Integrand size = 21, antiderivative size = 158 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {a^3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {b \left (3 a^2+9 a b+5 b^2\right ) \cosh ^3(c+d x)}{3 d}-\frac {b^2 (9 a+10 b) \cosh ^5(c+d x)}{5 d}+\frac {b^2 (3 a+10 b) \cosh ^7(c+d x)}{7 d}-\frac {5 b^3 \cosh ^9(c+d x)}{9 d}+\frac {b^3 \cosh ^{11}(c+d x)}{11 d} \]
-a^3*arctanh(cosh(d*x+c))/d-b*(3*a^2+3*a*b+b^2)*cosh(d*x+c)/d+1/3*b*(3*a^2 +9*a*b+5*b^2)*cosh(d*x+c)^3/d-1/5*b^2*(9*a+10*b)*cosh(d*x+c)^5/d+1/7*b^2*( 3*a+10*b)*cosh(d*x+c)^7/d-5/9*b^3*cosh(d*x+c)^9/d+1/11*b^3*cosh(d*x+c)^11/ d
Time = 5.95 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.98 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {-20790 b \left (384 a^2+280 a b+77 b^2\right ) \cosh (c+d x)+6930 b (8 a+5 b) (16 a+11 b) \cosh (3 (c+d x))-2079 b^2 (112 a+55 b) \cosh (5 (c+d x))+495 b^2 (48 a+55 b) \cosh (7 (c+d x))-4235 b^3 \cosh (9 (c+d x))+315 b^3 \cosh (11 (c+d x))-3548160 a^3 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+3548160 a^3 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{3548160 d} \]
(-20790*b*(384*a^2 + 280*a*b + 77*b^2)*Cosh[c + d*x] + 6930*b*(8*a + 5*b)* (16*a + 11*b)*Cosh[3*(c + d*x)] - 2079*b^2*(112*a + 55*b)*Cosh[5*(c + d*x) ] + 495*b^2*(48*a + 55*b)*Cosh[7*(c + d*x)] - 4235*b^3*Cosh[9*(c + d*x)] + 315*b^3*Cosh[11*(c + d*x)] - 3548160*a^3*Log[Cosh[(c + d*x)/2]] + 3548160 *a^3*Log[Sinh[(c + d*x)/2]])/(3548160*d)
Time = 0.35 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.89, 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, 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.
| \(\displaystyle \int \text {csch}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \left (a+b \sin (i c+i d x)^4\right )^3}{\sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\left (b \sin (i c+i d x)^4+a\right )^3}{\sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle -\frac {\int \frac {\left (b \cosh ^4(c+d x)-2 b \cosh ^2(c+d x)+a+b\right )^3}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle -\frac {\int \left (-b^3 \cosh ^{10}(c+d x)+5 b^3 \cosh ^8(c+d x)-b^2 (3 a+10 b) \cosh ^6(c+d x)+b^2 (9 a+10 b) \cosh ^4(c+d x)-b \left (3 a^2+9 b a+5 b^2\right ) \cosh ^2(c+d x)+b \left (3 a^2+3 b a+b^2\right )+\frac {a^3}{1-\cosh ^2(c+d x)}\right )d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \text {arctanh}(\cosh (c+d x))-\frac {1}{3} b \left (3 a^2+9 a b+5 b^2\right ) \cosh ^3(c+d x)+b \left (3 a^2+3 a b+b^2\right ) \cosh (c+d x)-\frac {1}{7} b^2 (3 a+10 b) \cosh ^7(c+d x)+\frac {1}{5} b^2 (9 a+10 b) \cosh ^5(c+d x)-\frac {1}{11} b^3 \cosh ^{11}(c+d x)+\frac {5}{9} b^3 \cosh ^9(c+d x)}{d}\) |
-((a^3*ArcTanh[Cosh[c + d*x]] + b*(3*a^2 + 3*a*b + b^2)*Cosh[c + d*x] - (b *(3*a^2 + 9*a*b + 5*b^2)*Cosh[c + d*x]^3)/3 + (b^2*(9*a + 10*b)*Cosh[c + d *x]^5)/5 - (b^2*(3*a + 10*b)*Cosh[c + d*x]^7)/7 + (5*b^3*Cosh[c + d*x]^9)/ 9 - (b^3*Cosh[c + d*x]^11)/11)/d)
3.3.10.3.1 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 = 6.76 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.91
| method | result | size |
