Integrand size = 21, antiderivative size = 143 \[ \int \sinh (c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {(a+b)^3 \cosh (c+d x)}{d}-\frac {2 b (a+b)^2 \cosh ^3(c+d x)}{d}+\frac {3 b (a+b) (a+5 b) \cosh ^5(c+d x)}{5 d}-\frac {4 b^2 (3 a+5 b) \cosh ^7(c+d x)}{7 d}+\frac {b^2 (a+5 b) \cosh ^9(c+d x)}{3 d}-\frac {6 b^3 \cosh ^{11}(c+d x)}{11 d}+\frac {b^3 \cosh ^{13}(c+d x)}{13 d} \]
(a+b)^3*cosh(d*x+c)/d-2*b*(a+b)^2*cosh(d*x+c)^3/d+3/5*b*(a+b)*(a+5*b)*cosh (d*x+c)^5/d-4/7*b^2*(3*a+5*b)*cosh(d*x+c)^7/d+1/3*b^2*(a+5*b)*cosh(d*x+c)^ 9/d-6/11*b^3*cosh(d*x+c)^11/d+1/13*b^3*cosh(d*x+c)^13/d
Time = 1.75 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.10 \[ \int \sinh (c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {60060 \left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right ) \cosh (c+d x)-15015 b \left (1280 a^2+1344 a b+429 b^2\right ) \cosh (3 (c+d x))+3003 b \left (768 a^2+1728 a b+715 b^2\right ) \cosh (5 (c+d x))-4290 b^2 (216 a+143 b) \cosh (7 (c+d x))+10010 b^2 (8 a+13 b) \cosh (9 (c+d x))-17745 b^3 \cosh (11 (c+d x))+1155 b^3 \cosh (13 (c+d x))}{61501440 d} \]
(60060*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*Cosh[c + d*x] - 1501 5*b*(1280*a^2 + 1344*a*b + 429*b^2)*Cosh[3*(c + d*x)] + 3003*b*(768*a^2 + 1728*a*b + 715*b^2)*Cosh[5*(c + d*x)] - 4290*b^2*(216*a + 143*b)*Cosh[7*(c + d*x)] + 10010*b^2*(8*a + 13*b)*Cosh[9*(c + d*x)] - 17745*b^3*Cosh[11*(c + d*x)] + 1155*b^3*Cosh[13*(c + d*x)])/(61501440*d)
Time = 0.37 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.88, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 26, 3694, 1403, 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.
| \(\displaystyle \int \sinh (c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -i \sin (i c+i d x) \left (a+b \sin (i c+i d x)^4\right )^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \sin (i c+i d x) \left (b \sin (i c+i d x)^4+a\right )^3dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle \frac {\int \left (b \cosh ^4(c+d x)-2 b \cosh ^2(c+d x)+a+b\right )^3d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1403 |
| \(\displaystyle \frac {\int \left (b^3 \cosh ^{12}(c+d x)-6 b^3 \cosh ^{10}(c+d x)+12 b^3 \left (\frac {a+b}{4 b}+1\right ) \cosh ^8(c+d x)-8 b^3 \left (\frac {3 (a+b)}{2 b}+1\right ) \cosh ^6(c+d x)+12 b^2 (a+b) \left (\frac {a+b}{4 b}+1\right ) \cosh ^4(c+d x)-6 b (a+b)^2 \cosh ^2(c+d x)+a^3 \left (\frac {b \left (3 a^2+3 b a+b^2\right )}{a^3}+1\right )\right )d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{3} b^2 (a+5 b) \cosh ^9(c+d x)-\frac {4}{7} b^2 (3 a+5 b) \cosh ^7(c+d x)+\frac {3}{5} b (a+b) (a+5 b) \cosh ^5(c+d x)-2 b (a+b)^2 \cosh ^3(c+d x)+(a+b)^3 \cosh (c+d x)+\frac {1}{13} b^3 \cosh ^{13}(c+d x)-\frac {6}{11} b^3 \cosh ^{11}(c+d x)}{d}\) |
