Integrand size = 23, antiderivative size = 129 \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=-\frac {3}{2} a b^2 x-\frac {3 a^2 b \text {arctanh}(\cosh (c+d x))}{d}+\frac {b^3 \cosh (c+d x)}{d}-\frac {2 b^3 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}+\frac {a^3 \coth (c+d x)}{d}-\frac {a^3 \coth ^3(c+d x)}{3 d}+\frac {3 a b^2 \cosh (c+d x) \sinh (c+d x)}{2 d} \]
-3/2*a*b^2*x-3*a^2*b*arctanh(cosh(d*x+c))/d+b^3*cosh(d*x+c)/d-2/3*b^3*cosh (d*x+c)^3/d+1/5*b^3*cosh(d*x+c)^5/d+a^3*coth(d*x+c)/d-1/3*a^3*coth(d*x+c)^ 3/d+3/2*a*b^2*cosh(d*x+c)*sinh(d*x+c)/d
Time = 4.84 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.44 \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {-360 a b^2 c-360 a b^2 d x+150 b^3 \cosh (c+d x)-25 b^3 \cosh (3 (c+d x))+3 b^3 \cosh (5 (c+d x))+80 a^3 \coth \left (\frac {1}{2} (c+d x)\right )-720 a^2 b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+720 a^2 b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+80 a^3 \text {csch}^3(c+d x) \sinh ^4\left (\frac {1}{2} (c+d x)\right )-5 a^3 \text {csch}^4\left (\frac {1}{2} (c+d x)\right ) \sinh (c+d x)+180 a b^2 \sinh (2 (c+d x))+80 a^3 \tanh \left (\frac {1}{2} (c+d x)\right )}{240 d} \]
(-360*a*b^2*c - 360*a*b^2*d*x + 150*b^3*Cosh[c + d*x] - 25*b^3*Cosh[3*(c + d*x)] + 3*b^3*Cosh[5*(c + d*x)] + 80*a^3*Coth[(c + d*x)/2] - 720*a^2*b*Lo g[Cosh[(c + d*x)/2]] + 720*a^2*b*Log[Sinh[(c + d*x)/2]] + 80*a^3*Csch[c + d*x]^3*Sinh[(c + d*x)/2]^4 - 5*a^3*Csch[(c + d*x)/2]^4*Sinh[c + d*x] + 180 *a*b^2*Sinh[2*(c + d*x)] + 80*a^3*Tanh[(c + d*x)/2])/(240*d)
Time = 0.35 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3042, 3699, 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.
| \(\displaystyle \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+i b \sin (i c+i d x)^3\right )^3}{\sin (i c+i d x)^4}dx\) |
| \(\Big \downarrow \) 3699 |
| \(\displaystyle \int \left (a^3 \text {csch}^4(c+d x)+3 a^2 b \text {csch}(c+d x)+3 a b^2 \sinh ^2(c+d x)+b^3 \sinh ^5(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \coth ^3(c+d x)}{3 d}+\frac {a^3 \coth (c+d x)}{d}-\frac {3 a^2 b \text {arctanh}(\cosh (c+d x))}{d}+\frac {3 a b^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {3}{2} a b^2 x+\frac {b^3 \cosh ^5(c+d x)}{5 d}-\frac {2 b^3 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh (c+d x)}{d}\) |
(-3*a*b^2*x)/2 - (3*a^2*b*ArcTanh[Cosh[c + d*x]])/d + (b^3*Cosh[c + d*x])/ d - (2*b^3*Cosh[c + d*x]^3)/(3*d) + (b^3*Cosh[c + d*x]^5)/(5*d) + (a^3*Cot h[c + d*x])/d - (a^3*Coth[c + d*x]^3)/(3*d) + (3*a*b^2*Cosh[c + d*x]*Sinh[ c + d*x])/(2*d)
3.2.67.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n) ^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4] || Gt Q[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Time = 0.74 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.78
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )-6 a^{2} b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+b^{3} \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )}{d}\) | \(101\) |
| default | \(\frac {a^{3} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )-6 a^{2} b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+b^{3} \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )}{d}\) | \(101\) |