| parallelrisch | \(\frac {4 a^{3} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 b \left (-\frac {\left (a +\frac {5 b}{8}\right ) \left (a +\frac {11 b}{16}\right ) \cosh \left (3 d x +3 c \right )}{9}+\frac {7 \left (a +\frac {55 b}{112}\right ) b \cosh \left (5 d x +5 c \right )}{240}-\frac {\left (a +\frac {55 b}{48}\right ) b \cosh \left (7 d x +7 c \right )}{336}-\frac {b^{2} \cosh \left (11 d x +11 c \right )}{25344}+\frac {11 b^{2} \cosh \left (9 d x +9 c \right )}{20736}+\left (a^{2}+\frac {35}{48} a b +\frac {77}{384} b^{2}\right ) \cosh \left (d x +c \right )+\frac {8 a^{2}}{9}+\frac {64 a b}{105}+\frac {1024 b^{2}}{6237}\right )}{4 d}\) | \(143\) |
| derivativedivides | \(\frac {-2 a^{3} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+3 a^{2} b \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+3 a \,b^{2} \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )+b^{3} \left (-\frac {256}{693}+\frac {\sinh \left (d x +c \right )^{10}}{11}-\frac {10 \sinh \left (d x +c \right )^{8}}{99}+\frac {80 \sinh \left (d x +c \right )^{6}}{693}-\frac {32 \sinh \left (d x +c \right )^{4}}{231}+\frac {128 \sinh \left (d x +c \right )^{2}}{693}\right ) \cosh \left (d x +c \right )}{d}\) | \(148\) |
| default | \(\frac {-2 a^{3} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+3 a^{2} b \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+3 a \,b^{2} \left (-\frac {16}{35}+\frac {\sinh \left (d x +c \right )^{6}}{7}-\frac {6 \sinh \left (d x +c \right )^{4}}{35}+\frac {8 \sinh \left (d x +c \right )^{2}}{35}\right ) \cosh \left (d x +c \right )+b^{3} \left (-\frac {256}{693}+\frac {\sinh \left (d x +c \right )^{10}}{11}-\frac {10 \sinh \left (d x +c \right )^{8}}{99}+\frac {80 \sinh \left (d x +c \right )^{6}}{693}-\frac {32 \sinh \left (d x +c \right )^{4}}{231}+\frac {128 \sinh \left (d x +c \right )^{2}}{693}\right ) \cosh \left (d x +c \right )}{d}\) | \(148\) |
| risch | \(\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}+\frac {b^{3} {\mathrm e}^{11 d x +11 c}}{22528 d}-\frac {11 b^{3} {\mathrm e}^{9 d x +9 c}}{18432 d}-\frac {11 b^{3} {\mathrm e}^{-9 d x -9 c}}{18432 d}+\frac {b^{3} {\mathrm e}^{-11 d x -11 c}}{22528 d}+\frac {55 b^{3} {\mathrm e}^{7 d x +7 c}}{14336 d}-\frac {33 b^{3} {\mathrm e}^{5 d x +5 c}}{2048 d}+\frac {55 \,{\mathrm e}^{3 d x +3 c} b^{3}}{1024 d}-\frac {231 \,{\mathrm e}^{d x +c} b^{3}}{1024 d}-\frac {231 \,{\mathrm e}^{-d x -c} b^{3}}{1024 d}+\frac {55 \,{\mathrm e}^{-3 d x -3 c} b^{3}}{1024 d}-\frac {33 b^{3} {\mathrm e}^{-5 d x -5 c}}{2048 d}+\frac {55 b^{3} {\mathrm e}^{-7 d x -7 c}}{14336 d}-\frac {9 \,{\mathrm e}^{-d x -c} a^{2} b}{8 d}-\frac {105 \,{\mathrm e}^{-d x -c} a \,b^{2}}{128 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{2} b}{8 d}+\frac {21 \,{\mathrm e}^{-3 d x -3 c} a \,b^{2}}{128 d}-\frac {21 b^{2} {\mathrm e}^{-5 d x -5 c} a}{640 d}-\frac {21 b^{2} {\mathrm e}^{5 d x +5 c} a}{640 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2} b}{8 d}+\frac {21 \,{\mathrm e}^{3 d x +3 c} a \,b^{2}}{128 d}-\frac {9 \,{\mathrm e}^{d x +c} a^{2} b}{8 d}-\frac {105 \,{\mathrm e}^{d x +c} a \,b^{2}}{128 d}+\frac {3 a \,b^{2} {\mathrm e}^{7 d x +7 c}}{896 d}+\frac {3 a \,b^{2} {\mathrm e}^{-7 d x -7 c}}{896 d}\) | \(425\) |
1/4*(4*a^3*ln(tanh(1/2*d*x+1/2*c))-9*b*(-1/9*(a+5/8*b)*(a+11/16*b)*cosh(3* d*x+3*c)+7/240*(a+55/112*b)*b*cosh(5*d*x+5*c)-1/336*(a+55/48*b)*b*cosh(7*d *x+7*c)-1/25344*b^2*cosh(11*d*x+11*c)+11/20736*b^2*cosh(9*d*x+9*c)+(a^2+35 /48*a*b+77/384*b^2)*cosh(d*x+c)+8/9*a^2+64/105*a*b+1024/6237*b^2))/d
Leaf count of result is larger than twice the leaf count of optimal. 3824 vs. \(2 (148) = 296\).
