Integrand size = 21, antiderivative size = 106 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\frac {3 a x}{8}-\frac {b \cosh (c+d x)}{d}+\frac {b \cosh ^3(c+d x)}{d}-\frac {3 b \cosh ^5(c+d x)}{5 d}+\frac {b \cosh ^7(c+d x)}{7 d}-\frac {3 a \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {a \cosh (c+d x) \sinh ^3(c+d x)}{4 d} \]
3/8*a*x-b*cosh(d*x+c)/d+b*cosh(d*x+c)^3/d-3/5*b*cosh(d*x+c)^5/d+1/7*b*cosh (d*x+c)^7/d-3/8*a*cosh(d*x+c)*sinh(d*x+c)/d+1/4*a*cosh(d*x+c)*sinh(d*x+c)^ 3/d
Time = 0.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\frac {840 a c+840 a d x-1225 b \cosh (c+d x)+245 b \cosh (3 (c+d x))-49 b \cosh (5 (c+d x))+5 b \cosh (7 (c+d x))-560 a \sinh (2 (c+d x))+70 a \sinh (4 (c+d x))}{2240 d} \]
(840*a*c + 840*a*d*x - 1225*b*Cosh[c + d*x] + 245*b*Cosh[3*(c + d*x)] - 49 *b*Cosh[5*(c + d*x)] + 5*b*Cosh[7*(c + d*x)] - 560*a*Sinh[2*(c + d*x)] + 7 0*a*Sinh[4*(c + d*x)])/(2240*d)
Time = 0.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 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.
\(\displaystyle \int \sinh ^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (i c+i d x)^4 \left (a+i b \sin (i c+i d x)^3\right )dx\) |
\(\Big \downarrow \) 3699 |
\(\displaystyle \int \left (a \sinh ^4(c+d x)+b \sinh ^7(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {3 a \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {3 a x}{8}+\frac {b \cosh ^7(c+d x)}{7 d}-\frac {3 b \cosh ^5(c+d x)}{5 d}+\frac {b \cosh ^3(c+d x)}{d}-\frac {b \cosh (c+d x)}{d}\) |
(3*a*x)/8 - (b*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/d - (3*b*Cosh[c + d* x]^5)/(5*d) + (b*Cosh[c + d*x]^7)/(7*d) - (3*a*Cosh[c + d*x]*Sinh[c + d*x] )/(8*d) + (a*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)
3.2.41.3.1 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 = 0.60 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {a \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}\) | \(82\) |
default | \(\frac {a \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+b \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}\) | \(82\) |
parallelrisch | \(\frac {840 a x d +5 b \cosh \left (7 d x +7 c \right )-49 b \cosh \left (5 d x +5 c \right )+245 b \cosh \left (3 d x +3 c \right )-1225 b \cosh \left (d x +c \right )+70 a \sinh \left (4 d x +4 c \right )-560 a \sinh \left (2 d x +2 c \right )-1024 b}{2240 d}\) | \(84\) |
parts | \(\frac {a \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {b \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}\) | \(84\) |
risch | \(\frac {3 a x}{8}+\frac {b \,{\mathrm e}^{7 d x +7 c}}{896 d}-\frac {7 b \,{\mathrm e}^{5 d x +5 c}}{640 d}+\frac {a \,{\mathrm e}^{4 d x +4 c}}{64 d}+\frac {7 b \,{\mathrm e}^{3 d x +3 c}}{128 d}-\frac {a \,{\mathrm e}^{2 d x +2 c}}{8 d}-\frac {35 b \,{\mathrm e}^{d x +c}}{128 d}-\frac {35 b \,{\mathrm e}^{-d x -c}}{128 d}+\frac {a \,{\mathrm e}^{-2 d x -2 c}}{8 d}+\frac {7 b \,{\mathrm e}^{-3 d x -3 c}}{128 d}-\frac {a \,{\mathrm e}^{-4 d x -4 c}}{64 d}-\frac {7 b \,{\mathrm e}^{-5 d x -5 c}}{640 d}+\frac {b \,{\mathrm e}^{-7 d x -7 c}}{896 d}\) | \(183\) |
1/d*(a*((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*( -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 ))
Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.77 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\frac {5 \, b \cosh \left (d x + c\right )^{7} + 35 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 49 \, b \cosh \left (d x + c\right )^{5} + 280 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 35 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 7 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 245 \, b \cosh \left (d x + c\right )^{3} + 840 \, a d x + 35 \, {\left (3 \, b \cosh \left (d x + c\right )^{5} - 14 \, b \cosh \left (d x + c\right )^{3} + 21 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 1225 \, b \cosh \left (d x + c\right ) + 280 \, {\left (a \cosh \left (d x + c\right )^{3} - 4 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2240 \, d} \]
