Integrand size = 23, antiderivative size = 144 \[ \int \text {csch}^{14}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {(a+b)^3 \coth (c+d x)}{d}+\frac {2 a (a+b)^2 \coth ^3(c+d x)}{d}-\frac {3 a (a+b) (5 a+b) \coth ^5(c+d x)}{5 d}+\frac {4 a^2 (5 a+3 b) \coth ^7(c+d x)}{7 d}-\frac {a^2 (5 a+b) \coth ^9(c+d x)}{3 d}+\frac {6 a^3 \coth ^{11}(c+d x)}{11 d}-\frac {a^3 \coth ^{13}(c+d x)}{13 d} \]
-(a+b)^3*coth(d*x+c)/d+2*a*(a+b)^2*coth(d*x+c)^3/d-3/5*a*(a+b)*(5*a+b)*cot h(d*x+c)^5/d+4/7*a^2*(5*a+3*b)*coth(d*x+c)^7/d-1/3*a^2*(5*a+b)*coth(d*x+c) ^9/d+6/11*a^3*coth(d*x+c)^11/d-1/13*a^3*coth(d*x+c)^13/d
Leaf count is larger than twice the leaf count of optimal. \(350\) vs. \(2(144)=288\).
Time = 3.47 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.43 \[ \int \text {csch}^{14}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {\left (8580 \left (1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right ) \cosh (c+d x)-6435 \left (1024 a^3+2944 a^2 b+2408 a b^2+693 b^3\right ) \cosh (3 (c+d x))+3660800 a^3 \cosh (5 (c+d x))+13087360 a^2 b \cosh (5 (c+d x))+13093080 a b^2 \cosh (5 (c+d x))+4129125 b^3 \cosh (5 (c+d x))-1464320 a^3 \cosh (7 (c+d x))-5234944 a^2 b \cosh (7 (c+d x))-6390384 a b^2 \cosh (7 (c+d x))-2312310 b^3 \cosh (7 (c+d x))+399360 a^3 \cosh (9 (c+d x))+1427712 a^2 b \cosh (9 (c+d x))+1873872 a b^2 \cosh (9 (c+d x))+810810 b^3 \cosh (9 (c+d x))-66560 a^3 \cosh (11 (c+d x))-237952 a^2 b \cosh (11 (c+d x))-312312 a b^2 \cosh (11 (c+d x))-165165 b^3 \cosh (11 (c+d x))+5120 a^3 \cosh (13 (c+d x))+18304 a^2 b \cosh (13 (c+d x))+24024 a b^2 \cosh (13 (c+d x))+15015 b^3 \cosh (13 (c+d x))\right ) \text {csch}^{13}(c+d x)}{61501440 d} \]
-1/61501440*((8580*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*Cosh[c + d*x] - 6435*(1024*a^3 + 2944*a^2*b + 2408*a*b^2 + 693*b^3)*Cosh[3*(c + d*x )] + 3660800*a^3*Cosh[5*(c + d*x)] + 13087360*a^2*b*Cosh[5*(c + d*x)] + 13 093080*a*b^2*Cosh[5*(c + d*x)] + 4129125*b^3*Cosh[5*(c + d*x)] - 1464320*a ^3*Cosh[7*(c + d*x)] - 5234944*a^2*b*Cosh[7*(c + d*x)] - 6390384*a*b^2*Cos h[7*(c + d*x)] - 2312310*b^3*Cosh[7*(c + d*x)] + 399360*a^3*Cosh[9*(c + d* x)] + 1427712*a^2*b*Cosh[9*(c + d*x)] + 1873872*a*b^2*Cosh[9*(c + d*x)] + 810810*b^3*Cosh[9*(c + d*x)] - 66560*a^3*Cosh[11*(c + d*x)] - 237952*a^2*b *Cosh[11*(c + d*x)] - 312312*a*b^2*Cosh[11*(c + d*x)] - 165165*b^3*Cosh[11 *(c + d*x)] + 5120*a^3*Cosh[13*(c + d*x)] + 18304*a^2*b*Cosh[13*(c + d*x)] + 24024*a*b^2*Cosh[13*(c + d*x)] + 15015*b^3*Cosh[13*(c + d*x)])*Csch[c + d*x]^13)/d
Time = 0.36 (sec) , antiderivative size = 127, 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, 25, 3696, 1433, 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.
