Integrand size = 23, antiderivative size = 120 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=-\frac {(a+b)^2 \cosh (c+d x)}{d}+\frac {(a+b) (a+5 b) \cosh ^3(c+d x)}{3 d}-\frac {2 b (3 a+5 b) \cosh ^5(c+d x)}{5 d}+\frac {2 b (a+5 b) \cosh ^7(c+d x)}{7 d}-\frac {5 b^2 \cosh ^9(c+d x)}{9 d}+\frac {b^2 \cosh ^{11}(c+d x)}{11 d} \]
-(a+b)^2*cosh(d*x+c)/d+1/3*(a+b)*(a+5*b)*cosh(d*x+c)^3/d-2/5*b*(3*a+5*b)*c osh(d*x+c)^5/d+2/7*b*(a+5*b)*cosh(d*x+c)^7/d-5/9*b^2*cosh(d*x+c)^9/d+1/11* b^2*cosh(d*x+c)^11/d
Time = 0.24 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.72 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=-\frac {3 a^2 \cosh (c+d x)}{4 d}-\frac {35 a b \cosh (c+d x)}{32 d}-\frac {231 b^2 \cosh (c+d x)}{512 d}+\frac {a^2 \cosh (3 (c+d x))}{12 d}+\frac {7 a b \cosh (3 (c+d x))}{32 d}+\frac {55 b^2 \cosh (3 (c+d x))}{512 d}-\frac {7 a b \cosh (5 (c+d x))}{160 d}-\frac {33 b^2 \cosh (5 (c+d x))}{1024 d}+\frac {a b \cosh (7 (c+d x))}{224 d}+\frac {55 b^2 \cosh (7 (c+d x))}{7168 d}-\frac {11 b^2 \cosh (9 (c+d x))}{9216 d}+\frac {b^2 \cosh (11 (c+d x))}{11264 d} \]
(-3*a^2*Cosh[c + d*x])/(4*d) - (35*a*b*Cosh[c + d*x])/(32*d) - (231*b^2*Co sh[c + d*x])/(512*d) + (a^2*Cosh[3*(c + d*x)])/(12*d) + (7*a*b*Cosh[3*(c + d*x)])/(32*d) + (55*b^2*Cosh[3*(c + d*x)])/(512*d) - (7*a*b*Cosh[5*(c + d *x)])/(160*d) - (33*b^2*Cosh[5*(c + d*x)])/(1024*d) + (a*b*Cosh[7*(c + d*x )])/(224*d) + (55*b^2*Cosh[7*(c + d*x)])/(7168*d) - (11*b^2*Cosh[9*(c + d* x)])/(9216*d) + (b^2*Cosh[11*(c + d*x)])/(11264*d)
Time = 0.34 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.88, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 26, 3694, 1467, 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.
