Integrand size = 23, antiderivative size = 182 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {3}{8} (a+b) \left (a^2+14 a b+21 b^2\right ) x-\frac {3 (a+b) \left (a^2+14 a b+21 b^2\right ) \tanh (c+d x)}{8 d}-\frac {b \left (6 a^2+35 a b+21 b^2\right ) \tanh ^3(c+d x)}{8 d}-\frac {3 b^2 (5 a+21 b) \tanh ^5(c+d x)}{40 d}-\frac {3 (a+3 b) \sinh ^2(c+d x) \tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^2}{8 d}+\frac {\cosh (c+d x) \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{4 d} \]
3/8*(a+b)*(a^2+14*a*b+21*b^2)*x-3/8*(a+b)*(a^2+14*a*b+21*b^2)*tanh(d*x+c)/ d-1/8*b*(6*a^2+35*a*b+21*b^2)*tanh(d*x+c)^3/d-3/40*b^2*(5*a+21*b)*tanh(d*x +c)^5/d-3/8*(a+3*b)*sinh(d*x+c)^2*tanh(d*x+c)*(a+b*tanh(d*x+c)^2)^2/d+1/4* cosh(d*x+c)*sinh(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3/d
Time = 5.62 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.69 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {60 \left (a^3+15 a^2 b+35 a b^2+21 b^3\right ) (c+d x)-40 (a+b)^2 (a+4 b) \sinh (2 (c+d x))+5 (a+b)^3 \sinh (4 (c+d x))-32 b \left (15 a^2+50 a b+36 b^2-b (5 a+7 b) \text {sech}^2(c+d x)+b^2 \text {sech}^4(c+d x)\right ) \tanh (c+d x)}{160 d} \]
(60*(a^3 + 15*a^2*b + 35*a*b^2 + 21*b^3)*(c + d*x) - 40*(a + b)^2*(a + 4*b )*Sinh[2*(c + d*x)] + 5*(a + b)^3*Sinh[4*(c + d*x)] - 32*b*(15*a^2 + 50*a* b + 36*b^2 - b*(5*a + 7*b)*Sech[c + d*x]^2 + b^2*Sech[c + d*x]^4)*Tanh[c + d*x])/(160*d)
Time = 0.42 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4146, 369, 27, 439, 25, 437, 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.
| \(\displaystyle \int \sinh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (i c+i d x)^4 \left (a-b \tan (i c+i d x)^2\right )^3dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\tanh ^4(c+d x) \left (b \tanh ^2(c+d x)+a\right )^3}{\left (1-\tanh ^2(c+d x)\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{4 \left (1-\tanh ^2(c+d x)\right )^2}-\frac {1}{4} \int \frac {3 \tanh ^2(c+d x) \left (b \tanh ^2(c+d x)+a\right )^2 \left (3 b \tanh ^2(c+d x)+a\right )}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{4 \left (1-\tanh ^2(c+d x)\right )^2}-\frac {3}{4} \int \frac {\tanh ^2(c+d x) \left (b \tanh ^2(c+d x)+a\right )^2 \left (3 b \tanh ^2(c+d x)+a\right )}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{4 \left (1-\tanh ^2(c+d x)\right )^2}-\frac {3}{4} \left (\frac {1}{2} \int -\frac {\tanh ^2(c+d x) \left (b \tanh ^2(c+d x)+a\right ) \left (b (5 a+21 b) \tanh ^2(c+d x)+a (a+9 b)\right )}{1-\tanh ^2(c+d x)}d\tanh (c+d x)+\frac {(a+3 b) \tanh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2}{2 \left (1-\tanh ^2(c+d x)\right )}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{4 \left (1-\tanh ^2(c+d x)\right )^2}-\frac {3}{4} \left (\frac {(a+3 b) \tanh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2}{2 \left (1-\tanh ^2(c+d x)\right )}-\frac {1}{2} \int \frac {\tanh ^2(c+d x) \left (b \tanh ^2(c+d x)+a\right ) \left (b (5 a+21 b) \tanh ^2(c+d x)+a (a+9 b)\right )}{1-\tanh ^2(c+d x)}d\tanh (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 437 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{4 \left (1-\tanh ^2(c+d x)\right )^2}-\frac {3}{4} \left (\frac {(a+3 b) \tanh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2}{2 \left (1-\tanh ^2(c+d x)\right )}-\frac {1}{2} \int \left (-b^2 (5 a+21 b) \tanh ^4(c+d x)-b \left (6 a^2+35 b a+21 b^2\right ) \tanh ^2(c+d x)-(a+b) \left (a^2+14 b a+21 b^2\right )+\frac {a^3+15 b a^2+35 b^2 a+21 b^3}{1-\tanh ^2(c+d x)}\right )d\tanh (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{4 \left (1-\tanh ^2(c+d x)\right )^2}-\frac {3}{4} \left (\frac {1}{2} \left (-(a+b) \left (a^2+14 a b+21 b^2\right ) \text {arctanh}(\tanh (c+d x))+\frac {1}{3} b \left (6 a^2+35 a b+21 b^2\right ) \tanh ^3(c+d x)+(a+b) \left (a^2+14 a b+21 b^2\right ) \tanh (c+d x)+\frac {1}{5} b^2 (5 a+21 b) \tanh ^5(c+d x)\right )+\frac {(a+3 b) \tanh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2}{2 \left (1-\tanh ^2(c+d x)\right )}\right )}{d}\) |
((Tanh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^3)/(4*(1 - Tanh[c + d*x]^2)^2) - (3*(((a + 3*b)*Tanh[c + d*x]^3*(a + b*Tanh[c + d*x]^2)^2)/(2*(1 - Tanh[c + d*x]^2)) + (-((a + b)*(a^2 + 14*a*b + 21*b^2)*ArcTanh[Tanh[c + d*x]]) + (a + b)*(a^2 + 14*a*b + 21*b^2)*Tanh[c + d*x] + (b*(6*a^2 + 35*a*b + 21*b^ 2)*Tanh[c + d*x]^3)/3 + (b^2*(5*a + 21*b)*Tanh[c + d*x]^5)/5)/2))/4)/d
3.1.17.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2* b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 0 ] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*( a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f , g, m}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*b*g*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*b*e*( p + 1) + (b*e - a*f)*(m + 1)) + d*(2*b*e*(p + 1) + (b*e - a*f)*(m + 2*q + 1 ))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && G tQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Time = 10.64 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.35
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{5}}{4 \cosh \left (d x +c \right )}-\frac {5 \sinh \left (d x +c \right )^{3}}{8 \cosh \left (d x +c \right )}+\frac {15 d x}{8}+\frac {15 c}{8}-\frac {15 \tanh \left (d x +c \right )}{8}\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{7}}{4 \cosh \left (d x +c \right )^{3}}-\frac {7 \sinh \left (d x +c \right )^{5}}{8 \cosh \left (d x +c \right )^{3}}+\frac {35 d x}{8}+\frac {35 c}{8}-\frac {35 \tanh \left (d x +c \right )}{8}-\frac {35 \tanh \left (d x +c \right )^{3}}{24}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{9}}{4 \cosh \left (d x +c \right )^{5}}-\frac {9 \sinh \left (d x +c \right )^{7}}{8 \cosh \left (d x +c \right )^{5}}+\frac {63 d x}{8}+\frac {63 c}{8}-\frac {63 \tanh \left (d x +c \right )}{8}-\frac {21 \tanh \left (d x +c \right )^{3}}{8}-\frac {63 \tanh \left (d x +c \right )^{5}}{40}\right )}{d}\) | \(246\) |