| \(\displaystyle \int \text {csch}^{14}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a+b \sin (i c+i d x)^4\right )^3}{\sin (i c+i d x)^{14}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (b \sin (i c+i d x)^4+a\right )^3}{\sin (i c+i d x)^{14}}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \coth ^{14}(c+d x) \left ((a+b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^3d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1433 |
| \(\displaystyle \frac {\int \left (a^3 \coth ^{14}(c+d x)-6 a^3 \coth ^{12}(c+d x)+3 a^2 (5 a+b) \coth ^{10}(c+d x)-4 a^2 (5 a+3 b) \coth ^8(c+d x)+3 a (a+b) (5 a+b) \coth ^6(c+d x)-6 a (a+b)^2 \coth ^4(c+d x)+(a+b)^3 \coth ^2(c+d x)\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{13} a^3 \coth ^{13}(c+d x)+\frac {6}{11} a^3 \coth ^{11}(c+d x)-\frac {1}{3} a^2 (5 a+b) \coth ^9(c+d x)+\frac {4}{7} a^2 (5 a+3 b) \coth ^7(c+d x)-\frac {3}{5} a (a+b) (5 a+b) \coth ^5(c+d x)+2 a (a+b)^2 \coth ^3(c+d x)-(a+b)^3 \coth (c+d x)}{d}\) |
(-((a + b)^3*Coth[c + d*x]) + 2*a*(a + b)^2*Coth[c + d*x]^3 - (3*a*(a + b) *(5*a + b)*Coth[c + d*x]^5)/5 + (4*a^2*(5*a + 3*b)*Coth[c + d*x]^7)/7 - (a ^2*(5*a + b)*Coth[c + d*x]^9)/3 + (6*a^3*Coth[c + d*x]^11)/11 - (a^3*Coth[ c + d*x]^13)/13)/d
3.3.25.3.1 Defintions of rubi rules used
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || !IntegerQ[(m + 1)/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 = 1.64 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {1024}{3003}-\frac {\operatorname {csch}\left (d x +c \right )^{12}}{13}+\frac {12 \operatorname {csch}\left (d x +c \right )^{10}}{143}-\frac {40 \operatorname {csch}\left (d x +c \right )^{8}}{429}+\frac {320 \operatorname {csch}\left (d x +c \right )^{6}}{3003}-\frac {128 \operatorname {csch}\left (d x +c \right )^{4}}{1001}+\frac {512 \operatorname {csch}\left (d x +c \right )^{2}}{3003}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (-\frac {128}{315}-\frac {\operatorname {csch}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {csch}\left (d x +c \right )^{6}}{63}-\frac {16 \operatorname {csch}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {csch}\left (d x +c \right )^{2}}{315}\right ) \coth \left (d x +c \right )+3 a \,b^{2} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )-b^{3} \coth \left (d x +c \right )}{d}\) | \(177\) |
| default | \(\frac {a^{3} \left (-\frac {1024}{3003}-\frac {\operatorname {csch}\left (d x +c \right )^{12}}{13}+\frac {12 \operatorname {csch}\left (d x +c \right )^{10}}{143}-\frac {40 \operatorname {csch}\left (d x +c \right )^{8}}{429}+\frac {320 \operatorname {csch}\left (d x +c \right )^{6}}{3003}-\frac {128 \operatorname {csch}\left (d x +c \right )^{4}}{1001}+\frac {512 \operatorname {csch}\left (d x +c \right )^{2}}{3003}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (-\frac {128}{315}-\frac {\operatorname {csch}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {csch}\left (d x +c \right )^{6}}{63}-\frac {16 \operatorname {csch}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {csch}\left (d x +c \right )^{2}}{315}\right ) \coth \left (d x +c \right )+3 a \,b^{2} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )-b^{3} \coth \left (d x +c \right )}{d}\) | \(177\) |