((a + b)^3*Cosh[c + d*x] - 2*b*(a + b)^2*Cosh[c + d*x]^3 + (3*b*(a + b)*(a + 5*b)*Cosh[c + d*x]^5)/5 - (4*b^2*(3*a + 5*b)*Cosh[c + d*x]^7)/7 + (b^2* (a + 5*b)*Cosh[c + d*x]^9)/3 - (6*b^3*Cosh[c + d*x]^11)/11 + (b^3*Cosh[c + d*x]^13)/13)/d
3.3.9.3.1 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[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandInte grand[(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a *c, 0] && IGtQ[p, 0]
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 = 9.73 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {a^{3} \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )+3 a \,b^{2} \left (\frac {128}{315}+\frac {\sinh \left (d x +c \right )^{8}}{9}-\frac {8 \sinh \left (d x +c \right )^{6}}{63}+\frac {16 \sinh \left (d x +c \right )^{4}}{105}-\frac {64 \sinh \left (d x +c \right )^{2}}{315}\right ) \cosh \left (d x +c \right )+b^{3} \left (\frac {1024}{3003}+\frac {\sinh \left (d x +c \right )^{12}}{13}-\frac {12 \sinh \left (d x +c \right )^{10}}{143}+\frac {40 \sinh \left (d x +c \right )^{8}}{429}-\frac {320 \sinh \left (d x +c \right )^{6}}{3003}+\frac {128 \sinh \left (d x +c \right )^{4}}{1001}-\frac {512 \sinh \left (d x +c \right )^{2}}{3003}\right ) \cosh \left (d x +c \right )}{d}\) | \(176\) |
| default | \(\frac {a^{3} \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )+3 a \,b^{2} \left (\frac {128}{315}+\frac {\sinh \left (d x +c \right )^{8}}{9}-\frac {8 \sinh \left (d x +c \right )^{6}}{63}+\frac {16 \sinh \left (d x +c \right )^{4}}{105}-\frac {64 \sinh \left (d x +c \right )^{2}}{315}\right ) \cosh \left (d x +c \right )+b^{3} \left (\frac {1024}{3003}+\frac {\sinh \left (d x +c \right )^{12}}{13}-\frac {12 \sinh \left (d x +c \right )^{10}}{143}+\frac {40 \sinh \left (d x +c \right )^{8}}{429}-\frac {320 \sinh \left (d x +c \right )^{6}}{3003}+\frac {128 \sinh \left (d x +c \right )^{4}}{1001}-\frac {512 \sinh \left (d x +c \right )^{2}}{3003}\right ) \cosh \left (d x +c \right )}{d}\) | \(176\) |
| parallelrisch | \(\frac {\left (-19219200 a^{2} b -20180160 a \,b^{2}-6441435 b^{3}\right ) \cosh \left (3 d x +3 c \right )+\left (2306304 a^{2} b +5189184 a \,b^{2}+2147145 b^{3}\right ) \cosh \left (5 d x +5 c \right )-926640 \left (a +\frac {143 b}{216}\right ) b^{2} \cosh \left (7 d x +7 c \right )+80080 \left (\frac {13 b}{8}+a \right ) b^{2} \cosh \left (9 d x +9 c \right )-17745 b^{3} \cosh \left (11 d x +11 c \right )+1155 b^{3} \cosh \left (13 d x +13 c \right )+\left (61501440 a^{3}+115315200 a^{2} b +90810720 a \,b^{2}+25765740 b^{3}\right ) \cosh \left (d x +c \right )+61501440 a^{3}+98402304 a^{2} b +74973184 a \,b^{2}+20971520 b^{3}}{61501440 d}\) | \(181\) |
