Integrand size = 23, antiderivative size = 156 \[ \int \text {csch}^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=3 a^2 b x-\frac {5 b^3 x}{16}+\frac {a^3 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {5 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b^3 \cosh (c+d x) \sinh ^5(c+d x)}{6 d} \]
3*a^2*b*x-5/16*b^3*x+1/2*a^3*arctanh(cosh(d*x+c))/d-3*a*b^2*cosh(d*x+c)/d+ a*b^2*cosh(d*x+c)^3/d-1/2*a^3*coth(d*x+c)*csch(d*x+c)/d+5/16*b^3*cosh(d*x+ c)*sinh(d*x+c)/d-5/24*b^3*cosh(d*x+c)*sinh(d*x+c)^3/d+1/6*b^3*cosh(d*x+c)* sinh(d*x+c)^5/d
Time = 5.14 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.06 \[ \int \text {csch}^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {576 a^2 b c-60 b^3 c+576 a^2 b d x-60 b^3 d x-432 a b^2 \cosh (c+d x)+48 a b^2 \cosh (3 (c+d x))-24 a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+96 a^3 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-96 a^3 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )-24 a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+45 b^3 \sinh (2 (c+d x))-9 b^3 \sinh (4 (c+d x))+b^3 \sinh (6 (c+d x))}{192 d} \]
(576*a^2*b*c - 60*b^3*c + 576*a^2*b*d*x - 60*b^3*d*x - 432*a*b^2*Cosh[c + d*x] + 48*a*b^2*Cosh[3*(c + d*x)] - 24*a^3*Csch[(c + d*x)/2]^2 + 96*a^3*Lo g[Cosh[(c + d*x)/2]] - 96*a^3*Log[Sinh[(c + d*x)/2]] - 24*a^3*Sech[(c + d* x)/2]^2 + 45*b^3*Sinh[2*(c + d*x)] - 9*b^3*Sinh[4*(c + d*x)] + b^3*Sinh[6* (c + d*x)])/(192*d)
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 26, 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.
| \(\displaystyle \int \text {csch}^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a+i b \sin (i c+i d x)^3\right )^3}{\sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\left (i b \sin (i c+i d x)^3+a\right )^3}{\sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 3699 |
| \(\displaystyle -i \int \left (i b^3 \sinh ^6(c+d x)+3 i a b^2 \sinh ^3(c+d x)+i a^3 \text {csch}^3(c+d x)+3 i a^2 b\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -i \left (\frac {i a^3 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {i a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+3 i a^2 b x+\frac {i a b^2 \cosh ^3(c+d x)}{d}-\frac {3 i a b^2 \cosh (c+d x)}{d}+\frac {i b^3 \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac {5 i b^3 \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac {5 i b^3 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {5}{16} i b^3 x\right )\) |
(-I)*((3*I)*a^2*b*x - ((5*I)/16)*b^3*x + ((I/2)*a^3*ArcTanh[Cosh[c + d*x]] )/d - ((3*I)*a*b^2*Cosh[c + d*x])/d + (I*a*b^2*Cosh[c + d*x]^3)/d - ((I/2) *a^3*Coth[c + d*x]*Csch[c + d*x])/d + (((5*I)/16)*b^3*Cosh[c + d*x]*Sinh[c + d*x])/d - (((5*I)/24)*b^3*Cosh[c + d*x]*Sinh[c + d*x]^3)/d + ((I/6)*b^3 *Cosh[c + d*x]*Sinh[c + d*x]^5)/d)
3.2.66.3.1 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[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 = 0.89 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (d x +c \right )+3 a \,b^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+b^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}\) | \(115\) |
| default | \(\frac {a^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (d x +c \right )+3 a \,b^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+b^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}\) | \(115\) |