| default | \(\frac {a^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{5}}{4 \cosh \left (d x +c \right )}-\frac {5 \sinh \left (d x +c \right )^{3}}{8 \cosh \left (d x +c \right )}+\frac {15 d x}{8}+\frac {15 c}{8}-\frac {15 \tanh \left (d x +c \right )}{8}\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{7}}{4 \cosh \left (d x +c \right )^{3}}-\frac {7 \sinh \left (d x +c \right )^{5}}{8 \cosh \left (d x +c \right )^{3}}+\frac {35 d x}{8}+\frac {35 c}{8}-\frac {35 \tanh \left (d x +c \right )}{8}-\frac {35 \tanh \left (d x +c \right )^{3}}{24}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{9}}{4 \cosh \left (d x +c \right )^{5}}-\frac {9 \sinh \left (d x +c \right )^{7}}{8 \cosh \left (d x +c \right )^{5}}+\frac {63 d x}{8}+\frac {63 c}{8}-\frac {63 \tanh \left (d x +c \right )}{8}-\frac {21 \tanh \left (d x +c \right )^{3}}{8}-\frac {63 \tanh \left (d x +c \right )^{5}}{40}\right )}{d}\) | \(246\) |
| risch | \(\frac {3 a^{3} x}{8}+\frac {45 b \,a^{2} x}{8}+\frac {105 a \,b^{2} x}{8}+\frac {63 b^{3} x}{8}+\frac {{\mathrm e}^{4 d x +4 c} a^{3}}{64 d}+\frac {3 \,{\mathrm e}^{4 d x +4 c} a^{2} b}{64 d}+\frac {3 \,{\mathrm e}^{4 d x +4 c} a \,b^{2}}{64 d}+\frac {{\mathrm e}^{4 d x +4 c} b^{3}}{64 d}-\frac {{\mathrm e}^{2 d x +2 c} a^{3}}{8 d}-\frac {3 \,{\mathrm e}^{2 d x +2 c} a^{2} b}{4 d}-\frac {9 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{8 d}-\frac {{\mathrm e}^{2 d x +2 c} b^{3}}{2 d}+\frac {{\mathrm e}^{-2 d x -2 c} a^{3}}{8 d}+\frac {3 \,{\mathrm e}^{-2 d x -2 c} a^{2} b}{4 d}+\frac {9 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{8 d}+\frac {{\mathrm e}^{-2 d x -2 c} b^{3}}{2 d}-\frac {{\mathrm e}^{-4 d x -4 c} a^{3}}{64 d}-\frac {3 \,{\mathrm e}^{-4 d x -4 c} a^{2} b}{64 d}-\frac {3 \,{\mathrm e}^{-4 d x -4 c} a \,b^{2}}{64 d}-\frac {{\mathrm e}^{-4 d x -4 c} b^{3}}{64 d}+\frac {2 b \left (15 a^{2} {\mathrm e}^{8 d x +8 c}+60 a b \,{\mathrm e}^{8 d x +8 c}+50 b^{2} {\mathrm e}^{8 d x +8 c}+60 a^{2} {\mathrm e}^{6 d x +6 c}+210 a b \,{\mathrm e}^{6 d x +6 c}+150 b^{2} {\mathrm e}^{6 d x +6 c}+90 a^{2} {\mathrm e}^{4 d x +4 c}+290 a b \,{\mathrm e}^{4 d x +4 c}+210 \,{\mathrm e}^{4 d x +4 c} b^{2}+60 a^{2} {\mathrm e}^{2 d x +2 c}+190 a b \,{\mathrm e}^{2 d x +2 c}+130 \,{\mathrm e}^{2 d x +2 c} b^{2}+15 a^{2}+50 a b +36 b^{2}\right )}{5 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{5}}\) | \(506\) |
1/d*(a^3*((1/4*sinh(d*x+c)^3-3/8*sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c)+3 *a^2*b*(1/4*sinh(d*x+c)^5/cosh(d*x+c)-5/8*sinh(d*x+c)^3/cosh(d*x+c)+15/8*d *x+15/8*c-15/8*tanh(d*x+c))+3*a*b^2*(1/4*sinh(d*x+c)^7/cosh(d*x+c)^3-7/8*s inh(d*x+c)^5/cosh(d*x+c)^3+35/8*d*x+35/8*c-35/8*tanh(d*x+c)-35/24*tanh(d*x +c)^3)+b^3*(1/4*sinh(d*x+c)^9/cosh(d*x+c)^5-9/8*sinh(d*x+c)^7/cosh(d*x+c)^ 5+63/8*d*x+63/8*c-63/8*tanh(d*x+c)-21/8*tanh(d*x+c)^3-63/40*tanh(d*x+c)^5) )
Leaf count of result is larger than twice the leaf count of optimal. 879 vs. \(2 (170) = 340\).
