Integrand size = 23, antiderivative size = 182 \[ \int \text {csch}^{16}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {(a+b)^3 \coth (c+d x)}{d}-\frac {(a+b)^2 (7 a+b) \coth ^3(c+d x)}{3 d}+\frac {3 a (a+b) (7 a+3 b) \coth ^5(c+d x)}{5 d}-\frac {a \left (35 a^2+30 a b+3 b^2\right ) \coth ^7(c+d x)}{7 d}+\frac {5 a^2 (7 a+3 b) \coth ^9(c+d x)}{9 d}-\frac {3 a^2 (7 a+b) \coth ^{11}(c+d x)}{11 d}+\frac {7 a^3 \coth ^{13}(c+d x)}{13 d}-\frac {a^3 \coth ^{15}(c+d x)}{15 d} \]
(a+b)^3*coth(d*x+c)/d-1/3*(a+b)^2*(7*a+b)*coth(d*x+c)^3/d+3/5*a*(a+b)*(7*a +3*b)*coth(d*x+c)^5/d-1/7*a*(35*a^2+30*a*b+3*b^2)*coth(d*x+c)^7/d+5/9*a^2* (7*a+3*b)*coth(d*x+c)^9/d-3/11*a^2*(7*a+b)*coth(d*x+c)^11/d+7/13*a^3*coth( d*x+c)^13/d-1/15*a^3*coth(d*x+c)^15/d
Leaf count is larger than twice the leaf count of optimal. \(404\) vs. \(2(182)=364\).
Time = 4.72 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.22 \[ \int \text {csch}^{16}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {\left (45045 \left (1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right ) \cosh (c+d x)-5005 \left (7168 a^3+20352 a^2 b+16632 a b^2+4785 b^3\right ) \cosh (3 (c+d x))+21525504 a^3 \cosh (5 (c+d x))+74954880 a^2 b \cosh (5 (c+d x))+74162088 a b^2 \cosh (5 (c+d x))+23288265 b^3 \cosh (5 (c+d x))-9784320 a^3 \cosh (7 (c+d x))-34070400 a^2 b \cosh (7 (c+d x))-39999960 a b^2 \cosh (7 (c+d x))-14189175 b^3 \cosh (7 (c+d x))+3261440 a^3 \cosh (9 (c+d x))+11356800 a^2 b \cosh (9 (c+d x))+14054040 a b^2 \cosh (9 (c+d x))+5720715 b^3 \cosh (9 (c+d x))-752640 a^3 \cosh (11 (c+d x))-2620800 a^2 b \cosh (11 (c+d x))-3243240 a b^2 \cosh (11 (c+d x))-1486485 b^3 \cosh (11 (c+d x))+107520 a^3 \cosh (13 (c+d x))+374400 a^2 b \cosh (13 (c+d x))+463320 a b^2 \cosh (13 (c+d x))+225225 b^3 \cosh (13 (c+d x))-7168 a^3 \cosh (15 (c+d x))-24960 a^2 b \cosh (15 (c+d x))-30888 a b^2 \cosh (15 (c+d x))-15015 b^3 \cosh (15 (c+d x))\right ) \text {csch}^{15}(c+d x)}{369008640 d} \]
-1/369008640*((45045*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*Cosh[c + d*x] - 5005*(7168*a^3 + 20352*a^2*b + 16632*a*b^2 + 4785*b^3)*Cosh[3*(c + d*x)] + 21525504*a^3*Cosh[5*(c + d*x)] + 74954880*a^2*b*Cosh[5*(c + d*x) ] + 74162088*a*b^2*Cosh[5*(c + d*x)] + 23288265*b^3*Cosh[5*(c + d*x)] - 97 84320*a^3*Cosh[7*(c + d*x)] - 34070400*a^2*b*Cosh[7*(c + d*x)] - 39999960* a*b^2*Cosh[7*(c + d*x)] - 14189175*b^3*Cosh[7*(c + d*x)] + 3261440*a^3*Cos h[9*(c + d*x)] + 11356800*a^2*b*Cosh[9*(c + d*x)] + 14054040*a*b^2*Cosh[9* (c + d*x)] + 5720715*b^3*Cosh[9*(c + d*x)] - 752640*a^3*Cosh[11*(c + d*x)] - 2620800*a^2*b*Cosh[11*(c + d*x)] - 3243240*a*b^2*Cosh[11*(c + d*x)] - 1 486485*b^3*Cosh[11*(c + d*x)] + 107520*a^3*Cosh[13*(c + d*x)] + 374400*a^2 *b*Cosh[13*(c + d*x)] + 463320*a*b^2*Cosh[13*(c + d*x)] + 225225*b^3*Cosh[ 13*(c + d*x)] - 7168*a^3*Cosh[15*(c + d*x)] - 24960*a^2*b*Cosh[15*(c + d*x )] - 30888*a*b^2*Cosh[15*(c + d*x)] - 15015*b^3*Cosh[15*(c + d*x)])*Csch[c + d*x]^15)/d
Time = 0.39 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3696, 1584, 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.