| parallelrisch | \(\frac {192 a^{3} \ln \left (\frac {1}{\sqrt {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}\right )+24 a^{3} \left (\operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-144 a^{3} \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+576 x \,a^{2} b d -60 x \,b^{3} d -432 a \,b^{2} \cosh \left (d x +c \right )+48 a \,b^{2} \cosh \left (3 d x +3 c \right )+b^{3} \sinh \left (6 d x +6 c \right )+45 b^{3} \sinh \left (2 d x +2 c \right )-9 b^{3} \sinh \left (4 d x +4 c \right )+384 a \,b^{2}}{192 d}\) | \(158\) |
| risch | \(3 a^{2} b x -\frac {5 b^{3} x}{16}+\frac {b^{3} {\mathrm e}^{6 d x +6 c}}{384 d}-\frac {3 \,{\mathrm e}^{4 d x +4 c} b^{3}}{128 d}+\frac {{\mathrm e}^{3 d x +3 c} a \,b^{2}}{8 d}+\frac {15 \,{\mathrm e}^{2 d x +2 c} b^{3}}{128 d}-\frac {9 \,{\mathrm e}^{d x +c} a \,b^{2}}{8 d}-\frac {9 \,{\mathrm e}^{-d x -c} a \,b^{2}}{8 d}-\frac {15 \,{\mathrm e}^{-2 d x -2 c} b^{3}}{128 d}+\frac {{\mathrm e}^{-3 d x -3 c} a \,b^{2}}{8 d}+\frac {3 \,{\mathrm e}^{-4 d x -4 c} b^{3}}{128 d}-\frac {b^{3} {\mathrm e}^{-6 d x -6 c}}{384 d}-\frac {a^{3} {\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}\) | \(258\) |
1/d*(a^3*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+3*a^2*b*(d*x+c )+3*a*b^2*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c)+b^3*((1/6*sinh(d*x+c)^5-5/2 4*sinh(d*x+c)^3+5/16*sinh(d*x+c))*cosh(d*x+c)-5/16*d*x-5/16*c))
Leaf count of result is larger than twice the leaf count of optimal. 3627 vs. \(2 (144) = 288\).
Time = 0.31 (sec) , antiderivative size = 3627, normalized size of antiderivative = 23.25 \[ \int \text {csch}^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \]
1/384*(b^3*cosh(d*x + c)^16 + 16*b^3*cosh(d*x + c)*sinh(d*x + c)^15 + b^3* sinh(d*x + c)^16 - 11*b^3*cosh(d*x + c)^14 + 48*a*b^2*cosh(d*x + c)^13 + 6 4*b^3*cosh(d*x + c)^12 + (120*b^3*cosh(d*x + c)^2 - 11*b^3)*sinh(d*x + c)^ 14 - 528*a*b^2*cosh(d*x + c)^11 + 2*(280*b^3*cosh(d*x + c)^3 - 77*b^3*cosh (d*x + c) + 24*a*b^2)*sinh(d*x + c)^13 + (1820*b^3*cosh(d*x + c)^4 - 1001* b^3*cosh(d*x + c)^2 + 624*a*b^2*cosh(d*x + c) + 64*b^3)*sinh(d*x + c)^12 + 4*(1092*b^3*cosh(d*x + c)^5 - 1001*b^3*cosh(d*x + c)^3 + 936*a*b^2*cosh(d *x + c)^2 + 192*b^3*cosh(d*x + c) - 132*a*b^2)*sinh(d*x + c)^11 - 48*(48*a ^2*b - 5*b^3)*d*x*cosh(d*x + c)^8 - 3*(33*b^3 - 8*(48*a^2*b - 5*b^3)*d*x)* cosh(d*x + c)^10 + (8008*b^3*cosh(d*x + c)^6 - 11011*b^3*cosh(d*x + c)^4 + 13728*a*b^2*cosh(d*x + c)^3 + 4224*b^3*cosh(d*x + c)^2 - 5808*a*b^2*cosh( d*x + c) - 99*b^3 + 24*(48*a^2*b - 5*b^3)*d*x)*sinh(d*x + c)^10 - 96*(4*a^ 3 - 5*a*b^2)*cosh(d*x + c)^9 + 2*(5720*b^3*cosh(d*x + c)^7 - 11011*b^3*cos h(d*x + c)^5 + 17160*a*b^2*cosh(d*x + c)^4 + 7040*b^3*cosh(d*x + c)^3 - 14 520*a*b^2*cosh(d*x + c)^2 - 192*a^3 + 240*a*b^2 - 15*(33*b^3 - 8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^9 + 3*(4290*b^3*cosh(d*x + c)^ 8 - 11011*b^3*cosh(d*x + c)^6 + 20592*a*b^2*cosh(d*x + c)^5 + 10560*b^3*co sh(d*x + c)^4 - 29040*a*b^2*cosh(d*x + c)^3 - 16*(48*a^2*b - 5*b^3)*d*x - 45*(33*b^3 - 8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c)^2 - 288*(4*a^3 - 5*a* b^2)*cosh(d*x + c))*sinh(d*x + c)^8 - 528*a*b^2*cosh(d*x + c)^5 - 96*(4...
