7}}{8}-\frac {7 \sinh \left (d x +c \right )^{5}}{48}+\frac {35 \sinh \left (d x +c \right )^{3}}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(122\) |
| parts | \(\frac {a^{2} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d}+\frac {b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{7}}{8}-\frac {7 \sinh \left (d x +c \right )^{5}}{48}+\frac {35 \sinh \left (d x +c \right )^{3}}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {2 a b \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )}{d}\) | \(127\) |
| parallelrisch | \(\frac {-7680 a^{2} d x +4200 b^{2} d x +15 b^{2} \sinh \left (8 d x +8 c \right )-160 b^{2} \sinh \left (6 d x +6 c \right )+840 b^{2} \sinh \left (4 d x +4 c \right )+3840 a^{2} \sinh \left (2 d x +2 c \right )-3360 b^{2} \sinh \left (2 d x +2 c \right )+384 a b \cosh \left (5 d x +5 c \right )-3200 a b \cosh \left (3 d x +3 c \right )+19200 a b \cosh \left (d x +c \right )+16384 a b}{15360 d}\) | \(131\) |
| risch | \(-\frac {a^{2} x}{2}+\frac {35 b^{2} x}{128}+\frac {b^{2} {\mathrm e}^{8 d x +8 c}}{2048 d}-\frac {b^{2} {\mathrm e}^{6 d x +6 c}}{192 d}+\frac {b \,{\mathrm e}^{5 d x +5 c} a}{80 d}+\frac {7 \,{\mathrm e}^{4 d x +4 c} b^{2}}{256 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} a b}{48 d}+\frac {{\mathrm e}^{2 d x +2 c} a^{2}}{8 d}-\frac {7 \,{\mathrm e}^{2 d x +2 c} b^{2}}{64 d}+\frac {5 \,{\mathrm e}^{d x +c} a b}{8 d}+\frac {5 \,{\mathrm e}^{-d x -c} a b}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{2}}{8 d}+\frac {7 \,{\mathrm e}^{-2 d x -2 c} b^{2}}{64 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} a b}{48 d}-\frac {7 \,{\mathrm e}^{-4 d x -4 c} b^{2}}{256 d}+\frac {b \,{\mathrm e}^{-5 d x -5 c} a}{80 d}+\frac {b^{2} {\mathrm e}^{-6 d x -6 c}}{192 d}-\frac {b^{2} {\mathrm e}^{-8 d x -8 c}}{2048 d}\) | \(277\) |
1/d*(a^2*(1/2*sinh(d*x+c)*cosh(d*x+c)-1/2*d*x-1/2*c)+2*a*b*(8/15+1/5*sinh( d*x+c)^4-4/15*sinh(d*x+c)^2)*cosh(d*x+c)+b^2*((1/8*sinh(d*x+c)^7-7/48*sinh (d*x+c)^5+35/192*sinh(d*x+c)^3-35/128*sinh(d*x+c))*cosh(d*x+c)+35/128*d*x+ 35/128*c))
Time = 0.26 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.52 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {15 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 48 \, a b \cosh \left (d x + c\right )^{5} + 240 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 15 \, {\left (7 \, b^{2} \cosh \left (d x + c\right )^{3} - 8 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} - 400 \, a b \cosh \left (d x + c\right )^{3} + 5 \, {\left (21 \, b^{2} \cosh \left (d x + c\right )^{5} - 80 \, b^{2} \cosh \left (d x + c\right )^{3} + 84 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 15 \, {\left (64 \, a^{2} - 35 \, b^{2}\right )} d x + 2400 \, a b \cosh \left (d x + c\right ) + 240 \, {\left (2 \, a b \cosh \left (d x + c\right )^{3} - 5 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 15 \, {\left (b^{2} \cosh \left (d x + c\right )^{7} - 8 \, b^{2} \cosh \left (d x + c\right )^{5} + 28 \, b^{2} \cosh \left (d x + c\right )^{3} + 8 \, {\left (8 \, a^{2} - 7 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{1920 \, d} \]
1/1920*(15*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + 48*a*b*cosh(d*x + c)^5 + 24 0*a*b*cosh(d*x + c)*sinh(d*x + c)^4 + 15*(7*b^2*cosh(d*x + c)^3 - 8*b^2*co sh(d*x + c))*sinh(d*x + c)^5 - 400*a*b*cosh(d*x + c)^3 + 5*(21*b^2*cosh(d* x + c)^5 - 80*b^2*cosh(d*x + c)^3 + 84*b^2*cosh(d*x + c))*sinh(d*x + c)^3 - 15*(64*a^2 - 35*b^2)*d*x + 2400*a*b*cosh(d*x + c) + 240*(2*a*b*cosh(d*x + c)^3 - 5*a*b*cosh(d*x + c))*sinh(d*x + c)^2 + 15*(b^2*cosh(d*x + c)^7 - 8*b^2*cosh(d*x + c)^5 + 28*b^2*cosh(d*x + c)^3 + 8*(8*a^2 - 7*b^2)*cosh(d* x + c))*sinh(d*x + c))/d