| parallelrisch | \(\frac {720 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -15 \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\cosh \left (d x +c \right )-\frac {\cosh \left (3 d x +3 c \right )}{3}\right ) a^{3} \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-25 b^{3} \cosh \left (3 d x +3 c \right )+3 b^{3} \cosh \left (5 d x +5 c \right )-360 \left (a x d -\frac {a \sinh \left (2 d x +2 c \right )}{2}-\frac {5 b \cosh \left (d x +c \right )}{12}-\frac {16 b}{45}\right ) b^{2}}{240 d}\) | \(130\) |
| risch | \(-\frac {3 a \,b^{2} x}{2}+\frac {b^{3} {\mathrm e}^{5 d x +5 c}}{160 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} b^{3}}{96 d}+\frac {3 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{8 d}+\frac {5 \,{\mathrm e}^{d x +c} b^{3}}{16 d}+\frac {5 \,{\mathrm e}^{-d x -c} b^{3}}{16 d}-\frac {3 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{8 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} b^{3}}{96 d}+\frac {b^{3} {\mathrm e}^{-5 d x -5 c}}{160 d}-\frac {4 a^{3} \left (3 \,{\mathrm e}^{2 d x +2 c}-1\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d}\) | \(214\) |
1/d*(a^3*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)-6*a^2*b*arctanh(exp(d*x+c))+3 *a*b^2*(1/2*sinh(d*x+c)*cosh(d*x+c)-1/2*d*x-1/2*c)+b^3*(8/15+1/5*sinh(d*x+ c)^4-4/15*sinh(d*x+c)^2)*cosh(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 3801 vs. \(2 (119) = 238\).
Time = 0.31 (sec) , antiderivative size = 3801, normalized size of antiderivative = 29.47 \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \]
1/480*(3*b^3*cosh(d*x + c)^16 + 48*b^3*cosh(d*x + c)*sinh(d*x + c)^15 + 3* b^3*sinh(d*x + c)^16 - 34*b^3*cosh(d*x + c)^14 + 180*a*b^2*cosh(d*x + c)^1 3 + 234*b^3*cosh(d*x + c)^12 + 2*(180*b^3*cosh(d*x + c)^2 - 17*b^3)*sinh(d *x + c)^14 + 4*(420*b^3*cosh(d*x + c)^3 - 119*b^3*cosh(d*x + c) + 45*a*b^2 )*sinh(d*x + c)^13 - 378*b^3*cosh(d*x + c)^10 + 26*(210*b^3*cosh(d*x + c)^ 4 - 119*b^3*cosh(d*x + c)^2 + 90*a*b^2*cosh(d*x + c) + 9*b^3)*sinh(d*x + c )^12 - 180*(4*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c)^11 + 4*(3276*b^3*cosh(d*x + c)^5 - 3094*b^3*cosh(d*x + c)^3 - 180*a*b^2*d*x + 3510*a*b^2*cosh(d*x + c)^2 + 702*b^3*cosh(d*x + c) - 135*a*b^2)*sinh(d*x + c)^11 + 2*(12012*b^3 *cosh(d*x + c)^6 - 17017*b^3*cosh(d*x + c)^4 + 25740*a*b^2*cosh(d*x + c)^3 + 7722*b^3*cosh(d*x + c)^2 - 189*b^3 - 990*(4*a*b^2*d*x + 3*a*b^2)*cosh(d *x + c))*sinh(d*x + c)^10 + 360*(6*a*b^2*d*x + a*b^2)*cosh(d*x + c)^9 + 4* (8580*b^3*cosh(d*x + c)^7 - 17017*b^3*cosh(d*x + c)^5 + 32175*a*b^2*cosh(d *x + c)^4 + 12870*b^3*cosh(d*x + c)^3 + 540*a*b^2*d*x - 945*b^3*cosh(d*x + c) + 90*a*b^2 - 2475*(4*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 378*b^3*cosh(d*x + c)^6 + 6*(6435*b^3*cosh(d*x + c)^8 - 17017*b^3*c osh(d*x + c)^6 + 38610*a*b^2*cosh(d*x + c)^5 + 19305*b^3*cosh(d*x + c)^4 - 2835*b^3*cosh(d*x + c)^2 - 4950*(4*a*b^2*d*x + 3*a*b^2)*cosh(d*x + c)^3 + 540*(6*a*b^2*d*x + a*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 - 120*(18*a*b^2* d*x + 16*a^3 - 3*a*b^2)*cosh(d*x + c)^7 + 24*(1430*b^3*cosh(d*x + c)^9 ...