| \(\displaystyle \int \text {csch}^{16}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin (i c+i d x)^4\right )^3}{\sin (i c+i d x)^{16}}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \coth ^{16}(c+d x) \left (1-\tanh ^2(c+d x)\right ) \left ((a+b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^3d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1584 |
| \(\displaystyle \frac {\int \left (a^3 \coth ^{16}(c+d x)-7 a^3 \coth ^{14}(c+d x)+3 a^2 (7 a+b) \coth ^{12}(c+d x)-5 a^2 (7 a+3 b) \coth ^{10}(c+d x)+a \left (35 a^2+30 b a+3 b^2\right ) \coth ^8(c+d x)+3 a (-7 a-3 b) (a+b) \coth ^6(c+d x)+(a+b)^2 (7 a+b) \coth ^4(c+d x)-(a+b)^3 \coth ^2(c+d x)\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{15} a^3 \coth ^{15}(c+d x)+\frac {7}{13} a^3 \coth ^{13}(c+d x)-\frac {1}{7} a \left (35 a^2+30 a b+3 b^2\right ) \coth ^7(c+d x)-\frac {3}{11} a^2 (7 a+b) \coth ^{11}(c+d x)+\frac {5}{9} a^2 (7 a+3 b) \coth ^9(c+d x)+\frac {3}{5} a (a+b) (7 a+3 b) \coth ^5(c+d x)-\frac {1}{3} (a+b)^2 (7 a+b) \coth ^3(c+d x)+(a+b)^3 \coth (c+d x)}{d}\) |
((a + b)^3*Coth[c + d*x] - ((a + b)^2*(7*a + b)*Coth[c + d*x]^3)/3 + (3*a* (a + b)*(7*a + 3*b)*Coth[c + d*x]^5)/5 - (a*(35*a^2 + 30*a*b + 3*b^2)*Coth [c + d*x]^7)/7 + (5*a^2*(7*a + 3*b)*Coth[c + d*x]^9)/9 - (3*a^2*(7*a + b)* Coth[c + d*x]^11)/11 + (7*a^3*Coth[c + d*x]^13)/13 - (a^3*Coth[c + d*x]^15 )/15)/d
3.3.26.3.1 Defintions of rubi rules used
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 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[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2) ^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] & & IntegerQ[m/2] && IntegerQ[p]
Time = 2.25 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.20
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {2048}{6435}-\frac {\operatorname {csch}\left (d x +c \right )^{14}}{15}+\frac {14 \operatorname {csch}\left (d x +c \right )^{12}}{195}-\frac {56 \operatorname {csch}\left (d x +c \right )^{10}}{715}+\frac {112 \operatorname {csch}\left (d x +c \right )^{8}}{1287}-\frac {128 \operatorname {csch}\left (d x +c \right )^{6}}{1287}+\frac {256 \operatorname {csch}\left (d x +c \right )^{4}}{2145}-\frac {1024 \operatorname {csch}\left (d x +c \right )^{2}}{6435}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (\frac {256}{693}-\frac {\operatorname {csch}\left (d x +c \right )^{10}}{11}+\frac {10 \operatorname {csch}\left (d x +c \right )^{8}}{99}-\frac {80 \operatorname {csch}\left (d x +c \right )^{6}}{693}+\frac {32 \operatorname {csch}\left (d x +c \right )^{4}}{231}-\frac {128 \operatorname {csch}\left (d x +c \right )^{2}}{693}\right ) \coth \left (d x +c \right )+3 a \,b^{2} \left (\frac {16}{35}-\frac {\operatorname {csch}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {csch}\left (d x +c \right )^{4}}{35}-\frac {8 \operatorname {csch}\left (d x +c \right )^{2}}{35}\right ) \coth \left (d x +c \right )+b^{3} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )}{d}\) | \(218\) |