1/2240*(5*b*cosh(d*x + c)^7 + 35*b*cosh(d*x + c)*sinh(d*x + c)^6 - 49*b*co sh(d*x + c)^5 + 280*a*cosh(d*x + c)*sinh(d*x + c)^3 + 35*(5*b*cosh(d*x + c )^3 - 7*b*cosh(d*x + c))*sinh(d*x + c)^4 + 245*b*cosh(d*x + c)^3 + 840*a*d *x + 35*(3*b*cosh(d*x + c)^5 - 14*b*cosh(d*x + c)^3 + 21*b*cosh(d*x + c))* sinh(d*x + c)^2 - 1225*b*cosh(d*x + c) + 280*(a*cosh(d*x + c)^3 - 4*a*cosh (d*x + c))*sinh(d*x + c))/d
Time = 0.46 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.81 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\begin {cases} \frac {3 a x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 a x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {5 a \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {3 a \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {b \sinh ^{6}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 b \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {8 b \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {16 b \cosh ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right ) \sinh ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((3*a*x*sinh(c + d*x)**4/8 - 3*a*x*sinh(c + d*x)**2*cosh(c + d*x) **2/4 + 3*a*x*cosh(c + d*x)**4/8 + 5*a*sinh(c + d*x)**3*cosh(c + d*x)/(8*d ) - 3*a*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) + b*sinh(c + d*x)**6*cosh(c + d*x)/d - 2*b*sinh(c + d*x)**4*cosh(c + d*x)**3/d + 8*b*sinh(c + d*x)**2*c osh(c + d*x)**5/(5*d) - 16*b*cosh(c + d*x)**7/(35*d), Ne(d, 0)), (x*(a + b *sinh(c)**3)*sinh(c)**4, True))
Time = 0.19 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.55 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\frac {1}{64} \, a {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{4480} \, b {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} \]
1/64*a*(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*((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)
Time = 0.32 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.72 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\frac {3}{8} \, a x + \frac {b e^{\left (7 \, d x + 7 \, c\right )}}{896 \, d} - \frac {7 \, b e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {a e^{\left (4 \, d x + 4 \, c\right )}}{64 \, d} + \frac {7 \, b e^{\left (3 \, d x + 3 \, c\right )}}{128 \, d} - \frac {a e^{\left (2 \, d x + 2 \, c\right )}}{8 \, d} - \frac {35 \, b e^{\left (d x + c\right )}}{128 \, d} - \frac {35 \, b e^{\left (-d x - c\right )}}{128 \, d} + \frac {a e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, d} + \frac {7 \, b e^{\left (-3 \, d x - 3 \, c\right )}}{128 \, d} - \frac {a e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} - \frac {7 \, b e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} + \frac {b e^{\left (-7 \, d x - 7 \, c\right )}}{896 \, d} \]
3/8*a*x + 1/896*b*e^(7*d*x + 7*c)/d - 7/640*b*e^(5*d*x + 5*c)/d + 1/64*a*e ^(4*d*x + 4*c)/d + 7/128*b*e^(3*d*x + 3*c)/d - 1/8*a*e^(2*d*x + 2*c)/d - 3 5/128*b*e^(d*x + c)/d - 35/128*b*e^(-d*x - c)/d + 1/8*a*e^(-2*d*x - 2*c)/d + 7/128*b*e^(-3*d*x - 3*c)/d - 1/64*a*e^(-4*d*x - 4*c)/d - 7/640*b*e^(-5* d*x - 5*c)/d + 1/896*b*e^(-7*d*x - 7*c)/d
Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.80 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\frac {280\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3-280\,b\,\mathrm {cosh}\left (c+d\,x\right )-168\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5+40\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^7-175\,a\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )+105\,a\,d\,x+70\,a\,{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )}{280\,d} \]