Timed out. \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (119) = 238\).
Time = 0.21 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.02 \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=-\frac {3}{8} \, a b^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + \frac {1}{480} \, b^{3} {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} - 3 \, a^{2} b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} + \frac {4}{3} \, a^{3} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} \]
-3/8*a*b^2*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) + 1/480*b^3*(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) - 3*a^2*b*(log(e^(- d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d) + 4/3*a^3*(3*e^(-2*d*x - 2*c)/( d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)) - 1/(d *(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)))
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (119) = 238\).
Time = 0.41 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.21 \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=-\frac {720 \, {\left (d x + c\right )} a b^{2} - 3 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 25 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 180 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 150 \, b^{3} e^{\left (d x + c\right )} + 1440 \, a^{2} b \log \left (e^{\left (d x + c\right )} + 1\right ) - 1440 \, a^{2} b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {{\left (150 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 180 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} - 475 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 528 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 234 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 180 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 34 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b^{3} - 60 \, {\left (32 \, a^{3} - 9 \, a b^{2}\right )} e^{\left (7 \, d x + 7 \, c\right )} + 20 \, {\left (32 \, a^{3} - 27 \, a b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )}^{3} {\left (e^{\left (d x + c\right )} - 1\right )}^{3}}}{480 \, d} \]
-1/480*(720*(d*x + c)*a*b^2 - 3*b^3*e^(5*d*x + 5*c) + 25*b^3*e^(3*d*x + 3* c) - 180*a*b^2*e^(2*d*x + 2*c) - 150*b^3*e^(d*x + c) + 1440*a^2*b*log(e^(d *x + c) + 1) - 1440*a^2*b*log(abs(e^(d*x + c) - 1)) - (150*b^3*e^(10*d*x + 10*c) - 180*a*b^2*e^(9*d*x + 9*c) - 475*b^3*e^(8*d*x + 8*c) + 528*b^3*e^( 6*d*x + 6*c) - 234*b^3*e^(4*d*x + 4*c) + 180*a*b^2*e^(3*d*x + 3*c) + 34*b^ 3*e^(2*d*x + 2*c) - 3*b^3 - 60*(32*a^3 - 9*a*b^2)*e^(7*d*x + 7*c) + 20*(32 *a^3 - 27*a*b^2)*e^(5*d*x + 5*c))*e^(-5*d*x - 5*c)/((e^(d*x + c) + 1)^3*(e ^(d*x + c) - 1)^3))/d
Time = 1.62 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.07 \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {5\,b^3\,{\mathrm {e}}^{c+d\,x}}{16\,d}-\frac {4\,a^3}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {5\,b^3\,{\mathrm {e}}^{-c-d\,x}}{16\,d}-\frac {5\,b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{96\,d}-\frac {5\,b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{96\,d}+\frac {b^3\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,d}+\frac {b^3\,{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,d}-\frac {8\,a^3}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {6\,\mathrm {atan}\left (\frac {a^2\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^4\,b^2}}\right )\,\sqrt {a^4\,b^2}}{\sqrt {-d^2}}-\frac {3\,a\,b^2\,x}{2}-\frac {3\,a\,b^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}+\frac {3\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d} \]
(5*b^3*exp(c + d*x))/(16*d) - (4*a^3)/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2 *d*x) + 1)) + (5*b^3*exp(- c - d*x))/(16*d) - (5*b^3*exp(- 3*c - 3*d*x))/( 96*d) - (5*b^3*exp(3*c + 3*d*x))/(96*d) + (b^3*exp(- 5*c - 5*d*x))/(160*d) + (b^3*exp(5*c + 5*d*x))/(160*d) - (8*a^3)/(3*d*(3*exp(2*c + 2*d*x) - 3*e xp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (6*atan((a^2*b*exp(d*x)*exp(c)* (-d^2)^(1/2))/(d*(a^4*b^2)^(1/2)))*(a^4*b^2)^(1/2))/(-d^2)^(1/2) - (3*a*b^ 2*x)/2 - (3*a*b^2*exp(- 2*c - 2*d*x))/(8*d) + (3*a*b^2*exp(2*c + 2*d*x))/( 8*d)