| \(\displaystyle \int \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int i \sin (i c+i d x)^3 \left (a+b \sin (i c+i d x)^4\right )^2dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \sin (i c+i d x)^3 \left (b \sin (i c+i d x)^4+a\right )^2dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^4(c+d x)-2 b \cosh ^2(c+d x)+a+b\right )^2d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle -\frac {\int \left (-b^2 \cosh ^{10}(c+d x)+5 b^2 \cosh ^8(c+d x)-2 b (a+5 b) \cosh ^6(c+d x)+2 b (3 a+5 b) \cosh ^4(c+d x)+(-a-5 b) (a+b) \cosh ^2(c+d x)+(a+b)^2\right )d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {2}{7} b (a+5 b) \cosh ^7(c+d x)+\frac {2}{5} b (3 a+5 b) \cosh ^5(c+d x)-\frac {1}{3} (a+b) (a+5 b) \cosh ^3(c+d x)+(a+b)^2 \cosh (c+d x)-\frac {1}{11} b^2 \cosh ^{11}(c+d x)+\frac {5}{9} b^2 \cosh ^9(c+d x)}{d}\) |
-(((a + b)^2*Cosh[c + d*x] - ((a + b)*(a + 5*b)*Cosh[c + d*x]^3)/3 + (2*b* (3*a + 5*b)*Cosh[c + d*x]^5)/5 - (2*b*(a + 5*b)*Cosh[c + d*x]^7)/7 + (5*b^ 2*Cosh[c + d*x]^9)/9 - (b^2*Cosh[c + d*x]^11)/11)/d)
3.2.96.3.1 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[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 3.50 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.08
| method | result | size |
| parallelrisch | \(\frac {\left (295680 a^{2}+776160 a b +381150 b^{2}\right ) \cosh \left (3 d x +3 c \right )-155232 \left (a +\frac {165 b}{224}\right ) b \cosh \left (5 d x +5 c \right )+15840 \left (a +\frac {55 b}{32}\right ) b \cosh \left (7 d x +7 c \right )+315 b^{2} \cosh \left (11 d x +11 c \right )-4235 b^{2} \cosh \left (9 d x +9 c \right )+\left (-2661120 a^{2}-3880800 a b -1600830 b^{2}\right ) \cosh \left (d x +c \right )-2365440 a^{2}-3244032 a b -1310720 b^{2}}{3548160 d}\) | \(130\) |
| derivativedivides | \(\frac {a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+2 a b \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )+b^{2} \left (-\frac {256}{693}+\frac {\sinh \left (d x +c \right )^{10}}{11}-\frac {10 \sinh \left (d x +c \right )^{8}}{99}+\frac {80 \sinh \left (d x +c \right )^{6}}{693}-\frac {32 \sinh \left (d x +c \right )^{4}}{231}+\frac {128 \sinh \left (d x +c \right )^{2}}{693}\right ) \cosh \left (d x +c \right )}{d}\) | \(132\) |
| default | \(\frac {a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+2 a b \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )+b^{2} \left (-\frac {256}{693}+\frac {\sinh \left (d x +c \right )^{10}}{11}-\frac {10 \sinh \left (d x +c \right )^{8}}{99}+\frac {80 \sinh \left (d x +c \right )^{6}}{693}-\frac {32 \sinh \left (d x +c \right )^{4}}{231}+\frac {128 \sinh \left (d x +c \right )^{2}}{693}\right ) \cosh \left (d x +c \right )}{d}\) | \(132\) |
| parts | \(\frac {a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}+\frac {b^{2} \left (-\frac {256}{693}+\frac {\sinh \left (d x +c \right )^{10}}{11}-\frac {10 \sinh \left (d x +c \right )^{8}}{99}+\frac {80 \sinh \left (d x +c \right )^{6}}{693}-\frac {32 \sinh \left (d x +c \right )^{4}}{231}+\frac {128 \sinh \left (d x +c \right )^{2}}{693}\right ) \cosh \left (d x +c \right )}{d}+\frac {2 a b \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )}{d}\) | \(137\) |