| default | \(\frac {a^{3} \left (\frac {2048}{6435}-\frac {\operatorname {csch}\left (d x +c \right )^{14}}{15}+\frac {14 \operatorname {csch}\left (d x +c \right )^{12}}{195}-\frac {56 \operatorname {csch}\left (d x +c \right )^{10}}{715}+\frac {112 \operatorname {csch}\left (d x +c \right )^{8}}{1287}-\frac {128 \operatorname {csch}\left (d x +c \right )^{6}}{1287}+\frac {256 \operatorname {csch}\left (d x +c \right )^{4}}{2145}-\frac {1024 \operatorname {csch}\left (d x +c \right )^{2}}{6435}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (\frac {256}{693}-\frac {\operatorname {csch}\left (d x +c \right )^{10}}{11}+\frac {10 \operatorname {csch}\left (d x +c \right )^{8}}{99}-\frac {80 \operatorname {csch}\left (d x +c \right )^{6}}{693}+\frac {32 \operatorname {csch}\left (d x +c \right )^{4}}{231}-\frac {128 \operatorname {csch}\left (d x +c \right )^{2}}{693}\right ) \coth \left (d x +c \right )+3 a \,b^{2} \left (\frac {16}{35}-\frac {\operatorname {csch}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {csch}\left (d x +c \right )^{4}}{35}-\frac {8 \operatorname {csch}\left (d x +c \right )^{2}}{35}\right ) \coth \left (d x +c \right )+b^{3} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )}{d}\) | \(218\) |
| parallelrisch | \(-\frac {7 \left (\left (-\frac {3267}{7168} b^{3}-\frac {3}{13} a^{3}-\frac {45}{56} a^{2} b -\frac {891}{896} a \,b^{2}\right ) \cosh \left (11 d x +11 c \right )+\left (\frac {495}{7168} b^{3}+\frac {3}{91} a^{3}+\frac {891}{6272} a \,b^{2}+\frac {45}{392} a^{2} b \right ) \cosh \left (13 d x +13 c \right )+\left (-\frac {33}{7168} b^{3}-\frac {1}{455} a^{3}-\frac {3}{392} a^{2} b -\frac {297}{31360} a \,b^{2}\right ) \cosh \left (15 d x +15 c \right )+\left (-11 a^{3}-\frac {1749}{56} a^{2} b -\frac {3267}{128} a \,b^{2}-\frac {52635}{7168} b^{3}\right ) \cosh \left (3 d x +3 c \right )+\left (\frac {33}{5} a^{3}+\frac {51183}{7168} b^{3}+\frac {1287}{56} a^{2} b +\frac {14553}{640} a \,b^{2}\right ) \cosh \left (5 d x +5 c \right )+\left (-\frac {4455}{1024} b^{3}-3 a^{3}-\frac {585}{56} a^{2} b -\frac {10989}{896} a \,b^{2}\right ) \cosh \left (7 d x +7 c \right )+\left (a^{3}+\frac {12573}{7168} b^{3}+\frac {3861}{896} a \,b^{2}+\frac {195}{56} a^{2} b \right ) \cosh \left (9 d x +9 c \right )+\frac {99 \cosh \left (d x +c \right ) \left (a^{3}+\frac {9}{8} a^{2} b +\frac {105}{128} a \,b^{2}+\frac {231}{1024} b^{3}\right )}{7}\right ) \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{15} \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{25952256 d}\) | \(287\) |