| parts | \(\frac {b^{3} \left (\frac {1024}{3003}+\frac {\sinh \left (d x +c \right )^{12}}{13}-\frac {12 \sinh \left (d x +c \right )^{10}}{143}+\frac {40 \sinh \left (d x +c \right )^{8}}{429}-\frac {320 \sinh \left (d x +c \right )^{6}}{3003}+\frac {128 \sinh \left (d x +c \right )^{4}}{1001}-\frac {512 \sinh \left (d x +c \right )^{2}}{3003}\right ) \cosh \left (d x +c \right )}{d}+\frac {a^{3} \cosh \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \left (\frac {128}{315}+\frac {\sinh \left (d x +c \right )^{8}}{9}-\frac {8 \sinh \left (d x +c \right )^{6}}{63}+\frac {16 \sinh \left (d x +c \right )^{4}}{105}-\frac {64 \sinh \left (d x +c \right )^{2}}{315}\right ) \cosh \left (d x +c \right )}{d}+\frac {3 a^{2} b \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )}{d}\) | \(184\) |
| risch | \(\frac {3 b \,{\mathrm e}^{-5 d x -5 c} a^{2}}{160 d}+\frac {3 b \,{\mathrm e}^{5 d x +5 c} a^{2}}{160 d}+\frac {b^{3} {\mathrm e}^{-13 d x -13 c}}{106496 d}+\frac {b^{3} {\mathrm e}^{13 d x +13 c}}{106496 d}-\frac {13 b^{3} {\mathrm e}^{11 d x +11 c}}{90112 d}+\frac {13 b^{3} {\mathrm e}^{9 d x +9 c}}{12288 d}+\frac {13 b^{3} {\mathrm e}^{-9 d x -9 c}}{12288 d}-\frac {13 b^{3} {\mathrm e}^{-11 d x -11 c}}{90112 d}-\frac {143 b^{3} {\mathrm e}^{7 d x +7 c}}{28672 d}+\frac {143 b^{3} {\mathrm e}^{5 d x +5 c}}{8192 d}-\frac {429 \,{\mathrm e}^{3 d x +3 c} b^{3}}{8192 d}+\frac {{\mathrm e}^{d x +c} a^{3}}{2 d}+\frac {429 \,{\mathrm e}^{d x +c} b^{3}}{2048 d}+\frac {{\mathrm e}^{-d x -c} a^{3}}{2 d}+\frac {429 \,{\mathrm e}^{-d x -c} b^{3}}{2048 d}-\frac {429 \,{\mathrm e}^{-3 d x -3 c} b^{3}}{8192 d}+\frac {143 b^{3} {\mathrm e}^{-5 d x -5 c}}{8192 d}-\frac {143 b^{3} {\mathrm e}^{-7 d x -7 c}}{28672 d}+\frac {15 \,{\mathrm e}^{-d x -c} a^{2} b}{16 d}+\frac {189 \,{\mathrm e}^{-d x -c} a \,b^{2}}{256 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} a^{2} b}{32 d}-\frac {21 \,{\mathrm e}^{-3 d x -3 c} a \,b^{2}}{128 d}+\frac {27 b^{2} {\mathrm e}^{-5 d x -5 c} a}{640 d}+\frac {27 b^{2} {\mathrm e}^{5 d x +5 c} a}{640 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} a^{2} b}{32 d}-\frac {21 \,{\mathrm e}^{3 d x +3 c} a \,b^{2}}{128 d}+\frac {15 \,{\mathrm e}^{d x +c} a^{2} b}{16 d}+\frac {189 \,{\mathrm e}^{d x +c} a \,b^{2}}{256 d}+\frac {b^{2} {\mathrm e}^{-9 d x -9 c} a}{1536 d}-\frac {27 a \,b^{2} {\mathrm e}^{7 d x +7 c}}{3584 d}-\frac {27 a \,b^{2} {\mathrm e}^{-7 d x -7 c}}{3584 d}+\frac {b^{2} {\mathrm e}^{9 d x +9 c} a}{1536 d}\) | \(522\) |
1/d*(a^3*cosh(d*x+c)+3*a^2*b*(8/15+1/5*sinh(d*x+c)^4-4/15*sinh(d*x+c)^2)*c osh(d*x+c)+3*a*b^2*(128/315+1/9*sinh(d*x+c)^8-8/63*sinh(d*x+c)^6+16/105*si nh(d*x+c)^4-64/315*sinh(d*x+c)^2)*cosh(d*x+c)+b^3*(1024/3003+1/13*sinh(d*x +c)^12-12/143*sinh(d*x+c)^10+40/429*sinh(d*x+c)^8-320/3003*sinh(d*x+c)^6+1 28/1001*sinh(d*x+c)^4-512/3003*sinh(d*x+c)^2)*cosh(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (133) = 266\).