| risch | \(\frac {b^{2} {\mathrm e}^{11 d x +11 c}}{22528 d}-\frac {11 b^{2} {\mathrm e}^{9 d x +9 c}}{18432 d}+\frac {b \,{\mathrm e}^{7 d x +7 c} a}{448 d}+\frac {55 b^{2} {\mathrm e}^{7 d x +7 c}}{14336 d}-\frac {7 b \,{\mathrm e}^{5 d x +5 c} a}{320 d}-\frac {33 b^{2} {\mathrm e}^{5 d x +5 c}}{2048 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2}}{24 d}+\frac {7 \,{\mathrm e}^{3 d x +3 c} a b}{64 d}+\frac {55 \,{\mathrm e}^{3 d x +3 c} b^{2}}{1024 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{2}}{8 d}-\frac {35 \,{\mathrm e}^{d x +c} a b}{64 d}-\frac {231 \,{\mathrm e}^{d x +c} b^{2}}{1024 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2}}{8 d}-\frac {35 \,{\mathrm e}^{-d x -c} a b}{64 d}-\frac {231 \,{\mathrm e}^{-d x -c} b^{2}}{1024 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{2}}{24 d}+\frac {7 \,{\mathrm e}^{-3 d x -3 c} a b}{64 d}+\frac {55 \,{\mathrm e}^{-3 d x -3 c} b^{2}}{1024 d}-\frac {7 b \,{\mathrm e}^{-5 d x -5 c} a}{320 d}-\frac {33 b^{2} {\mathrm e}^{-5 d x -5 c}}{2048 d}+\frac {b \,{\mathrm e}^{-7 d x -7 c} a}{448 d}+\frac {55 b^{2} {\mathrm e}^{-7 d x -7 c}}{14336 d}-\frac {11 b^{2} {\mathrm e}^{-9 d x -9 c}}{18432 d}+\frac {b^{2} {\mathrm e}^{-11 d x -11 c}}{22528 d}\) | \(372\) |
1/3548160*((295680*a^2+776160*a*b+381150*b^2)*cosh(3*d*x+3*c)-155232*(a+16 5/224*b)*b*cosh(5*d*x+5*c)+15840*(a+55/32*b)*b*cosh(7*d*x+7*c)+315*b^2*cos h(11*d*x+11*c)-4235*b^2*cosh(9*d*x+9*c)+(-2661120*a^2-3880800*a*b-1600830* b^2)*cosh(d*x+c)-2365440*a^2-3244032*a*b-1310720*b^2)/d
Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (110) = 220\).
Time = 0.27 (sec) , antiderivative size = 404, normalized size of antiderivative = 3.37 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\frac {315 \, b^{2} \cosh \left (d x + c\right )^{11} + 3465 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{10} - 4235 \, b^{2} \cosh \left (d x + c\right )^{9} + 3465 \, {\left (15 \, b^{2} \cosh \left (d x + c\right )^{3} - 11 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} + 495 \, {\left (32 \, a b + 55 \, b^{2}\right )} \cosh \left (d x + c\right )^{7} + 1155 \, {\left (126 \, b^{2} \cosh \left (d x + c\right )^{5} - 308 \, b^{2} \cosh \left (d x + c\right )^{3} + 3 \, {\left (32 \, a b + 55 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} - 693 \, {\left (224 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 3465 \, {\left (30 \, b^{2} \cosh \left (d x + c\right )^{7} - 154 \, b^{2} \cosh \left (d x + c\right )^{5} + 5 \, {\left (32 \, a b + 55 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (224 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 2310 \, {\left (128 \, a^{2} + 336 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3465 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{9} - 44 \, b^{2} \cosh \left (d x + c\right )^{7} + 3 \, {\left (32 \, a b + 55 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} - 2 \, {\left (224 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 2 \, {\left (128 \, a^{2} + 336 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 6930 \, {\left (384 \, a^{2} + 560 \, a b + 231 \, b^{2}\right )} \cosh \left (d x + c\right )}{3548160 \, d} \]
1/3548160*(315*b^2*cosh(d*x + c)^11 + 3465*b^2*cosh(d*x + c)*sinh(d*x + c) ^10 - 4235*b^2*cosh(d*x + c)^9 + 3465*(15*b^2*cosh(d*x + c)^3 - 11*b^2*cos h(d*x + c))*sinh(d*x + c)^8 + 495*(32*a*b + 55*b^2)*cosh(d*x + c)^7 + 1155 *(126*b^2*cosh(d*x + c)^5 - 308*b^2*cosh(d*x + c)^3 + 3*(32*a*b + 55*b^2)* cosh(d*x + c))*sinh(d*x + c)^6 - 693*(224*a*b + 165*b^2)*cosh(d*x + c)^5 + 3465*(30*b^2*cosh(d*x + c)^7 - 154*b^2*cosh(d*x + c)^5 + 5*(32*a*b + 55*b ^2)*cosh(d*x + c)^3 - (224*a*b + 165*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 2310*(128*a^2 + 336*a*b + 165*b^2)*cosh(d*x + c)^3 + 3465*(5*b^2*cosh(d*x + c)^9 - 44*b^2*cosh(d*x + c)^7 + 3*(32*a*b + 55*b^2)*cosh(d*x + c)^5 - 2 *(224*a*b + 165*b^2)*cosh(d*x + c)^3 + 2*(128*a^2 + 336*a*b + 165*b^2)*cos h(d*x + c))*sinh(d*x + c)^2 - 6930*(384*a^2 + 560*a*b + 231*b^2)*cosh(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (109) = 218\).