| risch | \(-\frac {4 \left (34189155 b^{3} {\mathrm e}^{10 d x +10 c}-9784320 a^{3} {\mathrm e}^{8 d x +8 c}+3261440 a^{3} {\mathrm e}^{6 d x +6 c}-555555 b^{3} {\mathrm e}^{24 d x +24 c}+3153150 b^{3} {\mathrm e}^{22 d x +22 c}-10900890 b^{3} {\mathrm e}^{20 d x +20 c}+25600575 b^{3} {\mathrm e}^{18 d x +18 c}-43108065 b^{3} {\mathrm e}^{16 d x +16 c}+53513460 b^{3} {\mathrm e}^{14 d x +14 c}-24960 a^{2} b -7168 a^{3}-15015 b^{3}-30888 a \,b^{2}-35875840 a^{3} {\mathrm e}^{12 d x +12 c}-49549500 b^{3} {\mathrm e}^{12 d x +12 c}+21525504 a^{3} {\mathrm e}^{10 d x +10 c}-17342325 b^{3} {\mathrm e}^{8 d x +8 c}-41081040 \,{\mathrm e}^{8 d x +8 c} a \,b^{2}+11356800 b \,a^{2} {\mathrm e}^{6 d x +6 c}+14054040 a \,b^{2} {\mathrm e}^{6 d x +6 c}+374400 a^{2} b \,{\mathrm e}^{2 d x +2 c}-2620800 a^{2} b \,{\mathrm e}^{4 d x +4 c}+463320 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+45045 b^{3} {\mathrm e}^{26 d x +26 c}+46126080 a^{3} {\mathrm e}^{14 d x +14 c}+107520 a^{3} {\mathrm e}^{2 d x +2 c}+225225 b^{3} {\mathrm e}^{2 d x +2 c}+6276270 b^{3} {\mathrm e}^{6 d x +6 c}-752640 a^{3} {\mathrm e}^{4 d x +4 c}-1531530 b^{3} {\mathrm e}^{4 d x +4 c}-3243240 \,{\mathrm e}^{4 d x +4 c} b^{2} a -9297288 a \,b^{2} {\mathrm e}^{20 d x +20 c}+35675640 a \,b^{2} {\mathrm e}^{18 d x +18 c}-54362880 a^{2} b \,{\mathrm e}^{16 d x +16 c}-80463240 a \,b^{2} {\mathrm e}^{16 d x +16 c}+106254720 a^{2} b \,{\mathrm e}^{14 d x +14 c}+1081080 a \,b^{2} {\mathrm e}^{22 d x +22 c}+11531520 a^{2} b \,{\mathrm e}^{18 d x +18 c}+118301040 a \,b^{2} {\mathrm e}^{14 d x +14 c}-113393280 a^{2} b \,{\mathrm e}^{12 d x +12 c}-118918800 a \,b^{2} {\mathrm e}^{12 d x +12 c}+74954880 a^{2} b \,{\mathrm e}^{10 d x +10 c}+83459376 a \,b^{2} {\mathrm e}^{10 d x +10 c}-34070400 a^{2} b \,{\mathrm e}^{8 d x +8 c}\right )}{45045 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{15}}\) | \(622\) |
1/d*(a^3*(2048/6435-1/15*csch(d*x+c)^14+14/195*csch(d*x+c)^12-56/715*csch( d*x+c)^10+112/1287*csch(d*x+c)^8-128/1287*csch(d*x+c)^6+256/2145*csch(d*x+ c)^4-1024/6435*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(256/693-1/11*csch(d*x+c )^10+10/99*csch(d*x+c)^8-80/693*csch(d*x+c)^6+32/231*csch(d*x+c)^4-128/693 *csch(d*x+c)^2)*coth(d*x+c)+3*a*b^2*(16/35-1/7*csch(d*x+c)^6+6/35*csch(d*x +c)^4-8/35*csch(d*x+c)^2)*coth(d*x+c)+b^3*(2/3-1/3*csch(d*x+c)^2)*coth(d*x +c))
Leaf count of result is larger than twice the leaf count of optimal. 2967 vs. \(2 (168) = 336\).
