Integrand size = 23, antiderivative size = 140 \[ \int \text {csch}^{10}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {b^3 x}{2}-\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}+\frac {2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}-\frac {3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac {4 a^3 \coth ^7(c+d x)}{7 d}-\frac {a^3 \coth ^9(c+d x)}{9 d}+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d} \]
-1/2*b^3*x-a*(a^2+3*a*b+3*b^2)*coth(d*x+c)/d+2/3*a^2*(2*a+3*b)*coth(d*x+c) ^3/d-3/5*a^2*(2*a+b)*coth(d*x+c)^5/d+4/7*a^3*coth(d*x+c)^7/d-1/9*a^3*coth( d*x+c)^9/d+1/2*b^3*cosh(d*x+c)*sinh(d*x+c)/d
Time = 2.60 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.82 \[ \int \text {csch}^{10}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {-4 a \coth (c+d x) \left (128 a^2+504 a b+945 b^2-4 a (16 a+63 b) \text {csch}^2(c+d x)+3 a (16 a+63 b) \text {csch}^4(c+d x)-40 a^2 \text {csch}^6(c+d x)+35 a^2 \text {csch}^8(c+d x)\right )+315 b^3 (-2 (c+d x)+\sinh (2 (c+d x)))}{1260 d} \]
(-4*a*Coth[c + d*x]*(128*a^2 + 504*a*b + 945*b^2 - 4*a*(16*a + 63*b)*Csch[ c + d*x]^2 + 3*a*(16*a + 63*b)*Csch[c + d*x]^4 - 40*a^2*Csch[c + d*x]^6 + 35*a^2*Csch[c + d*x]^8) + 315*b^3*(-2*(c + d*x) + Sinh[2*(c + d*x)]))/(126 0*d)
Time = 0.57 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 25, 3696, 1582, 25, 2333, 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.
| \(\displaystyle \int \text {csch}^{10}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a+b \sin (i c+i d x)^4\right )^3}{\sin (i c+i d x)^{10}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (b \sin (i c+i d x)^4+a\right )^3}{\sin (i c+i d x)^{10}}dx\) |
| \(\Big \downarrow \) 3696 |
| \(\displaystyle \frac {\int \frac {\coth ^{10}(c+d x) \left ((a+b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )^3}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1582 |
| \(\displaystyle \frac {\frac {b^3 \tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}-\frac {1}{2} \int -\frac {\coth ^{10}(c+d x) \left (-\left (\left (2 a^3+6 b a^2+6 b^2 a+b^3\right ) \tanh ^{10}(c+d x)\right )+2 a \left (5 a^2+9 b a+3 b^2\right ) \tanh ^8(c+d x)-2 a^2 (10 a+9 b) \tanh ^6(c+d x)+2 a^2 (10 a+3 b) \tanh ^4(c+d x)-10 a^3 \tanh ^2(c+d x)+2 a^3\right )}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {\coth ^{10}(c+d x) \left (-\left (\left (2 a^3+6 b a^2+6 b^2 a+b^3\right ) \tanh ^{10}(c+d x)\right )+2 a \left (5 a^2+9 b a+3 b^2\right ) \tanh ^8(c+d x)-2 a^2 (10 a+9 b) \tanh ^6(c+d x)+2 a^2 (10 a+3 b) \tanh ^4(c+d x)-10 a^3 \tanh ^2(c+d x)+2 a^3\right )}{1-\tanh ^2(c+d x)}d\tanh (c+d x)+\frac {b^3 \tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \frac {\frac {1}{2} \int \left (2 a^3 \coth ^{10}(c+d x)-8 a^3 \coth ^8(c+d x)+6 a^2 (2 a+b) \coth ^6(c+d x)-4 a^2 (2 a+3 b) \coth ^4(c+d x)+2 a \left (a^2+3 b a+3 b^2\right ) \coth ^2(c+d x)+\frac {b^3}{\tanh ^2(c+d x)-1}\right )d\tanh (c+d x)+\frac {b^3 \tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{2} \left (-\frac {2}{9} a^3 \coth ^9(c+d x)+\frac {8}{7} a^3 \coth ^7(c+d x)-2 a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)-\frac {6}{5} a^2 (2 a+b) \coth ^5(c+d x)+\frac {4}{3} a^2 (2 a+3 b) \coth ^3(c+d x)-b^3 \text {arctanh}(\tanh (c+d x))\right )+\frac {b^3 \tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}}{d}\) |
((-(b^3*ArcTanh[Tanh[c + d*x]]) - 2*a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x] + (4*a^2*(2*a + 3*b)*Coth[c + d*x]^3)/3 - (6*a^2*(2*a + b)*Coth[c + d*x]^5 )/5 + (8*a^3*Coth[c + d*x]^7)/7 - (2*a^3*Coth[c + d*x]^9)/9)/2 + (b^3*Tanh [c + d*x])/(2*(1 - Tanh[c + d*x]^2)))/d
