Integrand size = 21, antiderivative size = 130 \[ \int \sinh (c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {3 a b x}{4}+\frac {a^2 \cosh (c+d x)}{d}-\frac {b^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{d}-\frac {3 b^2 \cosh ^5(c+d x)}{5 d}+\frac {b^2 \cosh ^7(c+d x)}{7 d}-\frac {3 a b \cosh (c+d x) \sinh (c+d x)}{4 d}+\frac {a b \cosh (c+d x) \sinh ^3(c+d x)}{2 d} \]
3/4*a*b*x+a^2*cosh(d*x+c)/d-b^2*cosh(d*x+c)/d+b^2*cosh(d*x+c)^3/d-3/5*b^2* cosh(d*x+c)^5/d+1/7*b^2*cosh(d*x+c)^7/d-3/4*a*b*cosh(d*x+c)*sinh(d*x+c)/d+ 1/2*a*b*cosh(d*x+c)*sinh(d*x+c)^3/d
Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.71 \[ \int \sinh (c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {35 \left (64 a^2-35 b^2\right ) \cosh (c+d x)+b (245 b \cosh (3 (c+d x))-49 b \cosh (5 (c+d x))+5 b \cosh (7 (c+d x))+140 a (12 (c+d x)-8 \sinh (2 (c+d x))+\sinh (4 (c+d x))))}{2240 d} \]
(35*(64*a^2 - 35*b^2)*Cosh[c + d*x] + b*(245*b*Cosh[3*(c + d*x)] - 49*b*Co sh[5*(c + d*x)] + 5*b*Cosh[7*(c + d*x)] + 140*a*(12*(c + d*x) - 8*Sinh[2*( c + d*x)] + Sinh[4*(c + d*x)])))/(2240*d)
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 26, 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.
| \(\displaystyle \int \sinh (c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -i \sin (i c+i d x) \left (a+i b \sin (i c+i d x)^3\right )^2dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \sin (i c+i d x) \left (i b \sin (i c+i d x)^3+a\right )^2dx\) |
| \(\Big \downarrow \) 3699 |
| \(\displaystyle -i \int \left (i b^2 \sinh ^7(c+d x)+2 i a b \sinh ^4(c+d x)+i a^2 \sinh (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -i \left (\frac {i a^2 \cosh (c+d x)}{d}+\frac {i a b \sinh ^3(c+d x) \cosh (c+d x)}{2 d}-\frac {3 i a b \sinh (c+d x) \cosh (c+d x)}{4 d}+\frac {3}{4} i a b x+\frac {i b^2 \cosh ^7(c+d x)}{7 d}-\frac {3 i b^2 \cosh ^5(c+d x)}{5 d}+\frac {i b^2 \cosh ^3(c+d x)}{d}-\frac {i b^2 \cosh (c+d x)}{d}\right )\) |
(-I)*(((3*I)/4)*a*b*x + (I*a^2*Cosh[c + d*x])/d - (I*b^2*Cosh[c + d*x])/d + (I*b^2*Cosh[c + d*x]^3)/d - (((3*I)/5)*b^2*Cosh[c + d*x]^5)/d + ((I/7)*b ^2*Cosh[c + d*x]^7)/d - (((3*I)/4)*a*b*Cosh[c + d*x]*Sinh[c + d*x])/d + (( I/2)*a*b*Cosh[c + d*x]*Sinh[c + d*x]^3)/d)
3.2.52.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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 = 0.68 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(\frac {a^{2} \cosh \left (d x +c \right )+2 a b \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b^{2} \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}\) | \(96\) |
| default | \(\frac {a^{2} \cosh \left (d x +c \right )+2 a b \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b^{2} \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}\) | \(96\) |
| parts | \(\frac {b^{2} \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}+\frac {a^{2} \cosh \left (d x +c \right )}{d}+\frac {2 a b \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(101\) |