| parallelrisch | \(\frac {-1155 \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{13} a^{3}+17745 \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{11} a^{3}-130130 \left (\frac {8 b}{13}+a \right ) a^{2} \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+613470 \left (a +\frac {216 b}{143}\right ) a^{2} \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-2147145 a^{3}-5189184 a^{2} b -2306304 a \,b^{2}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (6441435 a^{3}+20180160 a^{2} b +19219200 a \,b^{2}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-25765740 a^{3}-90810720 a^{2} b -115315200 a \,b^{2}-61501440 b^{3}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )-1155 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} a^{3}-\frac {169 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} a^{3}}{11}+\frac {338 \left (\frac {8 b}{13}+a \right ) a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3}-\frac {3718 \left (a +\frac {216 b}{143}\right ) a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{7}+\left (\frac {9984}{5} a \,b^{2}+1859 a^{3}+\frac {22464}{5} a^{2} b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-5577 a^{3}-17472 a^{2} b -16640 a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+22308 a^{3}+78624 a^{2} b +99840 a \,b^{2}+53248 b^{3}\right )}{123002880 d}\) | \(341\) |
| risch | \(-\frac {2 \left (-11891880 b^{3} {\mathrm e}^{10 d x +10 c}+3660800 a^{3} {\mathrm e}^{8 d x +8 c}-1464320 a^{3} {\mathrm e}^{6 d x +6 c}+15015 b^{3} {\mathrm e}^{24 d x +24 c}-180180 b^{3} {\mathrm e}^{22 d x +22 c}+990990 b^{3} {\mathrm e}^{20 d x +20 c}-3303300 b^{3} {\mathrm e}^{18 d x +18 c}+7432425 b^{3} {\mathrm e}^{16 d x +16 c}-11891880 b^{3} {\mathrm e}^{14 d x +14 c}+18304 a^{2} b +5120 a^{3}+15015 b^{3}+24024 a \,b^{2}+8785920 a^{3} {\mathrm e}^{12 d x +12 c}+13873860 b^{3} {\mathrm e}^{12 d x +12 c}-6589440 a^{3} {\mathrm e}^{10 d x +10 c}+7432425 b^{3} {\mathrm e}^{8 d x +8 c}+15135120 \,{\mathrm e}^{8 d x +8 c} a \,b^{2}-5234944 b \,a^{2} {\mathrm e}^{6 d x +6 c}-6630624 a \,b^{2} {\mathrm e}^{6 d x +6 c}-237952 a^{2} b \,{\mathrm e}^{2 d x +2 c}+1427712 a^{2} b \,{\mathrm e}^{4 d x +4 c}-312312 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}-66560 a^{3} {\mathrm e}^{2 d x +2 c}-180180 b^{3} {\mathrm e}^{2 d x +2 c}-3303300 b^{3} {\mathrm e}^{6 d x +6 c}+399360 a^{3} {\mathrm e}^{4 d x +4 c}+990990 b^{3} {\mathrm e}^{4 d x +4 c}+1873872 \,{\mathrm e}^{4 d x +4 c} b^{2} a +240240 a \,b^{2} {\mathrm e}^{20 d x +20 c}-2042040 a \,b^{2} {\mathrm e}^{18 d x +18 c}+2306304 a^{2} b \,{\mathrm e}^{16 d x +16 c}+7711704 a \,b^{2} {\mathrm e}^{16 d x +16 c}-10762752 a^{2} b \,{\mathrm e}^{14 d x +14 c}-17008992 a \,b^{2} {\mathrm e}^{14 d x +14 c}+20646912 a^{2} b \,{\mathrm e}^{12 d x +12 c}+24216192 a \,b^{2} {\mathrm e}^{12 d x +12 c}-21250944 a^{2} b \,{\mathrm e}^{10 d x +10 c}-23207184 a \,b^{2} {\mathrm e}^{10 d x +10 c}+13087360 a^{2} b \,{\mathrm e}^{8 d x +8 c}\right )}{15015 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{13}}\) | \(564\) |
1/d*(a^3*(-1024/3003-1/13*csch(d*x+c)^12+12/143*csch(d*x+c)^10-40/429*csch (d*x+c)^8+320/3003*csch(d*x+c)^6-128/1001*csch(d*x+c)^4+512/3003*csch(d*x+ c)^2)*coth(d*x+c)+3*a^2*b*(-128/315-1/9*csch(d*x+c)^8+8/63*csch(d*x+c)^6-1 6/105*csch(d*x+c)^4+64/315*csch(d*x+c)^2)*coth(d*x+c)+3*a*b^2*(-8/15-1/5*c sch(d*x+c)^4+4/15*csch(d*x+c)^2)*coth(d*x+c)-b^3*coth(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 2323 vs. \(2 (134) = 268\).