3.3.23.3.1 Defintions of rubi rules used
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_.), x_Symbol] :> Simp[(-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Simp[(-d)^(m/2 - 1)/(2*e^ (2*p)*(q + 1)) Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e *x^2))*(2*(-d)^(-m/2 + 1)*e^(2*p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -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 = 1.38 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.93
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {128}{315}-\frac {\operatorname {csch}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {csch}\left (d x +c \right )^{6}}{63}-\frac {16 \operatorname {csch}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {csch}\left (d x +c \right )^{2}}{315}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )-3 a \,b^{2} \coth \left (d x +c \right )+b^{3} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d}\) | \(130\) |
| default | \(\frac {a^{3} \left (-\frac {128}{315}-\frac {\operatorname {csch}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {csch}\left (d x +c \right )^{6}}{63}-\frac {16 \operatorname {csch}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {csch}\left (d x +c \right )^{2}}{315}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )-3 a \,b^{2} \coth \left (d x +c \right )+b^{3} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d}\) | \(130\) |
| parallelrisch | \(\frac {-a^{3} \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{9} \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{9} \left (-84 \cosh \left (3 d x +3 c \right )+36 \cosh \left (5 d x +5 c \right )+126 \cosh \left (d x +c \right )+\cosh \left (9 d x +9 c \right )-9 \cosh \left (7 d x +7 c \right )\right )-10080 \left (\cosh \left (d x +c \right )-\frac {\cosh \left (3 d x +3 c \right )}{2}+\frac {\cosh \left (5 d x +5 c \right )}{10}\right ) a^{2} b \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+483840 \,\operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}-967680 \left (a \coth \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {b \left (-2 d x +\sinh \left (2 d x +2 c \right )\right )}{12}\right ) b^{2}}{322560 d}\) | \(200\) |
| risch | \(-\frac {b^{3} x}{2}+\frac {{\mathrm e}^{2 d x +2 c} b^{3}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} b^{3}}{8 d}-\frac {2 a \left (945 b^{2} {\mathrm e}^{16 d x +16 c}-7560 b^{2} {\mathrm e}^{14 d x +14 c}+5040 a b \,{\mathrm e}^{12 d x +12 c}+26460 b^{2} {\mathrm e}^{12 d x +12 c}-22680 a b \,{\mathrm e}^{10 d x +10 c}-52920 b^{2} {\mathrm e}^{10 d x +10 c}+16128 \,{\mathrm e}^{8 d x +8 c} a^{2}+40824 \,{\mathrm e}^{8 d x +8 c} a b +66150 b^{2} {\mathrm e}^{8 d x +8 c}-10752 a^{2} {\mathrm e}^{6 d x +6 c}-37296 \,{\mathrm e}^{6 d x +6 c} a b -52920 b^{2} {\mathrm e}^{6 d x +6 c}+4608 \,{\mathrm e}^{4 d x +4 c} a^{2}+18144 \,{\mathrm e}^{4 d x +4 c} a b +26460 b^{2} {\mathrm e}^{4 d x +4 c}-1152 \,{\mathrm e}^{2 d x +2 c} a^{2}-4536 \,{\mathrm e}^{2 d x +2 c} b a -7560 b^{2} {\mathrm e}^{2 d x +2 c}+128 a^{2}+504 a b +945 b^{2}\right )}{315 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{9}}\) | \(322\) |
1/d*(a^3*(-128/315-1/9*csch(d*x+c)^8+8/63*csch(d*x+c)^6-16/105*csch(d*x+c) ^4+64/315*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(-8/15-1/5*csch(d*x+c)^4+4/15 *csch(d*x+c)^2)*coth(d*x+c)-3*a*b^2*coth(d*x+c)+b^3*(1/2*sinh(d*x+c)*cosh( d*x+c)-1/2*d*x-1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 1314 vs. \(2 (128) = 256\).