Time = 0.27 (sec) , antiderivative size = 2967, normalized size of antiderivative = 16.30 \[ \int \text {csch}^{16}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
8/45045*((3584*a^3 + 12480*a^2*b + 15444*a*b^2 - 15015*b^3)*cosh(d*x + c)^ 13 + 13*(3584*a^3 + 12480*a^2*b + 15444*a*b^2 - 15015*b^3)*cosh(d*x + c)*s inh(d*x + c)^12 - 2*(1792*a^3 + 6240*a^2*b + 7722*a*b^2 + 15015*b^3)*sinh( d*x + c)^13 - 15*(3584*a^3 + 12480*a^2*b + 15444*a*b^2 - 11011*b^3)*cosh(d *x + c)^11 + 6*(8960*a^3 + 31200*a^2*b + 38610*a*b^2 + 65065*b^3 - 26*(179 2*a^3 + 6240*a^2*b + 7722*a*b^2 + 15015*b^3)*cosh(d*x + c)^2)*sinh(d*x + c )^11 + 11*(26*(3584*a^3 + 12480*a^2*b + 15444*a*b^2 - 15015*b^3)*cosh(d*x + c)^3 - 15*(3584*a^3 + 12480*a^2*b + 15444*a*b^2 - 11011*b^3)*cosh(d*x + c))*sinh(d*x + c)^10 + 210*(1792*a^3 + 6240*a^2*b + 5148*a*b^2 - 3861*b^3) *cosh(d*x + c)^9 - 10*(143*(1792*a^3 + 6240*a^2*b + 7722*a*b^2 + 15015*b^3 )*cosh(d*x + c)^4 + 37632*a^3 + 131040*a^2*b + 216216*a*b^2 + 234234*b^3 - 165*(1792*a^3 + 6240*a^2*b + 7722*a*b^2 + 13013*b^3)*cosh(d*x + c)^2)*sin h(d*x + c)^9 + 9*(143*(3584*a^3 + 12480*a^2*b + 15444*a*b^2 - 15015*b^3)*c osh(d*x + c)^5 - 275*(3584*a^3 + 12480*a^2*b + 15444*a*b^2 - 11011*b^3)*co sh(d*x + c)^3 + 210*(1792*a^3 + 6240*a^2*b + 5148*a*b^2 - 3861*b^3)*cosh(d *x + c))*sinh(d*x + c)^8 - 182*(8960*a^3 + 31200*a^2*b + 13068*a*b^2 - 127 05*b^3)*cosh(d*x + c)^7 - 4*(858*(1792*a^3 + 6240*a^2*b + 7722*a*b^2 + 150 15*b^3)*cosh(d*x + c)^6 - 2475*(1792*a^3 + 6240*a^2*b + 7722*a*b^2 + 13013 *b^3)*cosh(d*x + c)^4 - 407680*a^3 - 1419600*a^2*b - 2918916*a*b^2 - 21471 45*b^3 + 3780*(896*a^3 + 3120*a^2*b + 5148*a*b^2 + 5577*b^3)*cosh(d*x +...
Timed out. \[ \int \text {csch}^{16}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2731 vs. \(2 (168) = 336\).
Time = 0.20 (sec) , antiderivative size = 2731, normalized size of antiderivative = 15.01 \[ \int \text {csch}^{16}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
4096/6435*a^3*(15*e^(-2*d*x - 2*c)/(d*(15*e^(-2*d*x - 2*c) - 105*e^(-4*d*x - 4*c) + 455*e^(-6*d*x - 6*c) - 1365*e^(-8*d*x - 8*c) + 3003*e^(-10*d*x - 10*c) - 5005*e^(-12*d*x - 12*c) + 6435*e^(-14*d*x - 14*c) - 6435*e^(-16*d *x - 16*c) + 5005*e^(-18*d*x - 18*c) - 3003*e^(-20*d*x - 20*c) + 1365*e^(- 22*d*x - 22*c) - 455*e^(-24*d*x - 24*c) + 105*e^(-26*d*x - 26*c) - 15*e^(- 28*d*x - 28*c) + e^(-30*d*x - 30*c) - 1)) - 105*e^(-4*d*x - 4*c)/(d*(15*e^ (-2*d*x - 2*c) - 105*e^(-4*d*x - 4*c) + 455*e^(-6*d*x - 6*c) - 1365*e^(-8* d*x - 8*c) + 3003*e^(-10*d*x - 10*c) - 5005*e^(-12*d*x - 12*c) + 6435*e^(- 14*d*x - 14*c) - 6435*e^(-16*d*x - 16*c) + 5005*e^(-18*d*x - 18*c) - 3003* e^(-20*d*x - 20*c) + 1365*e^(-22*d*x - 22*c) - 455*e^(-24*d*x - 24*c) + 10 5*e^(-26*d*x - 26*c) - 15*e^(-28*d*x - 28*c) + e^(-30*d*x - 30*c) - 1)) + 455*e^(-6*d*x - 6*c)/(d*(15*e^(-2*d*x - 2*c) - 105*e^(-4*d*x - 4*c) + 455* e^(-6*d*x - 6*c) - 1365*e^(-8*d*x - 8*c) + 3003*e^(-10*d*x - 10*c) - 5005* e^(-12*d*x - 12*c) + 6435*e^(-14*d*x - 14*c) - 6435*e^(-16*d*x - 16*c) + 5 005*e^(-18*d*x - 18*c) - 3003*e^(-20*d*x - 20*c) + 1365*e^(-22*d*x - 22*c) - 455*e^(-24*d*x - 24*c) + 105*e^(-26*d*x - 26*c) - 15*e^(-28*d*x - 28*c) + e^(-30*d*x - 30*c) - 1)) - 1365*e^(-8*d*x - 8*c)/(d*(15*e^(-2*d*x - 2*c ) - 105*e^(-4*d*x - 4*c) + 455*e^(-6*d*x - 6*c) - 1365*e^(-8*d*x - 8*c) + 3003*e^(-10*d*x - 10*c) - 5005*e^(-12*d*x - 12*c) + 6435*e^(-14*d*x - 14*c ) - 6435*e^(-16*d*x - 16*c) + 5005*e^(-18*d*x - 18*c) - 3003*e^(-20*d*x...