Time = 0.26 (sec) , antiderivative size = 594, normalized size of antiderivative = 4.15 \[ \int \sinh (c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {1155 \, b^{3} \cosh \left (d x + c\right )^{13} + 15015 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{12} - 17745 \, b^{3} \cosh \left (d x + c\right )^{11} + 15015 \, {\left (22 \, b^{3} \cosh \left (d x + c\right )^{3} - 13 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{10} + 10010 \, {\left (8 \, a b^{2} + 13 \, b^{3}\right )} \cosh \left (d x + c\right )^{9} + 45045 \, {\left (33 \, b^{3} \cosh \left (d x + c\right )^{5} - 65 \, b^{3} \cosh \left (d x + c\right )^{3} + 2 \, {\left (8 \, a b^{2} + 13 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} - 4290 \, {\left (216 \, a b^{2} + 143 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 30030 \, {\left (66 \, b^{3} \cosh \left (d x + c\right )^{7} - 273 \, b^{3} \cosh \left (d x + c\right )^{5} + 28 \, {\left (8 \, a b^{2} + 13 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - {\left (216 \, a b^{2} + 143 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 3003 \, {\left (768 \, a^{2} b + 1728 \, a b^{2} + 715 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 15015 \, {\left (55 \, b^{3} \cosh \left (d x + c\right )^{9} - 390 \, b^{3} \cosh \left (d x + c\right )^{7} + 84 \, {\left (8 \, a b^{2} + 13 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 10 \, {\left (216 \, a b^{2} + 143 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (768 \, a^{2} b + 1728 \, a b^{2} + 715 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} - 15015 \, {\left (1280 \, a^{2} b + 1344 \, a b^{2} + 429 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 15015 \, {\left (6 \, b^{3} \cosh \left (d x + c\right )^{11} - 65 \, b^{3} \cosh \left (d x + c\right )^{9} + 24 \, {\left (8 \, a b^{2} + 13 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} - 6 \, {\left (216 \, a b^{2} + 143 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 2 \, {\left (768 \, a^{2} b + 1728 \, a b^{2} + 715 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (1280 \, a^{2} b + 1344 \, a b^{2} + 429 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 60060 \, {\left (1024 \, a^{3} + 1920 \, a^{2} b + 1512 \, a b^{2} + 429 \, b^{3}\right )} \cosh \left (d x + c\right )}{61501440 \, d} \]
1/61501440*(1155*b^3*cosh(d*x + c)^13 + 15015*b^3*cosh(d*x + c)*sinh(d*x + c)^12 - 17745*b^3*cosh(d*x + c)^11 + 15015*(22*b^3*cosh(d*x + c)^3 - 13*b ^3*cosh(d*x + c))*sinh(d*x + c)^10 + 10010*(8*a*b^2 + 13*b^3)*cosh(d*x + c )^9 + 45045*(33*b^3*cosh(d*x + c)^5 - 65*b^3*cosh(d*x + c)^3 + 2*(8*a*b^2 + 13*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 - 4290*(216*a*b^2 + 143*b^3)*cosh (d*x + c)^7 + 30030*(66*b^3*cosh(d*x + c)^7 - 273*b^3*cosh(d*x + c)^5 + 28 *(8*a*b^2 + 13*b^3)*cosh(d*x + c)^3 - (216*a*b^2 + 143*b^3)*cosh(d*x + c)) *sinh(d*x + c)^6 + 3003*(768*a^2*b + 1728*a*b^2 + 715*b^3)*cosh(d*x + c)^5 + 15015*(55*b^3*cosh(d*x + c)^9 - 390*b^3*cosh(d*x + c)^7 + 84*(8*a*b^2 + 13*b^3)*cosh(d*x + c)^5 - 10*(216*a*b^2 + 143*b^3)*cosh(d*x + c)^3 + (768 *a^2*b + 1728*a*b^2 + 715*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 15015*(128 0*a^2*b + 1344*a*b^2 + 429*b^3)*cosh(d*x + c)^3 + 15015*(6*b^3*cosh(d*x + c)^11 - 65*b^3*cosh(d*x + c)^9 + 24*(8*a*b^2 + 13*b^3)*cosh(d*x + c)^7 - 6 *(216*a*b^2 + 143*b^3)*cosh(d*x + c)^5 + 2*(768*a^2*b + 1728*a*b^2 + 715*b ^3)*cosh(d*x + c)^3 - 3*(1280*a^2*b + 1344*a*b^2 + 429*b^3)*cosh(d*x + c)) *sinh(d*x + c)^2 + 60060*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*co sh(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (131) = 262\).
Time = 3.48 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.64 \[ \int \sinh (c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\begin {cases} \frac {a^{3} \cosh {\left (c + d x \right )}}{d} + \frac {3 a^{2} b \sinh ^{4}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {8 a^{2} b \cosh ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 a b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {8 a b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {48 a b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac {192 a b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac {128 a b^{2} \cosh ^{9}{\left (c + d x \right )}}{105 d} + \frac {b^{3} \sinh ^{12}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{d} - \frac {4 b^{3} \sinh ^{10}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac {8 b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{d} - \frac {64 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{7 d} + \frac {128 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{21 d} - \frac {512 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{231 d} + \frac {1024 b^{3} \cosh ^{13}{\left (c + d x \right )}}{3003 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{3} \sinh {\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a**3*cosh(c + d*x)/d + 3*a**2*b*sinh(c + d*x)**4*cosh(c + d*x)/ d - 4*a**2*b*sinh(c + d*x)**2*cosh(c + d*x)**3/d + 8*a**2*b*cosh(c + d*x)* *5/(5*d) + 3*a*b**2*sinh(c + d*x)**8*cosh(c + d*x)/d - 8*a*b**2*sinh(c + d *x)**6*cosh(c + d*x)**3/d + 48*a*b**2*sinh(c + d*x)**4*cosh(c + d*x)**5/(5 *d) - 192*a*b**2*sinh(c + d*x)**2*cosh(c + d*x)**7/(35*d) + 128*a*b**2*cos h(c + d*x)**9/(105*d) + b**3*sinh(c + d*x)**12*cosh(c + d*x)/d - 4*b**3*si nh(c + d*x)**10*cosh(c + d*x)**3/d + 8*b**3*sinh(c + d*x)**8*cosh(c + d*x) **5/d - 64*b**3*sinh(c + d*x)**6*cosh(c + d*x)**7/(7*d) + 128*b**3*sinh(c + d*x)**4*cosh(c + d*x)**9/(21*d) - 512*b**3*sinh(c + d*x)**2*cosh(c + d*x )**11/(231*d) + 1024*b**3*cosh(c + d*x)**13/(3003*d), Ne(d, 0)), (x*(a + b *sinh(c)**4)**3*sinh(c), True))
Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (133) = 266\).