Time = 0.64 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.89 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\begin {cases} \frac {a^{2} x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {a^{2} x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {a^{2} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} + \frac {2 a b \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 a b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {16 a b \cosh ^{5}{\left (c + d x \right )}}{15 d} + \frac {35 b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac {35 b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac {105 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac {35 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac {35 b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac {93 b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} - \frac {511 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{384 d} + \frac {385 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{384 d} - \frac {35 b^{2} \sinh {\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right )^{2} \sinh ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a**2*x*sinh(c + d*x)**2/2 - a**2*x*cosh(c + d*x)**2/2 + a**2*si nh(c + d*x)*cosh(c + d*x)/(2*d) + 2*a*b*sinh(c + d*x)**4*cosh(c + d*x)/d - 8*a*b*sinh(c + d*x)**2*cosh(c + d*x)**3/(3*d) + 16*a*b*cosh(c + d*x)**5/( 15*d) + 35*b**2*x*sinh(c + d*x)**8/128 - 35*b**2*x*sinh(c + d*x)**6*cosh(c + d*x)**2/32 + 105*b**2*x*sinh(c + d*x)**4*cosh(c + d*x)**4/64 - 35*b**2* x*sinh(c + d*x)**2*cosh(c + d*x)**6/32 + 35*b**2*x*cosh(c + d*x)**8/128 + 93*b**2*sinh(c + d*x)**7*cosh(c + d*x)/(128*d) - 511*b**2*sinh(c + d*x)**5 *cosh(c + d*x)**3/(384*d) + 385*b**2*sinh(c + d*x)**3*cosh(c + d*x)**5/(38 4*d) - 35*b**2*sinh(c + d*x)*cosh(c + d*x)**7/(128*d), Ne(d, 0)), (x*(a + b*sinh(c)**3)**2*sinh(c)**2, True))
Time = 0.21 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.32 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=-\frac {1}{8} \, a^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{6144} \, b^{2} {\left (\frac {{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac {1680 \, {\left (d x + c\right )}}{d} - \frac {672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} + \frac {1}{240} \, a b {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} \]
-1/8*a^2*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/6144*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) + 1/240*a*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 = 0.31 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.44 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=-\frac {1}{128} \, {\left (64 \, a^{2} - 35 \, b^{2}\right )} x + \frac {b^{2} e^{\left (8 \, d x + 8 \, c\right )}}{2048 \, d} - \frac {b^{2} e^{\left (6 \, d x + 6 \, c\right )}}{192 \, d} + \frac {a b e^{\left (5 \, d x + 5 \, c\right )}}{80 \, d} + \frac {7 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )}}{256 \, d} - \frac {5 \, a b e^{\left (3 \, d x + 3 \, c\right )}}{48 \, d} + \frac {5 \, a b e^{\left (d x + c\right )}}{8 \, d} + \frac {5 \, a b e^{\left (-d x - c\right )}}{8 \, d} - \frac {5 \, a b e^{\left (-3 \, d x - 3 \, c\right )}}{48 \, d} - \frac {7 \, b^{2} e^{\left (-4 \, d x - 4 \, c\right )}}{256 \, d} + \frac {a b e^{\left (-5 \, d x - 5 \, c\right )}}{80 \, d} + \frac {b^{2} e^{\left (-6 \, d x - 6 \, c\right )}}{192 \, d} - \frac {b^{2} e^{\left (-8 \, d x - 8 \, c\right )}}{2048 \, d} + \frac {{\left (8 \, a^{2} - 7 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{64 \, d} - \frac {{\left (8 \, a^{2} - 7 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{64 \, d} \]
-1/128*(64*a^2 - 35*b^2)*x + 1/2048*b^2*e^(8*d*x + 8*c)/d - 1/192*b^2*e^(6 *d*x + 6*c)/d + 1/80*a*b*e^(5*d*x + 5*c)/d + 7/256*b^2*e^(4*d*x + 4*c)/d - 5/48*a*b*e^(3*d*x + 3*c)/d + 5/8*a*b*e^(d*x + c)/d + 5/8*a*b*e^(-d*x - c) /d - 5/48*a*b*e^(-3*d*x - 3*c)/d - 7/256*b^2*e^(-4*d*x - 4*c)/d + 1/80*a*b *e^(-5*d*x - 5*c)/d + 1/192*b^2*e^(-6*d*x - 6*c)/d - 1/2048*b^2*e^(-8*d*x - 8*c)/d + 1/64*(8*a^2 - 7*b^2)*e^(2*d*x + 2*c)/d - 1/64*(8*a^2 - 7*b^2)*e ^(-2*d*x - 2*c)/d
Time = 2.47 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.70 \[ \int \sinh ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {480\,a^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )-420\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+105\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-20\,b^2\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+\frac {15\,b^2\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}+2400\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )-400\,a\,b\,\mathrm {cosh}\left (3\,c+3\,d\,x\right )+48\,a\,b\,\mathrm {cosh}\left (5\,c+5\,d\,x\right )-960\,a^2\,d\,x+525\,b^2\,d\,x}{1920\,d} \]