Leaf count of result is larger than twice the leaf count of optimal. 621 vs. \(2 (168) = 336\).
Time = 0.52 (sec) , antiderivative size = 621, normalized size of antiderivative = 3.41 \[ \int \text {csch}^{16}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {4 \, {\left (45045 \, b^{3} e^{\left (26 \, d x + 26 \, c\right )} - 555555 \, b^{3} e^{\left (24 \, d x + 24 \, c\right )} + 1081080 \, a b^{2} e^{\left (22 \, d x + 22 \, c\right )} + 3153150 \, b^{3} e^{\left (22 \, d x + 22 \, c\right )} - 9297288 \, a b^{2} e^{\left (20 \, d x + 20 \, c\right )} - 10900890 \, b^{3} e^{\left (20 \, d x + 20 \, c\right )} + 11531520 \, a^{2} b e^{\left (18 \, d x + 18 \, c\right )} + 35675640 \, a b^{2} e^{\left (18 \, d x + 18 \, c\right )} + 25600575 \, b^{3} e^{\left (18 \, d x + 18 \, c\right )} - 54362880 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} - 80463240 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} - 43108065 \, b^{3} e^{\left (16 \, d x + 16 \, c\right )} + 46126080 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 106254720 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 118301040 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 53513460 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} - 35875840 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} - 113393280 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} - 118918800 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 49549500 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 21525504 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 74954880 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 83459376 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 34189155 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 9784320 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 34070400 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 41081040 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 17342325 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 3261440 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 11356800 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 14054040 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 6276270 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 752640 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 2620800 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 3243240 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 1531530 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 107520 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 374400 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 463320 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 225225 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 7168 \, a^{3} - 24960 \, a^{2} b - 30888 \, a b^{2} - 15015 \, b^{3}\right )}}{45045 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{15}} \]