Time = 0.19 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.79 \[ \int \sinh (c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {1}{24600576} \, b^{3} {\left (\frac {{\left (3549 \, e^{\left (-2 \, d x - 2 \, c\right )} - 26026 \, e^{\left (-4 \, d x - 4 \, c\right )} + 122694 \, e^{\left (-6 \, d x - 6 \, c\right )} - 429429 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1288287 \, e^{\left (-10 \, d x - 10 \, c\right )} - 5153148 \, e^{\left (-12 \, d x - 12 \, c\right )} - 231\right )} e^{\left (13 \, d x + 13 \, c\right )}}{d} - \frac {5153148 \, e^{\left (-d x - c\right )} - 1288287 \, e^{\left (-3 \, d x - 3 \, c\right )} + 429429 \, e^{\left (-5 \, d x - 5 \, c\right )} - 122694 \, e^{\left (-7 \, d x - 7 \, c\right )} + 26026 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3549 \, e^{\left (-11 \, d x - 11 \, c\right )} + 231 \, e^{\left (-13 \, d x - 13 \, c\right )}}{d}\right )} - \frac {1}{53760} \, a b^{2} {\left (\frac {{\left (405 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2268 \, e^{\left (-4 \, d x - 4 \, c\right )} + 8820 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39690 \, e^{\left (-8 \, d x - 8 \, c\right )} - 35\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac {39690 \, e^{\left (-d x - c\right )} - 8820 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2268 \, e^{\left (-5 \, d x - 5 \, c\right )} - 405 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d}\right )} + \frac {1}{160} \, a^{2} b {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {a^{3} \cosh \left (d x + c\right )}{d} \]
-1/24600576*b^3*((3549*e^(-2*d*x - 2*c) - 26026*e^(-4*d*x - 4*c) + 122694* e^(-6*d*x - 6*c) - 429429*e^(-8*d*x - 8*c) + 1288287*e^(-10*d*x - 10*c) - 5153148*e^(-12*d*x - 12*c) - 231)*e^(13*d*x + 13*c)/d - (5153148*e^(-d*x - c) - 1288287*e^(-3*d*x - 3*c) + 429429*e^(-5*d*x - 5*c) - 122694*e^(-7*d* x - 7*c) + 26026*e^(-9*d*x - 9*c) - 3549*e^(-11*d*x - 11*c) + 231*e^(-13*d *x - 13*c))/d) - 1/53760*a*b^2*((405*e^(-2*d*x - 2*c) - 2268*e^(-4*d*x - 4 *c) + 8820*e^(-6*d*x - 6*c) - 39690*e^(-8*d*x - 8*c) - 35)*e^(9*d*x + 9*c) /d - (39690*e^(-d*x - c) - 8820*e^(-3*d*x - 3*c) + 2268*e^(-5*d*x - 5*c) - 405*e^(-7*d*x - 7*c) + 35*e^(-9*d*x - 9*c))/d) + 1/160*a^2*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) + a^3*cosh(d*x + c)/d
Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (133) = 266\).