Time = 0.27 (sec) , antiderivative size = 879, normalized size of antiderivative = 4.83 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]
1/320*(5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^9 - 15*(a^3 + 11*a^ 2*b + 19*a*b^2 + 9*b^3 - 12*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^ 2)*sinh(d*x + c)^7 + 8*(120*a^2*b + 400*a*b^2 + 288*b^3 + 15*(a^3 + 15*a^2 *b + 35*a*b^2 + 21*b^3)*d*x)*cosh(d*x + c)^5 + 40*(120*a^2*b + 400*a*b^2 + 288*b^3 + 15*(a^3 + 15*a^2*b + 35*a*b^2 + 21*b^3)*d*x)*cosh(d*x + c)*sinh (d*x + c)^4 + (630*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 - 150*a ^3 - 2010*a^2*b - 4850*a*b^2 - 3054*b^3 - 315*(a^3 + 11*a^2*b + 19*a*b^2 + 9*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 40*(120*a^2*b + 400*a*b^2 + 288 *b^3 + 15*(a^3 + 15*a^2*b + 35*a*b^2 + 21*b^3)*d*x)*cosh(d*x + c)^3 + 5*(8 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 - 105*(a^3 + 11*a^2*b + 19*a*b^2 + 9*b^3)*cosh(d*x + c)^4 - 62*a^3 - 978*a^2*b - 2282*a*b^2 - 1302 *b^3 - 4*(75*a^3 + 1005*a^2*b + 2425*a*b^2 + 1527*b^3)*cosh(d*x + c)^2)*si nh(d*x + c)^3 + 40*(2*(120*a^2*b + 400*a*b^2 + 288*b^3 + 15*(a^3 + 15*a^2* b + 35*a*b^2 + 21*b^3)*d*x)*cosh(d*x + c)^3 + 3*(120*a^2*b + 400*a*b^2 + 2 88*b^3 + 15*(a^3 + 15*a^2*b + 35*a*b^2 + 21*b^3)*d*x)*cosh(d*x + c))*sinh( d*x + c)^2 + 80*(120*a^2*b + 400*a*b^2 + 288*b^3 + 15*(a^3 + 15*a^2*b + 35 *a*b^2 + 21*b^3)*d*x)*cosh(d*x + c) + 5*(9*(a^3 + 3*a^2*b + 3*a*b^2 + b^3) *cosh(d*x + c)^8 - 21*(a^3 + 11*a^2*b + 19*a*b^2 + 9*b^3)*cosh(d*x + c)^6 - 2*(75*a^3 + 1005*a^2*b + 2425*a*b^2 + 1527*b^3)*cosh(d*x + c)^4 - 36*a^3 - 612*a^2*b - 1372*a*b^2 - 924*b^3 - 6*(31*a^3 + 489*a^2*b + 1141*a*b^...
\[ \int \sinh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \sinh ^{4}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (170) = 340\).
Time = 0.20 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.64 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {1}{64} \, a^{3} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {1}{320} \, b^{3} {\left (\frac {2520 \, {\left (d x + c\right )}}{d} + \frac {5 \, {\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )}\right )}}{d} - \frac {135 \, e^{\left (-2 \, d x - 2 \, c\right )} + 5358 \, e^{\left (-4 \, d x - 4 \, c\right )} + 18190 \, e^{\left (-6 \, d x - 6 \, c\right )} + 28455 \, e^{\left (-8 \, d x - 8 \, c\right )} + 19995 \, e^{\left (-10 \, d x - 10 \, c\right )} + 6560 \, e^{\left (-12 \, d x - 12 \, c\right )} - 5}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} + 5 \, e^{\left (-6 \, d x - 6 \, c\right )} + 10 \, e^{\left (-8 \, d x - 8 \, c\right )} + 10 \, e^{\left (-10 \, d x - 10 \, c\right )} + 5 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )}\right )}}\right )} + \frac {1}{64} \, a b^{2} {\left (\frac {840 \, {\left (d x + c\right )}}{d} + \frac {3 \, {\left (24 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )}\right )}}{d} - \frac {63 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1487 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2517 \, e^{\left (-6 \, d x - 6 \, c\right )} + 1608 \, e^{\left (-8 \, d x - 8 \, c\right )} - 3}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )}\right )}}\right )} + \frac {3}{64} \, a^{2} b {\left (\frac {120 \, {\left (d x + c\right )}}{d} + \frac {16 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )}}{d} - \frac {15 \, e^{\left (-2 \, d x - 2 \, c\right )} + 144 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}\right )} \]
1/64*a^3*(24*x + e^(4*d*x + 4*c)/d - 8*e^(2*d*x + 2*c)/d + 8*e^(-2*d*x - 2 *c)/d - e^(-4*d*x - 4*c)/d) + 1/320*b^3*(2520*(d*x + c)/d + 5*(32*e^(-2*d* x - 2*c) - e^(-4*d*x - 4*c))/d - (135*e^(-2*d*x - 2*c) + 5358*e^(-4*d*x - 4*c) + 18190*e^(-6*d*x - 6*c) + 28455*e^(-8*d*x - 8*c) + 19995*e^(-10*d*x - 10*c) + 6560*e^(-12*d*x - 12*c) - 5)/(d*(e^(-4*d*x - 4*c) + 5*e^(-6*d*x - 6*c) + 10*e^(-8*d*x - 8*c) + 10*e^(-10*d*x - 10*c) + 5*e^(-12*d*x - 12*c ) + e^(-14*d*x - 14*c)))) + 1/64*a*b^2*(840*(d*x + c)/d + 3*(24*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c))/d - (63*e^(-2*d*x - 2*c) + 1487*e^(-4*d*x - 4*c ) + 2517*e^(-6*d*x - 6*c) + 1608*e^(-8*d*x - 8*c) - 3)/(d*(e^(-4*d*x - 4*c ) + 3*e^(-6*d*x - 6*c) + 3*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c)))) + 3/64 *a^2*b*(120*(d*x + c)/d + (16*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c))/d - (15 *e^(-2*d*x - 2*c) + 144*e^(-4*d*x - 4*c) - 1)/(d*(e^(-4*d*x - 4*c) + e^(-6 *d*x - 6*c))))
Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (170) = 340\).