-4/45045*(45045*b^3*e^(26*d*x + 26*c) - 555555*b^3*e^(24*d*x + 24*c) + 108 1080*a*b^2*e^(22*d*x + 22*c) + 3153150*b^3*e^(22*d*x + 22*c) - 9297288*a*b ^2*e^(20*d*x + 20*c) - 10900890*b^3*e^(20*d*x + 20*c) + 11531520*a^2*b*e^( 18*d*x + 18*c) + 35675640*a*b^2*e^(18*d*x + 18*c) + 25600575*b^3*e^(18*d*x + 18*c) - 54362880*a^2*b*e^(16*d*x + 16*c) - 80463240*a*b^2*e^(16*d*x + 1 6*c) - 43108065*b^3*e^(16*d*x + 16*c) + 46126080*a^3*e^(14*d*x + 14*c) + 1 06254720*a^2*b*e^(14*d*x + 14*c) + 118301040*a*b^2*e^(14*d*x + 14*c) + 535 13460*b^3*e^(14*d*x + 14*c) - 35875840*a^3*e^(12*d*x + 12*c) - 113393280*a ^2*b*e^(12*d*x + 12*c) - 118918800*a*b^2*e^(12*d*x + 12*c) - 49549500*b^3* e^(12*d*x + 12*c) + 21525504*a^3*e^(10*d*x + 10*c) + 74954880*a^2*b*e^(10* d*x + 10*c) + 83459376*a*b^2*e^(10*d*x + 10*c) + 34189155*b^3*e^(10*d*x + 10*c) - 9784320*a^3*e^(8*d*x + 8*c) - 34070400*a^2*b*e^(8*d*x + 8*c) - 410 81040*a*b^2*e^(8*d*x + 8*c) - 17342325*b^3*e^(8*d*x + 8*c) + 3261440*a^3*e ^(6*d*x + 6*c) + 11356800*a^2*b*e^(6*d*x + 6*c) + 14054040*a*b^2*e^(6*d*x + 6*c) + 6276270*b^3*e^(6*d*x + 6*c) - 752640*a^3*e^(4*d*x + 4*c) - 262080 0*a^2*b*e^(4*d*x + 4*c) - 3243240*a*b^2*e^(4*d*x + 4*c) - 1531530*b^3*e^(4 *d*x + 4*c) + 107520*a^3*e^(2*d*x + 2*c) + 374400*a^2*b*e^(2*d*x + 2*c) + 463320*a*b^2*e^(2*d*x + 2*c) + 225225*b^3*e^(2*d*x + 2*c) - 7168*a^3 - 249 60*a^2*b - 30888*a*b^2 - 15015*b^3)/(d*(e^(2*d*x + 2*c) - 1)^15)
Time = 1.87 (sec) , antiderivative size = 3823, normalized size of antiderivative = 21.01 \[ \int \text {csch}^{16}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
((32*b^3)/(455*d) - (8*b^3*exp(2*c + 2*d*x))/(105*d))/(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) - ((8*exp(8*c + 8*d*x)*(840*a *b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(39*d) - (352*b^3*exp(18*c + 18*d *x))/(91*d) + (44*b^3*exp(20*c + 20*d*x))/(105*d) + (4*b^2*(8*a + 11*b))/( 455*d) - (64*b*exp(6*c + 6*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(91*d) - (12 8*b*exp(10*c + 10*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(65*d) + (4*b*exp(4*c + 4*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(91*d) + (24*b*exp(12*c + 12*d*x) *(448*a*b + 256*a^2 + 165*b^2))/(65*d) + (132*b^2*exp(16*c + 16*d*x)*(8*a + 11*b))/(91*d) - (32*b^2*exp(2*c + 2*d*x)*(96*a + 55*b))/(1365*d) - (64*b ^2*exp(14*c + 14*d*x)*(96*a + 55*b))/(91*d))/(66*exp(4*c + 4*d*x) - 12*exp (2*c + 2*d*x) - 220*exp(6*c + 6*d*x) + 495*exp(8*c + 8*d*x) - 792*exp(10*c + 10*d*x) + 924*exp(12*c + 12*d*x) - 792*exp(14*c + 14*d*x) + 495*exp(16* c + 16*d*x) - 220*exp(18*c + 18*d*x) + 66*exp(20*c + 20*d*x) - 12*exp(22*c + 22*d*x) + exp(24*c + 24*d*x) + 1) + ((192*b^3*exp(4*c + 4*d*x))/(455*d) - (16*b^3*exp(6*c + 6*d*x))/(105*d) + (32*b^2*(96*a + 55*b))/(15015*d) - (16*b^2*exp(2*c + 2*d*x)*(8*a + 11*b))/(455*d))/(5*exp(2*c + 2*d*x) - 10*e xp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) - 5*exp(8*c + 8*d*x) + exp(10*c + 10 *d*x) - 1) - ((4*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(6435*d) - (96*b^3*exp(10*c + 10*d*x))/(65*d) + (4*b^3*exp(12*c + 12*d*x))/(15*d) - (64*b*exp(2*c + 2*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(2145*d) + (12*b*e...