| parallelrisch | \(\frac {245 b^{2} \cosh \left (3 d x +3 c \right )-49 b^{2} \cosh \left (5 d x +5 c \right )+5 b^{2} \cosh \left (7 d x +7 c \right )-1120 a b \sinh \left (2 d x +2 c \right )+140 a b \sinh \left (4 d x +4 c \right )+\left (2240 a^{2}-1225 b^{2}\right ) \cosh \left (d x +c \right )+1680 a b d x +2240 a^{2}-1024 b^{2}}{2240 d}\) | \(109\) |
| risch | \(\frac {3 a b x}{4}+\frac {b^{2} {\mathrm e}^{7 d x +7 c}}{896 d}-\frac {7 b^{2} {\mathrm e}^{5 d x +5 c}}{640 d}+\frac {{\mathrm e}^{4 d x +4 c} a b}{32 d}+\frac {7 \,{\mathrm e}^{3 d x +3 c} b^{2}}{128 d}-\frac {{\mathrm e}^{2 d x +2 c} a b}{4 d}+\frac {{\mathrm e}^{d x +c} a^{2}}{2 d}-\frac {35 \,{\mathrm e}^{d x +c} b^{2}}{128 d}+\frac {{\mathrm e}^{-d x -c} a^{2}}{2 d}-\frac {35 \,{\mathrm e}^{-d x -c} b^{2}}{128 d}+\frac {{\mathrm e}^{-2 d x -2 c} a b}{4 d}+\frac {7 \,{\mathrm e}^{-3 d x -3 c} b^{2}}{128 d}-\frac {{\mathrm e}^{-4 d x -4 c} a b}{32 d}-\frac {7 b^{2} {\mathrm e}^{-5 d x -5 c}}{640 d}+\frac {b^{2} {\mathrm e}^{-7 d x -7 c}}{896 d}\) | \(235\) |
1/d*(a^2*cosh(d*x+c)+2*a*b*((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^2*(-16/35+1/7*sinh(d*x+c)^6-6/35*sinh(d*x+c)^4+8/35*sin h(d*x+c)^2)*cosh(d*x+c))
Time = 0.27 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.69 \[ \int \sinh (c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {5 \, b^{2} \cosh \left (d x + c\right )^{7} + 35 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 49 \, b^{2} \cosh \left (d x + c\right )^{5} + 560 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 245 \, b^{2} \cosh \left (d x + c\right )^{3} + 35 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} - 7 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 1680 \, a b d x + 35 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{5} - 14 \, b^{2} \cosh \left (d x + c\right )^{3} + 21 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, {\left (64 \, a^{2} - 35 \, b^{2}\right )} \cosh \left (d x + c\right ) + 560 \, {\left (a b \cosh \left (d x + c\right )^{3} - 4 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2240 \, d} \]
1/2240*(5*b^2*cosh(d*x + c)^7 + 35*b^2*cosh(d*x + c)*sinh(d*x + c)^6 - 49* b^2*cosh(d*x + c)^5 + 560*a*b*cosh(d*x + c)*sinh(d*x + c)^3 + 245*b^2*cosh (d*x + c)^3 + 35*(5*b^2*cosh(d*x + c)^3 - 7*b^2*cosh(d*x + c))*sinh(d*x + c)^4 + 1680*a*b*d*x + 35*(3*b^2*cosh(d*x + c)^5 - 14*b^2*cosh(d*x + c)^3 + 21*b^2*cosh(d*x + c))*sinh(d*x + c)^2 + 35*(64*a^2 - 35*b^2)*cosh(d*x + c ) + 560*(a*b*cosh(d*x + c)^3 - 4*a*b*cosh(d*x + c))*sinh(d*x + c))/d
Time = 0.45 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.68 \[ \int \sinh (c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\begin {cases} \frac {a^{2} \cosh {\left (c + d x \right )}}{d} + \frac {3 a b x \sinh ^{4}{\left (c + d x \right )}}{4} - \frac {3 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {3 a b x \cosh ^{4}{\left (c + d x \right )}}{4} + \frac {5 a b \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{4 d} - \frac {3 a b \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{4 d} + \frac {b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {8 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {16 