Time = 0.41 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.60 \[ \int \sinh (c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {b^{3} e^{\left (13 \, d x + 13 \, c\right )}}{106496 \, d} - \frac {13 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )}}{90112 \, d} - \frac {13 \, b^{3} e^{\left (-11 \, d x - 11 \, c\right )}}{90112 \, d} + \frac {b^{3} e^{\left (-13 \, d x - 13 \, c\right )}}{106496 \, d} + \frac {{\left (8 \, a b^{2} + 13 \, b^{3}\right )} e^{\left (9 \, d x + 9 \, c\right )}}{12288 \, d} - \frac {{\left (216 \, a b^{2} + 143 \, b^{3}\right )} e^{\left (7 \, d x + 7 \, c\right )}}{28672 \, d} + \frac {{\left (768 \, a^{2} b + 1728 \, a b^{2} + 715 \, b^{3}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{40960 \, d} - \frac {{\left (1280 \, a^{2} b + 1344 \, a b^{2} + 429 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{8192 \, d} + \frac {{\left (1024 \, a^{3} + 1920 \, a^{2} b + 1512 \, a b^{2} + 429 \, b^{3}\right )} e^{\left (d x + c\right )}}{2048 \, d} + \frac {{\left (1024 \, a^{3} + 1920 \, a^{2} b + 1512 \, a b^{2} + 429 \, b^{3}\right )} e^{\left (-d x - c\right )}}{2048 \, d} - \frac {{\left (1280 \, a^{2} b + 1344 \, a b^{2} + 429 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{8192 \, d} + \frac {{\left (768 \, a^{2} b + 1728 \, a b^{2} + 715 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{40960 \, d} - \frac {{\left (216 \, a b^{2} + 143 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{28672 \, d} + \frac {{\left (8 \, a b^{2} + 13 \, b^{3}\right )} e^{\left (-9 \, d x - 9 \, c\right )}}{12288 \, d} \]
1/106496*b^3*e^(13*d*x + 13*c)/d - 13/90112*b^3*e^(11*d*x + 11*c)/d - 13/9 0112*b^3*e^(-11*d*x - 11*c)/d + 1/106496*b^3*e^(-13*d*x - 13*c)/d + 1/1228 8*(8*a*b^2 + 13*b^3)*e^(9*d*x + 9*c)/d - 1/28672*(216*a*b^2 + 143*b^3)*e^( 7*d*x + 7*c)/d + 1/40960*(768*a^2*b + 1728*a*b^2 + 715*b^3)*e^(5*d*x + 5*c )/d - 1/8192*(1280*a^2*b + 1344*a*b^2 + 429*b^3)*e^(3*d*x + 3*c)/d + 1/204 8*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*e^(d*x + c)/d + 1/2048*(1 024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*e^(-d*x - c)/d - 1/8192*(1280 *a^2*b + 1344*a*b^2 + 429*b^3)*e^(-3*d*x - 3*c)/d + 1/40960*(768*a^2*b + 1 728*a*b^2 + 715*b^3)*e^(-5*d*x - 5*c)/d - 1/28672*(216*a*b^2 + 143*b^3)*e^ (-7*d*x - 7*c)/d + 1/12288*(8*a*b^2 + 13*b^3)*e^(-9*d*x - 9*c)/d
Time = 1.83 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.48 \[ \int \sinh (c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {a^3\,\mathrm {cosh}\left (c+d\,x\right )+\frac {3\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-2\,a^2\,b\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,a^2\,b\,\mathrm {cosh}\left (c+d\,x\right )+\frac {a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{3}-\frac {12\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+\frac {18\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^5}{5}-4\,a\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3+3\,a\,b^2\,\mathrm {cosh}\left (c+d\,x\right )+\frac {b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{13}}{13}-\frac {6\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^{11}}{11}+\frac {5\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^9}{3}-\frac {20\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^7}{7}+3\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^5-2\,b^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3+b^3\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]
(a^3*cosh(c + d*x) + b^3*cosh(c + d*x) - 2*b^3*cosh(c + d*x)^3 + 3*b^3*cos h(c + d*x)^5 - (20*b^3*cosh(c + d*x)^7)/7 + (5*b^3*cosh(c + d*x)^9)/3 - (6 *b^3*cosh(c + d*x)^11)/11 + (b^3*cosh(c + d*x)^13)/13 - 4*a*b^2*cosh(c + d *x)^3 - 2*a^2*b*cosh(c + d*x)^3 + (18*a*b^2*cosh(c + d*x)^5)/5 + (3*a^2*b* cosh(c + d*x)^5)/5 - (12*a*b^2*cosh(c + d*x)^7)/7 + (a*b^2*cosh(c + d*x)^9 )/3 + 3*a*b^2*cosh(c + d*x) + 3*a^2*b*cosh(c + d*x))/d