Integrand size = 14, antiderivative size = 128 \[ \int \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {1}{16} (2 a-b) \left (8 a^2-8 a b+5 b^2\right ) x+\frac {b \left (64 a^2-54 a b+15 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{48 d}+\frac {5 (2 a-b) b^2 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d} \]
1/16*(2*a-b)*(8*a^2-8*a*b+5*b^2)*x+1/48*b*(64*a^2-54*a*b+15*b^2)*cosh(d*x+ c)*sinh(d*x+c)/d+5/24*(2*a-b)*b^2*cosh(d*x+c)*sinh(d*x+c)^3/d+1/6*b*cosh(d *x+c)*sinh(d*x+c)*(a+b*sinh(d*x+c)^2)^2/d
Time = 0.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.74 \[ \int \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {12 (2 a-b) \left (8 a^2-8 a b+5 b^2\right ) (c+d x)+9 b \left (16 a^2-16 a b+5 b^2\right ) \sinh (2 (c+d x))+9 (2 a-b) b^2 \sinh (4 (c+d x))+b^3 \sinh (6 (c+d x))}{192 d} \]
(12*(2*a - b)*(8*a^2 - 8*a*b + 5*b^2)*(c + d*x) + 9*b*(16*a^2 - 16*a*b + 5 *b^2)*Sinh[2*(c + d*x)] + 9*(2*a - b)*b^2*Sinh[4*(c + d*x)] + b^3*Sinh[6*( c + d*x)])/(192*d)
Time = 0.38 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3659, 3042, 3648}
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.
| \(\displaystyle \int \left (a+b \sinh ^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-b \sin (i c+i d x)^2\right )^3dx\) |
| \(\Big \downarrow \) 3659 |
| \(\displaystyle \frac {1}{6} \int \left (b \sinh ^2(c+d x)+a\right ) \left (5 (2 a-b) b \sinh ^2(c+d x)+a (6 a-b)\right )dx+\frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d}+\frac {1}{6} \int \left (a-b \sin (i c+i d x)^2\right ) \left (a (6 a-b)-5 (2 a-b) b \sin (i c+i d x)^2\right )dx\) |
| \(\Big \downarrow \) 3648 |
| \(\displaystyle \frac {1}{6} \left (\frac {b \left (64 a^2-54 a b+15 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {3}{8} x (2 a-b) \left (8 a^2-8 a b+5 b^2\right )+\frac {5 b^2 (2 a-b) \sinh ^3(c+d x) \cosh (c+d x)}{4 d}\right )+\frac {b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d}\) |
(b*Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^2)/(6*d) + ((3*(2*a - b)*(8*a^2 - 8*a*b + 5*b^2)*x)/8 + (b*(64*a^2 - 54*a*b + 15*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (5*(2*a - b)*b^2*Cosh[c + d*x]*Sinh[c + d*x] ^3)/(4*d))/6
3.1.23.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)*((A_.) + (B_.)*sin[(e_.) + (f_ .)*(x_)]^2), x_Symbol] :> Simp[(4*A*(2*a + b) + B*(4*a + 3*b))*(x/8), x] + (-Simp[b*B*Cos[e + f*x]*(Sin[e + f*x]^3/(4*f)), x] - Simp[(4*A*b + B*(4*a + 3*b))*Cos[e + f*x]*(Sin[e + f*x]/(8*f)), x]) /; FreeQ[{a, b, e, f, A, B}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p - 1)/(2*f*p)), x] + Sim p[1/(2*p) Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(2*a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[ a + b, 0] && GtQ[p, 1]
Time = 0.50 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.72
| method | result | size |
| parallelrisch | \(\frac {\left (144 a^{2} b -144 a \,b^{2}+45 b^{3}\right ) \sinh \left (2 d x +2 c \right )+\left (18 a \,b^{2}-9 b^{3}\right ) \sinh \left (4 d x +4 c \right )+b^{3} \sinh \left (6 d x +6 c \right )+192 d \left (a^{2}-a b +\frac {5}{8} b^{2}\right ) x \left (a -\frac {b}{2}\right )}{192 d}\) | \(92\) |
| derivativedivides | \(\frac {b^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+3 a \,b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+a^{3} \left (d x +c \right )}{d}\) | \(131\) |
| default | \(\frac {b^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+3 a \,b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+a^{3} \left (d x +c \right )}{d}\) | \(131\) |
| parts | \(a^{3} x +\frac {b^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}+\frac {3 a^{2} b \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d}+\frac {3 a \,b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(132\) |
| risch | \(a^{3} x -\frac {3 a^{2} b x}{2}+\frac {9 a \,b^{2} x}{8}-\frac {5 b^{3} x}{16}+\frac {b^{3} {\mathrm e}^{6 d x +6 c}}{384 d}+\frac {3 \,{\mathrm e}^{4 d x +4 c} a \,b^{2}}{64 d}-\frac {3 \,{\mathrm e}^{4 d x +4 c} b^{3}}{128 d}+\frac {3 \,{\mathrm e}^{2 d x +2 c} a^{2} b}{8 d}-\frac {3 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{8 d}+\frac {15 \,{\mathrm e}^{2 d x +2 c} b^{3}}{128 d}-\frac {3 \,{\mathrm e}^{-2 d x -2 c} a^{2} b}{8 d}+\frac {3 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{8 d}-\frac {15 \,{\mathrm e}^{-2 d x -2 c} b^{3}}{128 d}-\frac {3 \,{\mathrm e}^{-4 d x -4 c} a \,b^{2}}{64 d}+\frac {3 \,{\mathrm e}^{-4 d x -4 c} b^{3}}{128 d}-\frac {b^{3} {\mathrm e}^{-6 d x -6 c}}{384 d}\) | \(237\) |
1/192*((144*a^2*b-144*a*b^2+45*b^3)*sinh(2*d*x+2*c)+(18*a*b^2-9*b^3)*sinh( 4*d*x+4*c)+b^3*sinh(6*d*x+6*c)+192*d*(a^2-a*b+5/8*b^2)*x*(a-1/2*b))/d
Time = 0.27 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.29 \[ \int \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {3 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{3} + 9 \, {\left (2 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (16 \, a^{3} - 24 \, a^{2} b + 18 \, a b^{2} - 5 \, b^{3}\right )} d x + 3 \, {\left (b^{3} \cosh \left (d x + c\right )^{5} + 6 \, {\left (2 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (16 \, a^{2} b - 16 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \]
1/96*(3*b^3*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(5*b^3*cosh(d*x + c)^3 + 9*( 2*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(16*a^3 - 24*a^2*b + 18* a*b^2 - 5*b^3)*d*x + 3*(b^3*cosh(d*x + c)^5 + 6*(2*a*b^2 - b^3)*cosh(d*x + c)^3 + 3*(16*a^2*b - 16*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (119) = 238\).