b^{2} \cosh ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right )^{2} \sinh {\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a**2*cosh(c + d*x)/d + 3*a*b*x*sinh(c + d*x)**4/4 - 3*a*b*x*sin h(c + d*x)**2*cosh(c + d*x)**2/2 + 3*a*b*x*cosh(c + d*x)**4/4 + 5*a*b*sinh (c + d*x)**3*cosh(c + d*x)/(4*d) - 3*a*b*sinh(c + d*x)*cosh(c + d*x)**3/(4 *d) + b**2*sinh(c + d*x)**6*cosh(c + d*x)/d - 2*b**2*sinh(c + d*x)**4*cosh (c + d*x)**3/d + 8*b**2*sinh(c + d*x)**2*cosh(c + d*x)**5/(5*d) - 16*b**2* cosh(c + d*x)**7/(35*d), Ne(d, 0)), (x*(a + b*sinh(c)**3)**2*sinh(c), True ))
Time = 0.21 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.38 \[ \int \sinh (c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {1}{32} \, a b {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{4480} \, b^{2} {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {a^{2} \cosh \left (d x + c\right )}{d} \]
1/32*a*b*(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^2*((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 ) + a^2*cosh(d*x + c)/d
Time = 0.34 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.68 \[ \int \sinh (c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {3}{4} \, a b x + \frac {b^{2} e^{\left (7 \, d x + 7 \, c\right )}}{896 \, d} - \frac {7 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {a b e^{\left (4 \, d x + 4 \, c\right )}}{32 \, d} + \frac {7 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )}}{128 \, d} - \frac {a b e^{\left (2 \, d x + 2 \, c\right )}}{4 \, d} + \frac {a b e^{\left (-2 \, d x - 2 \, c\right )}}{4 \, d} + \frac {7 \, b^{2} e^{\left (-3 \, d x - 3 \, c\right )}}{128 \, d} - \frac {a b e^{\left (-4 \, d x - 4 \, c\right )}}{32 \, d} - \frac {7 \, b^{2} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} + \frac {b^{2} e^{\left (-7 \, d x - 7 \, c\right )}}{896 \, d} + \frac {{\left (64 \, a^{2} - 35 \, b^{2}\right )} e^{\left (d x + c\right )}}{128 \, d} + \frac {{\left (64 \, a^{2} - 35 \, b^{2}\right )} e^{\left (-d x - c\right )}}{128 \, d} \]
3/4*a*b*x + 1/896*b^2*e^(7*d*x + 7*c)/d - 7/640*b^2*e^(5*d*x + 5*c)/d + 1/ 32*a*b*e^(4*d*x + 4*c)/d + 7/128*b^2*e^(3*d*x + 3*c)/d - 1/4*a*b*e^(2*d*x + 2*c)/d + 1/4*a*b*e^(-2*d*x - 2*c)/d + 7/128*b^2*e^(-3*d*x - 3*c)/d - 1/3 2*a*b*e^(-4*d*x - 4*c)/d - 7/640*b^2*e^(-5*d*x - 5*c)/d + 1/896*b^2*e^(-7* d*x - 7*c)/d + 1/128*(64*a^2 - 35*b^2)*e^(d*x + c)/d + 1/128*(64*a^2 - 35* b^2)*e^(-d*x - c)/d
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.80 \[ \int \sinh (c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {a^2\,\mathrm {cosh}\left (c+d\,x\right )+\frac {\mathrm {sinh}\left (c+d\,x\right )\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{2}-\frac {5\,\mathrm {sinh}\left (c+d\,x\right )\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )}{4}+\frac {3\,d\,x\,a\,b}{4}+\frac {b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}-\frac {3\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}+b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3-b^2\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]