Time = 0.29 (sec) , antiderivative size = 1314, normalized size of antiderivative = 9.39 \[ \int \text {csch}^{10}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
1/2520*(315*b^3*cosh(d*x + c)^11 + 3465*b^3*cosh(d*x + c)*sinh(d*x + c)^10 - (1024*a^3 + 4032*a^2*b + 7560*a*b^2 + 2835*b^3)*cosh(d*x + c)^9 - 4*(31 5*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*sinh(d*x + c)^9 + 9*(5775*b ^3*cosh(d*x + c)^3 - (1024*a^3 + 4032*a^2*b + 7560*a*b^2 + 2835*b^3)*cosh( d*x + c))*sinh(d*x + c)^8 + 9*(1024*a^3 + 4032*a^2*b + 5880*a*b^2 + 1225*b ^3)*cosh(d*x + c)^7 + 36*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2 - 4*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^2)*sin h(d*x + c)^7 + 21*(6930*b^3*cosh(d*x + c)^5 - 4*(1024*a^3 + 4032*a^2*b + 7 560*a*b^2 + 2835*b^3)*cosh(d*x + c)^3 + 3*(1024*a^3 + 4032*a^2*b + 5880*a* b^2 + 1225*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 9*(4096*a^3 + 16128*a^2*b + 16800*a*b^2 + 2625*b^3)*cosh(d*x + c)^5 - 36*(1260*b^3*d*x + 14*(315*b^ 3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^4 - 1024*a^3 - 40 32*a^2*b - 7560*a*b^2 - 21*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^ 2)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 9*(11550*b^3*cosh(d*x + c)^7 - 14*(1 024*a^3 + 4032*a^2*b + 7560*a*b^2 + 2835*b^3)*cosh(d*x + c)^5 + 35*(1024*a ^3 + 4032*a^2*b + 5880*a*b^2 + 1225*b^3)*cosh(d*x + c)^3 - 5*(4096*a^3 + 1 6128*a^2*b + 16800*a*b^2 + 2625*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 42*( 2048*a^3 + 6144*a^2*b + 5040*a*b^2 + 675*b^3)*cosh(d*x + c)^3 - 12*(28*(31 5*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^6 - 8820*b^3* d*x - 105*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + ...
Timed out. \[ \int \text {csch}^{10}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 842 vs. \(2 (128) = 256\).
Time = 0.19 (sec) , antiderivative size = 842, normalized size of antiderivative = 6.01 \[ \int \text {csch}^{10}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
-1/8*b^3*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 256/315*a^3*(9*e ^(-2*d*x - 2*c)/(d*(9*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d* x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) - 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) - 1)) - 36*e^(-4*d*x - 4*c)/(d*(9*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) - 84 *e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-1 8*d*x - 18*c) - 1)) + 84*e^(-6*d*x - 6*c)/(d*(9*e^(-2*d*x - 2*c) - 36*e^(- 4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) - 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) - 1)) - 126*e^(-8*d*x - 8*c)/(d*(9*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) - 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9 *e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) - 1)) - 1/(d*(9*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e ^(-10*d*x - 10*c) - 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(- 16*d*x - 16*c) + e^(-18*d*x - 18*c) - 1))) - 16/5*a^2*b*(5*e^(-2*d*x - 2*c )/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e ^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 10*e^(-4*d*x - 4*c)/(d*(5*e^( -2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x...
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (128) = 256\).