Time = 0.26 (sec) , antiderivative size = 2323, normalized size of antiderivative = 16.13 \[ \int \text {csch}^{14}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
-4/15015*((2560*a^3 + 9152*a^2*b + 12012*a*b^2 + 15015*b^3)*cosh(d*x + c)^ 12 - 48*(640*a^3 + 2288*a^2*b + 3003*a*b^2)*cosh(d*x + c)*sinh(d*x + c)^11 + (2560*a^3 + 9152*a^2*b + 12012*a*b^2 + 15015*b^3)*sinh(d*x + c)^12 - 52 *(640*a^3 + 2288*a^2*b + 3003*a*b^2 + 3465*b^3)*cosh(d*x + c)^10 - 2*(1664 0*a^3 + 59488*a^2*b + 78078*a*b^2 + 90090*b^3 - 33*(2560*a^3 + 9152*a^2*b + 12012*a*b^2 + 15015*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 - 40*(22*(640 *a^3 + 2288*a^2*b + 3003*a*b^2)*cosh(d*x + c)^3 - 13*(640*a^3 + 2288*a^2*b + 3003*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 + 78*(2560*a^3 + 9152*a^2*b + 13552*a*b^2 + 12705*b^3)*cosh(d*x + c)^8 + 3*(165*(2560*a^3 + 9152*a^2*b + 12012*a*b^2 + 15015*b^3)*cosh(d*x + c)^4 + 66560*a^3 + 237952*a^2*b + 3 52352*a*b^2 + 330330*b^3 - 780*(640*a^3 + 2288*a^2*b + 3003*a*b^2 + 3465*b ^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 - 96*(33*(640*a^3 + 2288*a^2*b + 3003 *a*b^2)*cosh(d*x + c)^5 - 65*(640*a^3 + 2288*a^2*b + 3003*a*b^2)*cosh(d*x + c)^3 + 52*(320*a^3 + 1144*a^2*b + 1309*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 572*(1280*a^3 + 4576*a^2*b + 7581*a*b^2 + 5775*b^3)*cosh(d*x + c)^6 + 4*(231*(2560*a^3 + 9152*a^2*b + 12012*a*b^2 + 15015*b^3)*cosh(d*x + c)^ 6 - 2730*(640*a^3 + 2288*a^2*b + 3003*a*b^2 + 3465*b^3)*cosh(d*x + c)^4 - 183040*a^3 - 654368*a^2*b - 1084083*a*b^2 - 825825*b^3 + 546*(2560*a^3 + 9 152*a^2*b + 13552*a*b^2 + 12705*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 - 24 *(132*(640*a^3 + 2288*a^2*b + 3003*a*b^2)*cosh(d*x + c)^7 - 546*(640*a^...
Timed out. \[ \int \text {csch}^{14}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1916 vs. \(2 (134) = 268\).