Time = 0.34 (sec) , antiderivative size = 3824, normalized size of antiderivative = 24.20 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
1/7096320*(315*b^3*cosh(d*x + c)^22 + 6930*b^3*cosh(d*x + c)*sinh(d*x + c) ^21 + 315*b^3*sinh(d*x + c)^22 - 4235*b^3*cosh(d*x + c)^20 + 385*(189*b^3* cosh(d*x + c)^2 - 11*b^3)*sinh(d*x + c)^20 + 7700*(63*b^3*cosh(d*x + c)^3 - 11*b^3*cosh(d*x + c))*sinh(d*x + c)^19 + 495*(48*a*b^2 + 55*b^3)*cosh(d* x + c)^18 + 55*(41895*b^3*cosh(d*x + c)^4 - 14630*b^3*cosh(d*x + c)^2 + 43 2*a*b^2 + 495*b^3)*sinh(d*x + c)^18 + 330*(25137*b^3*cosh(d*x + c)^5 - 146 30*b^3*cosh(d*x + c)^3 + 27*(48*a*b^2 + 55*b^3)*cosh(d*x + c))*sinh(d*x + c)^17 - 2079*(112*a*b^2 + 55*b^3)*cosh(d*x + c)^16 + 33*(712215*b^3*cosh(d *x + c)^6 - 621775*b^3*cosh(d*x + c)^4 - 7056*a*b^2 - 3465*b^3 + 2295*(48* a*b^2 + 55*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^16 + 528*(101745*b^3*cosh(d *x + c)^7 - 124355*b^3*cosh(d*x + c)^5 + 765*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^3 - 63*(112*a*b^2 + 55*b^3)*cosh(d*x + c))*sinh(d*x + c)^15 + 6930*(1 28*a^2*b + 168*a*b^2 + 55*b^3)*cosh(d*x + c)^14 + 330*(305235*b^3*cosh(d*x + c)^8 - 497420*b^3*cosh(d*x + c)^6 + 4590*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^4 + 2688*a^2*b + 3528*a*b^2 + 1155*b^3 - 756*(112*a*b^2 + 55*b^3)*cosh (d*x + c)^2)*sinh(d*x + c)^14 + 4620*(33915*b^3*cosh(d*x + c)^9 - 71060*b^ 3*cosh(d*x + c)^7 + 918*(48*a*b^2 + 55*b^3)*cosh(d*x + c)^5 - 252*(112*a*b ^2 + 55*b^3)*cosh(d*x + c)^3 + 21*(128*a^2*b + 168*a*b^2 + 55*b^3)*cosh(d* x + c))*sinh(d*x + c)^13 - 20790*(384*a^2*b + 280*a*b^2 + 77*b^3)*cosh(d*x + c)^12 + 2310*(88179*b^3*cosh(d*x + c)^10 - 230945*b^3*cosh(d*x + c)^...
Timed out. \[ \int \text {csch}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (148) = 296\).
Time = 0.19 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.07 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=-\frac {1}{1419264} \, b^{3} {\left (\frac {{\left (847 \, e^{\left (-2 \, d x - 2 \, c\right )} - 5445 \, e^{\left (-4 \, d x - 4 \, c\right )} + 22869 \, e^{\left (-6 \, d x - 6 \, c\right )} - 76230 \, e^{\left (-8 \, d x - 8 \, c\right )} + 320166 \, e^{\left (-10 \, d x - 10 \, c\right )} - 63\right )} e^{\left (11 \, d x + 11 \, c\right )}}{d} + \frac {320166 \, e^{\left (-d x - c\right )} - 76230 \, e^{\left (-3 \, d x - 3 \, c\right )} + 22869 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5445 \, e^{\left (-7 \, d x - 7 \, c\right )} + 847 \, e^{\left (-9 \, d x - 9 \, c\right )} - 63 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d}\right )} - \frac {3}{4480} \, a b^{2} {\left (\frac {{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{8} \, a^{2} b {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {a^{3} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
-1/1419264*b^3*((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) - 3/4480*a*b^2*((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) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/d) + 1/8* a^2*b*(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) + a^3*log(tanh(1/2*d*x + 1/2*c))/d
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (148) = 296\).