Time = 1.86 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.33 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\begin {cases} \frac {a^{2} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {2 a^{2} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {2 a b \sinh ^{6}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 a b \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {16 a b \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {32 a b \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {b^{2} \sinh ^{10}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {10 b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {16 b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{3 d} - \frac {32 b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{7 d} + \frac {128 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{63 d} - \frac {256 b^{2} \cosh ^{11}{\left (c + d x \right )}}{693 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{2} \sinh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a**2*sinh(c + d*x)**2*cosh(c + d*x)/d - 2*a**2*cosh(c + d*x)**3 /(3*d) + 2*a*b*sinh(c + d*x)**6*cosh(c + d*x)/d - 4*a*b*sinh(c + d*x)**4*c osh(c + d*x)**3/d + 16*a*b*sinh(c + d*x)**2*cosh(c + d*x)**5/(5*d) - 32*a* b*cosh(c + d*x)**7/(35*d) + b**2*sinh(c + d*x)**10*cosh(c + d*x)/d - 10*b* *2*sinh(c + d*x)**8*cosh(c + d*x)**3/(3*d) + 16*b**2*sinh(c + d*x)**6*cosh (c + d*x)**5/(3*d) - 32*b**2*sinh(c + d*x)**4*cosh(c + d*x)**7/(7*d) + 128 *b**2*sinh(c + d*x)**2*cosh(c + d*x)**9/(63*d) - 256*b**2*cosh(c + d*x)**1 1/(693*d), Ne(d, 0)), (x*(a + b*sinh(c)**4)**2*sinh(c)**3, True))
Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (110) = 220\).
Time = 0.22 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.56 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=-\frac {1}{1419264} \, b^{2} {\left (\frac {{\left (847 \, e^{\left (-2 \, d x - 2 \, c\right )} - 5445 \, e^{\left (-4 \, d x - 4 \, c\right )} + 22869 \, e^{\left (-6 \, d x - 6 \, c\right )} - 76230 \, e^{\left (-8 \, d x - 8 \, c\right )} + 320166 \, e^{\left (-10 \, d x - 10 \, c\right )} - 63\right )} e^{\left (11 \, d x + 11 \, c\right )}}{d} + \frac {320166 \, e^{\left (-d x - c\right )} - 76230 \, e^{\left (-3 \, d x - 3 \, c\right )} + 22869 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5445 \, e^{\left (-7 \, d x - 7 \, c\right )} + 847 \, e^{\left (-9 \, d x - 9 \, c\right )} - 63 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d}\right )} - \frac {1}{2240} \, a b {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{24} \, a^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
-1/1419264*b^2*((847*e^(-2*d*x - 2*c) - 5445*e^(-4*d*x - 4*c) + 22869*e^(- 6*d*x - 6*c) - 76230*e^(-8*d*x - 8*c) + 320166*e^(-10*d*x - 10*c) - 63)*e^ (11*d*x + 11*c)/d + (320166*e^(-d*x - c) - 76230*e^(-3*d*x - 3*c) + 22869* e^(-5*d*x - 5*c) - 5445*e^(-7*d*x - 7*c) + 847*e^(-9*d*x - 9*c) - 63*e^(-1 1*d*x - 11*c))/d) - 1/2240*a*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) - 24 5*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/d) + 1/24*a ^2*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3 *c)/d)
Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (110) = 220\).