Timed out. \[ \int \text {csch}^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.56 \[ \int \text {csch}^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=3 \, a^{2} b x - \frac {1}{384} \, b^{3} {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} + \frac {1}{8} \, a b^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {1}{2} \, a^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
3*a^2*b*x - 1/384*b^3*((9*e^(-2*d*x - 2*c) - 45*e^(-4*d*x - 4*c) - 1)*e^(6 *d*x + 6*c)/d + 120*(d*x + c)/d + (45*e^(-2*d*x - 2*c) - 9*e^(-4*d*x - 4*c ) + e^(-6*d*x - 6*c))/d) + 1/8*a*b^2*(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) + 1/2*a^3*(log(e^(-d*x - c) + 1)/ d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^ (-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1)))
Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (144) = 288\).
Time = 0.40 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.85 \[ \int \text {csch}^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=\frac {b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 45 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 432 \, a b^{2} e^{\left (d x + c\right )} + 192 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) - 192 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + 24 \, {\left (48 \, a^{2} b - 5 \, b^{3}\right )} {\left (d x + c\right )} - \frac {{\left (45 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 99 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 528 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 64 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 48 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 11 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + b^{3} + 48 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} e^{\left (9 \, d x + 9 \, c\right )} + 48 \, {\left (8 \, a^{3} - 19 \, a b^{2}\right )} e^{\left (7 \, d x + 7 \, c\right )}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )}^{2} {\left (e^{\left (d x + c\right )} - 1\right )}^{2}}}{384 \, d} \]
1/384*(b^3*e^(6*d*x + 6*c) - 9*b^3*e^(4*d*x + 4*c) + 48*a*b^2*e^(3*d*x + 3 *c) + 45*b^3*e^(2*d*x + 2*c) - 432*a*b^2*e^(d*x + c) + 192*a^3*log(e^(d*x + c) + 1) - 192*a^3*log(abs(e^(d*x + c) - 1)) + 24*(48*a^2*b - 5*b^3)*(d*x + c) - (45*b^3*e^(8*d*x + 8*c) - 99*b^3*e^(6*d*x + 6*c) + 528*a*b^2*e^(5* d*x + 5*c) + 64*b^3*e^(4*d*x + 4*c) - 48*a*b^2*e^(3*d*x + 3*c) - 11*b^3*e^ (2*d*x + 2*c) + b^3 + 48*(8*a^3 + 9*a*b^2)*e^(9*d*x + 9*c) + 48*(8*a^3 - 1 9*a*b^2)*e^(7*d*x + 7*c))*e^(-6*d*x - 6*c)/((e^(d*x + c) + 1)^2*(e^(d*x + c) - 1)^2))/d
Time = 1.68 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.86 \[ \int \text {csch}^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx=x\,\left (3\,a^2\,b-\frac {5\,b^3}{16}\right )+\frac {\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^6}}\right )\,\sqrt {a^6}}{\sqrt {-d^2}}-\frac {15\,b^3\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{128\,d}+\frac {15\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{128\,d}+\frac {3\,b^3\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{128\,d}-\frac {3\,b^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{128\,d}-\frac {b^3\,{\mathrm {e}}^{-6\,c-6\,d\,x}}{384\,d}+\frac {b^3\,{\mathrm {e}}^{6\,c+6\,d\,x}}{384\,d}-\frac {9\,a\,b^2\,{\mathrm {e}}^{-c-d\,x}}{8\,d}+\frac {a\,b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{8\,d}+\frac {a\,b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{8\,d}-\frac {9\,a\,b^2\,{\mathrm {e}}^{c+d\,x}}{8\,d}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
x*(3*a^2*b - (5*b^3)/16) + (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) - (15*b^3*exp(- 2*c - 2*d*x))/(128*d) + (15*b^3*exp(2*c + 2*d*x))/(128*d) + (3*b^3*exp(- 4*c - 4*d*x))/(128*d) - (3*b^3*exp(4*c + 4*d*x))/(128*d) - (b^3*exp(- 6*c - 6*d*x))/(384*d) + (b ^3*exp(6*c + 6*d*x))/(384*d) - (9*a*b^2*exp(- c - d*x))/(8*d) + (a*b^2*exp (- 3*c - 3*d*x))/(8*d) + (a*b^2*exp(3*c + 3*d*x))/(8*d) - (9*a*b^2*exp(c + d*x))/(8*d) - (a^3*exp(c + d*x))/(d*(exp(2*c + 2*d*x) - 1)) - (2*a^3*exp( c + d*x))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))