Time = 0.21 (sec) , antiderivative size = 1916, normalized size of antiderivative = 13.31 \[ \int \text {csch}^{14}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
-2048/3003*a^3*(13*e^(-2*d*x - 2*c)/(d*(13*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4*c) + 286*e^(-6*d*x - 6*c) - 715*e^(-8*d*x - 8*c) + 1287*e^(-10*d*x - 10*c) - 1716*e^(-12*d*x - 12*c) + 1716*e^(-14*d*x - 14*c) - 1287*e^(-16*d* x - 16*c) + 715*e^(-18*d*x - 18*c) - 286*e^(-20*d*x - 20*c) + 78*e^(-22*d* x - 22*c) - 13*e^(-24*d*x - 24*c) + e^(-26*d*x - 26*c) - 1)) - 78*e^(-4*d* x - 4*c)/(d*(13*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4*c) + 286*e^(-6*d*x - 6 *c) - 715*e^(-8*d*x - 8*c) + 1287*e^(-10*d*x - 10*c) - 1716*e^(-12*d*x - 1 2*c) + 1716*e^(-14*d*x - 14*c) - 1287*e^(-16*d*x - 16*c) + 715*e^(-18*d*x - 18*c) - 286*e^(-20*d*x - 20*c) + 78*e^(-22*d*x - 22*c) - 13*e^(-24*d*x - 24*c) + e^(-26*d*x - 26*c) - 1)) + 286*e^(-6*d*x - 6*c)/(d*(13*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4*c) + 286*e^(-6*d*x - 6*c) - 715*e^(-8*d*x - 8*c) + 1287*e^(-10*d*x - 10*c) - 1716*e^(-12*d*x - 12*c) + 1716*e^(-14*d*x - 1 4*c) - 1287*e^(-16*d*x - 16*c) + 715*e^(-18*d*x - 18*c) - 286*e^(-20*d*x - 20*c) + 78*e^(-22*d*x - 22*c) - 13*e^(-24*d*x - 24*c) + e^(-26*d*x - 26*c ) - 1)) - 715*e^(-8*d*x - 8*c)/(d*(13*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4* c) + 286*e^(-6*d*x - 6*c) - 715*e^(-8*d*x - 8*c) + 1287*e^(-10*d*x - 10*c) - 1716*e^(-12*d*x - 12*c) + 1716*e^(-14*d*x - 14*c) - 1287*e^(-16*d*x - 1 6*c) + 715*e^(-18*d*x - 18*c) - 286*e^(-20*d*x - 20*c) + 78*e^(-22*d*x - 2 2*c) - 13*e^(-24*d*x - 24*c) + e^(-26*d*x - 26*c) - 1)) + 1287*e^(-10*d*x - 10*c)/(d*(13*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4*c) + 286*e^(-6*d*x -...
Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (134) = 268\).
Time = 0.51 (sec) , antiderivative size = 563, normalized size of antiderivative = 3.91 \[ \int \text {csch}^{14}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {2 \, {\left (15015 \, b^{3} e^{\left (24 \, d x + 24 \, c\right )} - 180180 \, b^{3} e^{\left (22 \, d x + 22 \, c\right )} + 240240 \, a b^{2} e^{\left (20 \, d x + 20 \, c\right )} + 990990 \, b^{3} e^{\left (20 \, d x + 20 \, c\right )} - 2042040 \, a b^{2} e^{\left (18 \, d x + 18 \, c\right )} - 3303300 \, b^{3} e^{\left (18 \, d x + 18 \, c\right )} + 2306304 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 7711704 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} + 7432425 \, b^{3} e^{\left (16 \, d x + 16 \, c\right )} - 10762752 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} - 17008992 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} - 11891880 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 8785920 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 20646912 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 24216192 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 13873860 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} - 6589440 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} - 21250944 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 23207184 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 11891880 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 3660800 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 13087360 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 15135120 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 7432425 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 1464320 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 5234944 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 6630624 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 