Time = 0.34 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.32 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=\frac {b^{2} e^{\left (11 \, d x + 11 \, c\right )}}{22528 \, d} - \frac {11 \, b^{2} e^{\left (9 \, d x + 9 \, c\right )}}{18432 \, d} - \frac {11 \, b^{2} e^{\left (-9 \, d x - 9 \, c\right )}}{18432 \, d} + \frac {b^{2} e^{\left (-11 \, d x - 11 \, c\right )}}{22528 \, d} + \frac {{\left (32 \, a b + 55 \, b^{2}\right )} e^{\left (7 \, d x + 7 \, c\right )}}{14336 \, d} - \frac {{\left (224 \, a b + 165 \, b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{10240 \, d} + \frac {{\left (128 \, a^{2} + 336 \, a b + 165 \, b^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{3072 \, d} - \frac {{\left (384 \, a^{2} + 560 \, a b + 231 \, b^{2}\right )} e^{\left (d x + c\right )}}{1024 \, d} - \frac {{\left (384 \, a^{2} + 560 \, a b + 231 \, b^{2}\right )} e^{\left (-d x - c\right )}}{1024 \, d} + \frac {{\left (128 \, a^{2} + 336 \, a b + 165 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{3072 \, d} - \frac {{\left (224 \, a b + 165 \, b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{10240 \, d} + \frac {{\left (32 \, a b + 55 \, b^{2}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{14336 \, d} \]
1/22528*b^2*e^(11*d*x + 11*c)/d - 11/18432*b^2*e^(9*d*x + 9*c)/d - 11/1843 2*b^2*e^(-9*d*x - 9*c)/d + 1/22528*b^2*e^(-11*d*x - 11*c)/d + 1/14336*(32* a*b + 55*b^2)*e^(7*d*x + 7*c)/d - 1/10240*(224*a*b + 165*b^2)*e^(5*d*x + 5 *c)/d + 1/3072*(128*a^2 + 336*a*b + 165*b^2)*e^(3*d*x + 3*c)/d - 1/1024*(3 84*a^2 + 560*a*b + 231*b^2)*e^(d*x + c)/d - 1/1024*(384*a^2 + 560*a*b + 23 1*b^2)*e^(-d*x - c)/d + 1/3072*(128*a^2 + 336*a*b + 165*b^2)*e^(-3*d*x - 3 *c)/d - 1/10240*(224*a*b + 165*b^2)*e^(-5*d*x - 5*c)/d + 1/14336*(32*a*b + 55*b^2)*e^(-7*d*x - 7*c)/d
Time = 0.37 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.25 \[ \int \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx=-\frac {-\frac {a^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+a^2\,\mathrm {cosh}\left (c+d\,x\right )-\frac {2\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {6\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-2\,a\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3+2\,a\,b\,\mathrm {cosh}\left (c+d\,x\right )-\frac {b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}+\frac {5\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{9}-\frac {10\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+2\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5-\frac {5\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3}{3}+b^2\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]
-(a^2*cosh(c + d*x) + b^2*cosh(c + d*x) - (a^2*cosh(c + d*x)^3)/3 - (5*b^2 *cosh(c + d*x)^3)/3 + 2*b^2*cosh(c + d*x)^5 - (10*b^2*cosh(c + d*x)^7)/7 + (5*b^2*cosh(c + d*x)^9)/9 - (b^2*cosh(c + d*x)^11)/11 + 2*a*b*cosh(c + d* x) - 2*a*b*cosh(c + d*x)^3 + (6*a*b*cosh(c + d*x)^5)/5 - (2*a*b*cosh(c + d *x)^7)/7)/d