Time = 0.50 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.57 \[ \int \text {csch}^{10}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {1260 \, {\left (d x + c\right )} b^{3} - 315 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 315 \, {\left (2 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + \frac {16 \, {\left (945 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} - 7560 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 5040 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 26460 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 22680 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 52920 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 16128 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 40824 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 66150 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 10752 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 37296 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 52920 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 4608 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 18144 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 26460 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 1152 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 4536 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 7560 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 128 \, a^{3} + 504 \, a^{2} b + 945 \, a b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{9}}}{2520 \, d} \]
-1/2520*(1260*(d*x + c)*b^3 - 315*b^3*e^(2*d*x + 2*c) - 315*(2*b^3*e^(2*d* x + 2*c) - b^3)*e^(-2*d*x - 2*c) + 16*(945*a*b^2*e^(16*d*x + 16*c) - 7560* a*b^2*e^(14*d*x + 14*c) + 5040*a^2*b*e^(12*d*x + 12*c) + 26460*a*b^2*e^(12 *d*x + 12*c) - 22680*a^2*b*e^(10*d*x + 10*c) - 52920*a*b^2*e^(10*d*x + 10* c) + 16128*a^3*e^(8*d*x + 8*c) + 40824*a^2*b*e^(8*d*x + 8*c) + 66150*a*b^2 *e^(8*d*x + 8*c) - 10752*a^3*e^(6*d*x + 6*c) - 37296*a^2*b*e^(6*d*x + 6*c) - 52920*a*b^2*e^(6*d*x + 6*c) + 4608*a^3*e^(4*d*x + 4*c) + 18144*a^2*b*e^ (4*d*x + 4*c) + 26460*a*b^2*e^(4*d*x + 4*c) - 1152*a^3*e^(2*d*x + 2*c) - 4 536*a^2*b*e^(2*d*x + 2*c) - 7560*a*b^2*e^(2*d*x + 2*c) + 128*a^3 + 504*a^2 *b + 945*a*b^2)/(e^(2*d*x + 2*c) - 1)^9)/d
Time = 1.72 (sec) , antiderivative size = 1500, normalized size of antiderivative = 10.71 \[ \int \text {csch}^{10}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
((2*a*b^2)/(3*d) - (2*exp(2*c + 2*d*x)*(7*a*b^2 + 4*a^2*b))/(3*d) + (2*exp (4*c + 4*d*x)*(7*a*b^2 + 8*a^2*b))/d + (10*exp(8*c + 8*d*x)*(7*a*b^2 + 8*a ^2*b))/(3*d) - (2*exp(10*c + 10*d*x)*(7*a*b^2 + 4*a^2*b))/d - (2*exp(6*c + 6*d*x)*(105*a*b^2 + 144*a^2*b + 128*a^3))/(9*d) + (14*a*b^2*exp(12*c + 12 *d*x))/(3*d) - (2*a*b^2*exp(14*c + 14*d*x))/(3*d))/(28*exp(4*c + 4*d*x) - 8*exp(2*c + 2*d*x) - 56*exp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x) - 56*exp(10 *c + 10*d*x) + 28*exp(12*c + 12*d*x) - 8*exp(14*c + 14*d*x) + exp(16*c + 1 6*d*x) + 1) - ((2*(105*a*b^2 + 144*a^2*b + 128*a^3))/(315*d) - (8*exp(2*c + 2*d*x)*(7*a*b^2 + 8*a^2*b))/(21*d) + (4*exp(4*c + 4*d*x)*(7*a*b^2 + 4*a^ 2*b))/(7*d) - (8*a*b^2*exp(6*c + 6*d*x))/(3*d) + (2*a*b^2*exp(8*c + 8*d*x) )/(3*d))/(5*exp(2*c + 2*d*x) - 10*exp(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) - ((2*(7*a*b^2 + 4*a^2*b))/( 21*d) - (4*a*b^2*exp(2*c + 2*d*x))/(3*d) + (2*a*b^2*exp(4*c + 4*d*x))/(3*d ))/(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) - ((2* a*b^2)/(3*d) + (8*exp(4*c + 4*d*x)*(7*a*b^2 + 4*a^2*b))/(3*d) - (16*exp(6* c + 6*d*x)*(7*a*b^2 + 8*a^2*b))/(3*d) - (16*exp(10*c + 10*d*x)*(7*a*b^2 + 8*a^2*b))/(3*d) + (8*exp(12*c + 12*d*x)*(7*a*b^2 + 4*a^2*b))/(3*d) + (4*ex p(8*c + 8*d*x)*(105*a*b^2 + 144*a^2*b + 128*a^3))/(9*d) - (16*a*b^2*exp(2* c + 2*d*x))/(3*d) - (16*a*b^2*exp(14*c + 14*d*x))/(3*d) + (2*a*b^2*exp(16* c + 16*d*x))/(3*d))/(9*exp(2*c + 2*d*x) - 36*exp(4*c + 4*d*x) + 84*exp(...