Time = 0.35 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.73 \[ \int \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\begin {cases} a^{3} x + \frac {3 a^{2} b x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {3 a^{2} b x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {3 a^{2} b \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} + \frac {9 a b^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {9 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {9 a b^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {15 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {9 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {5 b^{3} x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {15 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {15 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {5 b^{3} x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {11 b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac {5 b^{3} \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{3} & \text {otherwise} \end {cases} \]
Piecewise((a**3*x + 3*a**2*b*x*sinh(c + d*x)**2/2 - 3*a**2*b*x*cosh(c + d* x)**2/2 + 3*a**2*b*sinh(c + d*x)*cosh(c + d*x)/(2*d) + 9*a*b**2*x*sinh(c + d*x)**4/8 - 9*a*b**2*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 9*a*b**2*x*c osh(c + d*x)**4/8 + 15*a*b**2*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) - 9*a*b **2*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) + 5*b**3*x*sinh(c + d*x)**6/16 - 15*b**3*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 15*b**3*x*sinh(c + d*x)** 2*cosh(c + d*x)**4/16 - 5*b**3*x*cosh(c + d*x)**6/16 + 11*b**3*sinh(c + d* x)**5*cosh(c + d*x)/(16*d) - 5*b**3*sinh(c + d*x)**3*cosh(c + d*x)**3/(6*d ) + 5*b**3*sinh(c + d*x)*cosh(c + d*x)**5/(16*d), Ne(d, 0)), (x*(a + b*sin h(c)**2)**3, True))
Time = 0.19 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.54 \[ \int \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {3}{64} \, a b^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {3}{8} \, a^{2} b {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a^{3} x - \frac {1}{384} \, b^{3} {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \]
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) - 3/8*a^2*b*(4*x - e^(2*d*x + 2*c)/d + e^(-2 *d*x - 2*c)/d) + a^3*x - 1/384*b^3*((9*e^(-2*d*x - 2*c) - 45*e^(-4*d*x - 4 *c) - 1)*e^(6*d*x + 6*c)/d + 120*(d*x + c)/d + (45*e^(-2*d*x - 2*c) - 9*e^ (-4*d*x - 4*c) + e^(-6*d*x - 6*c))/d)
Time = 0.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.38 \[ \int \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {b^{3} e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} - \frac {b^{3} e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} + \frac {1}{16} \, {\left (16 \, a^{3} - 24 \, a^{2} b + 18 \, a b^{2} - 5 \, b^{3}\right )} x + \frac {3 \, {\left (2 \, a b^{2} - b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} + \frac {3 \, {\left (16 \, a^{2} b - 16 \, a b^{2} + 5 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {3 \, {\left (16 \, a^{2} b - 16 \, a b^{2} + 5 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} - \frac {3 \, {\left (2 \, a b^{2} - b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{128 \, d} \]
1/384*b^3*e^(6*d*x + 6*c)/d - 1/384*b^3*e^(-6*d*x - 6*c)/d + 1/16*(16*a^3 - 24*a^2*b + 18*a*b^2 - 5*b^3)*x + 3/128*(2*a*b^2 - b^3)*e^(4*d*x + 4*c)/d + 3/128*(16*a^2*b - 16*a*b^2 + 5*b^3)*e^(2*d*x + 2*c)/d - 3/128*(16*a^2*b - 16*a*b^2 + 5*b^3)*e^(-2*d*x - 2*c)/d - 3/128*(2*a*b^2 - b^3)*e^(-4*d*x - 4*c)/d
Time = 0.26 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.96 \[ \int \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {\frac {45\,b^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}-\frac {9\,b^3\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+\frac {b^3\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}-36\,a\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+36\,a^2\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {9\,a\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{2}+48\,a^3\,d\,x-15\,b^3\,d\,x+54\,a\,b^2\,d\,x-72\,a^2\,b\,d\,x}{48\,d} \]