3303300 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 399360 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 1427712 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 1873872 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 990990 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 66560 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 237952 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 312312 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 180180 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5120 \, a^{3} + 18304 \, a^{2} b + 24024 \, a b^{2} + 15015 \, b^{3}\right )}}{15015 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{13}} \]
-2/15015*(15015*b^3*e^(24*d*x + 24*c) - 180180*b^3*e^(22*d*x + 22*c) + 240 240*a*b^2*e^(20*d*x + 20*c) + 990990*b^3*e^(20*d*x + 20*c) - 2042040*a*b^2 *e^(18*d*x + 18*c) - 3303300*b^3*e^(18*d*x + 18*c) + 2306304*a^2*b*e^(16*d *x + 16*c) + 7711704*a*b^2*e^(16*d*x + 16*c) + 7432425*b^3*e^(16*d*x + 16* c) - 10762752*a^2*b*e^(14*d*x + 14*c) - 17008992*a*b^2*e^(14*d*x + 14*c) - 11891880*b^3*e^(14*d*x + 14*c) + 8785920*a^3*e^(12*d*x + 12*c) + 20646912 *a^2*b*e^(12*d*x + 12*c) + 24216192*a*b^2*e^(12*d*x + 12*c) + 13873860*b^3 *e^(12*d*x + 12*c) - 6589440*a^3*e^(10*d*x + 10*c) - 21250944*a^2*b*e^(10* d*x + 10*c) - 23207184*a*b^2*e^(10*d*x + 10*c) - 11891880*b^3*e^(10*d*x + 10*c) + 3660800*a^3*e^(8*d*x + 8*c) + 13087360*a^2*b*e^(8*d*x + 8*c) + 151 35120*a*b^2*e^(8*d*x + 8*c) + 7432425*b^3*e^(8*d*x + 8*c) - 1464320*a^3*e^ (6*d*x + 6*c) - 5234944*a^2*b*e^(6*d*x + 6*c) - 6630624*a*b^2*e^(6*d*x + 6 *c) - 3303300*b^3*e^(6*d*x + 6*c) + 399360*a^3*e^(4*d*x + 4*c) + 1427712*a ^2*b*e^(4*d*x + 4*c) + 1873872*a*b^2*e^(4*d*x + 4*c) + 990990*b^3*e^(4*d*x + 4*c) - 66560*a^3*e^(2*d*x + 2*c) - 237952*a^2*b*e^(2*d*x + 2*c) - 31231 2*a*b^2*e^(2*d*x + 2*c) - 180180*b^3*e^(2*d*x + 2*c) + 5120*a^3 + 18304*a^ 2*b + 24024*a*b^2 + 15015*b^3)/(d*(e^(2*d*x + 2*c) - 1)^13)
Time = 1.76 (sec) , antiderivative size = 3138, normalized size of antiderivative = 21.79 \[ \int \text {csch}^{14}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
((6*b^3*exp(4*c + 4*d*x))/(13*d) - (2*b^3*exp(6*c + 6*d*x))/(13*d) + (2*b^ 2*(96*a + 55*b))/(715*d) - (6*b^2*exp(2*c + 2*d*x)*(8*a + 11*b))/(143*d))/ (6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1) - ((2*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(3003*d) - (12*b^3*exp(10*c + 10*d*x))/(13*d) + (2*b^3*exp(12*c + 12*d*x))/(13*d) - (4*b*exp(2*c + 2*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(143*d) + (2*b*exp(4* c + 4*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(143*d) + (30*b^2*exp(8*c + 8*d* x)*(8*a + 11*b))/(143*d) - (8*b^2*exp(6*c + 6*d*x)*(96*a + 55*b))/(143*d)) /(7*exp(2*c + 2*d*x) - 21*exp(4*c + 4*d*x) + 35*exp(6*c + 6*d*x) - 35*exp( 8*c + 8*d*x) + 21*exp(10*c + 10*d*x) - 7*exp(12*c + 12*d*x) + exp(14*c + 1 4*d*x) - 1) - ((8*exp(6*c + 6*d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 23 1*b^3))/(143*d) - (18*b^3*exp(16*c + 16*d*x))/(13*d) + (2*b^3*exp(18*c + 1 8*d*x))/(13*d) - (2*b^2*(96*a + 55*b))/(715*d) - (24*b*exp(4*c + 4*d*x)*(1 12*a*b + 128*a^2 + 33*b^2))/(143*d) - (84*b*exp(8*c + 8*d*x)*(112*a*b + 12 8*a^2 + 33*b^2))/(143*d) + (6*b*exp(2*c + 2*d*x)*(448*a*b + 256*a^2 + 165* b^2))/(715*d) + (84*b*exp(10*c + 10*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(7 15*d) + (72*b^2*exp(14*c + 14*d*x)*(8*a + 11*b))/(143*d) - (168*b^2*exp(12 *c + 12*d*x)*(96*a + 55*b))/(715*d))/(45*exp(4*c + 4*d*x) - 10*exp(2*c + 2 *d*x) - 120*exp(6*c + 6*d*x) + 210*exp(8*c + 8*d*x) - 252*exp(10*c + 10*d* x) + 210*exp(12*c + 12*d*x) - 120*exp(14*c + 14*d*x) + 45*exp(16*c + 16...