Time = 0.46 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.39 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {315 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} - 4235 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} + 23760 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 27225 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 232848 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 114345 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 887040 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 1164240 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 381150 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 7983360 \, a^{2} b e^{\left (d x + c\right )} - 5821200 \, a b^{2} e^{\left (d x + c\right )} - 1600830 \, b^{3} e^{\left (d x + c\right )} - 7096320 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) + 7096320 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - {\left (7983360 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 5821200 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1600830 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 887040 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 1164240 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 381150 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 232848 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 114345 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 23760 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 27225 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 4235 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 315 \, b^{3}\right )} e^{\left (-11 \, d x - 11 \, c\right )}}{7096320 \, d} \]
1/7096320*(315*b^3*e^(11*d*x + 11*c) - 4235*b^3*e^(9*d*x + 9*c) + 23760*a* b^2*e^(7*d*x + 7*c) + 27225*b^3*e^(7*d*x + 7*c) - 232848*a*b^2*e^(5*d*x + 5*c) - 114345*b^3*e^(5*d*x + 5*c) + 887040*a^2*b*e^(3*d*x + 3*c) + 1164240 *a*b^2*e^(3*d*x + 3*c) + 381150*b^3*e^(3*d*x + 3*c) - 7983360*a^2*b*e^(d*x + c) - 5821200*a*b^2*e^(d*x + c) - 1600830*b^3*e^(d*x + c) - 7096320*a^3* log(e^(d*x + c) + 1) + 7096320*a^3*log(abs(e^(d*x + c) - 1)) - (7983360*a^ 2*b*e^(10*d*x + 10*c) + 5821200*a*b^2*e^(10*d*x + 10*c) + 1600830*b^3*e^(1 0*d*x + 10*c) - 887040*a^2*b*e^(8*d*x + 8*c) - 1164240*a*b^2*e^(8*d*x + 8* c) - 381150*b^3*e^(8*d*x + 8*c) + 232848*a*b^2*e^(6*d*x + 6*c) + 114345*b^ 3*e^(6*d*x + 6*c) - 23760*a*b^2*e^(4*d*x + 4*c) - 27225*b^3*e^(4*d*x + 4*c ) + 4235*b^3*e^(2*d*x + 2*c) - 315*b^3)*e^(-11*d*x - 11*c))/d
Time = 0.69 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.06 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (128\,a^2\,b+168\,a\,b^2+55\,b^3\right )}{1024\,d}-\frac {2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^6}}\right )\,\sqrt {a^6}}{\sqrt {-d^2}}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (128\,a^2\,b+168\,a\,b^2+55\,b^3\right )}{1024\,d}-\frac {11\,b^3\,{\mathrm {e}}^{-9\,c-9\,d\,x}}{18432\,d}-\frac {11\,b^3\,{\mathrm {e}}^{9\,c+9\,d\,x}}{18432\,d}+\frac {b^3\,{\mathrm {e}}^{-11\,c-11\,d\,x}}{22528\,d}+\frac {b^3\,{\mathrm {e}}^{11\,c+11\,d\,x}}{22528\,d}-\frac {3\,b\,{\mathrm {e}}^{-c-d\,x}\,\left (384\,a^2+280\,a\,b+77\,b^2\right )}{1024\,d}+\frac {b^2\,{\mathrm {e}}^{-7\,c-7\,d\,x}\,\left (48\,a+55\,b\right )}{14336\,d}+\frac {b^2\,{\mathrm {e}}^{7\,c+7\,d\,x}\,\left (48\,a+55\,b\right )}{14336\,d}-\frac {3\,b^2\,{\mathrm {e}}^{-5\,c-5\,d\,x}\,\left (112\,a+55\,b\right )}{10240\,d}-\frac {3\,b^2\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (112\,a+55\,b\right )}{10240\,d}-\frac {3\,b\,{\mathrm {e}}^{c+d\,x}\,\left (384\,a^2+280\,a\,b+77\,b^2\right )}{1024\,d} \]
(exp(- 3*c - 3*d*x)*(168*a*b^2 + 128*a^2*b + 55*b^3))/(1024*d) - (2*atan(( a^3*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d*(a^6)^(1/2)))*(a^6)^(1/2))/(-d^2)^(1/ 2) + (exp(3*c + 3*d*x)*(168*a*b^2 + 128*a^2*b + 55*b^3))/(1024*d) - (11*b^ 3*exp(- 9*c - 9*d*x))/(18432*d) - (11*b^3*exp(9*c + 9*d*x))/(18432*d) + (b ^3*exp(- 11*c - 11*d*x))/(22528*d) + (b^3*exp(11*c + 11*d*x))/(22528*d) - (3*b*exp(- c - d*x)*(280*a*b + 384*a^2 + 77*b^2))/(1024*d) + (b^2*exp(- 7* c - 7*d*x)*(48*a + 55*b))/(14336*d) + (b^2*exp(7*c + 7*d*x)*(48*a + 55*b)) /(14336*d) - (3*b^2*exp(- 5*c - 5*d*x)*(112*a + 55*b))/(10240*d) - (3*b^2* exp(5*c + 5*d*x)*(112*a + 55*b))/(10240*d) - (3*b*exp(c + d*x)*(280*a*b + 384*a^2 + 77*b^2))/(1024*d)