Time = 0.55 (sec) , antiderivative size = 505, normalized size of antiderivative = 2.77 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {5 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 5 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 40 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 240 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 360 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 160 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 120 \, {\left (a^{3} + 15 \, a^{2} b + 35 \, a b^{2} + 21 \, b^{3}\right )} {\left (d x + c\right )} - 5 \, {\left (18 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 270 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 630 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 378 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 48 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 72 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 32 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + \frac {128 \, {\left (15 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 60 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 50 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 60 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 210 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 150 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 290 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 210 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 60 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 190 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 130 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 15 \, a^{2} b + 50 \, a b^{2} + 36 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{320 \, d} \]
1/320*(5*a^3*e^(4*d*x + 4*c) + 15*a^2*b*e^(4*d*x + 4*c) + 15*a*b^2*e^(4*d* x + 4*c) + 5*b^3*e^(4*d*x + 4*c) - 40*a^3*e^(2*d*x + 2*c) - 240*a^2*b*e^(2 *d*x + 2*c) - 360*a*b^2*e^(2*d*x + 2*c) - 160*b^3*e^(2*d*x + 2*c) + 120*(a ^3 + 15*a^2*b + 35*a*b^2 + 21*b^3)*(d*x + c) - 5*(18*a^3*e^(4*d*x + 4*c) + 270*a^2*b*e^(4*d*x + 4*c) + 630*a*b^2*e^(4*d*x + 4*c) + 378*b^3*e^(4*d*x + 4*c) - 8*a^3*e^(2*d*x + 2*c) - 48*a^2*b*e^(2*d*x + 2*c) - 72*a*b^2*e^(2* d*x + 2*c) - 32*b^3*e^(2*d*x + 2*c) + a^3 + 3*a^2*b + 3*a*b^2 + b^3)*e^(-4 *d*x - 4*c) + 128*(15*a^2*b*e^(8*d*x + 8*c) + 60*a*b^2*e^(8*d*x + 8*c) + 5 0*b^3*e^(8*d*x + 8*c) + 60*a^2*b*e^(6*d*x + 6*c) + 210*a*b^2*e^(6*d*x + 6* c) + 150*b^3*e^(6*d*x + 6*c) + 90*a^2*b*e^(4*d*x + 4*c) + 290*a*b^2*e^(4*d *x + 4*c) + 210*b^3*e^(4*d*x + 4*c) + 60*a^2*b*e^(2*d*x + 2*c) + 190*a*b^2 *e^(2*d*x + 2*c) + 130*b^3*e^(2*d*x + 2*c) + 15*a^2*b + 50*a*b^2 + 36*b^3) /(e^(2*d*x + 2*c) + 1)^5)/d
Time = 2.12 (sec) , antiderivative size = 730, normalized size of antiderivative = 4.01 \[ \int \sinh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {\frac {2\,\left (3\,a^2\,b+12\,a\,b^2+10\,b^3\right )}{5\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b+9\,a\,b^2+5\,b^3\right )}{5\,d}+\frac {12\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,a^2\,b+8\,a\,b^2+6\,b^3\right )}{5\,d}+\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (3\,a^2\,b+9\,a\,b^2+5\,b^3\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (3\,a^2\,b+12\,a\,b^2+10\,b^3\right )}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1}+\frac {\frac {2\,\left (3\,a^2\,b+9\,a\,b^2+5\,b^3\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b+12\,a\,b^2+10\,b^3\right )}{5\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+x\,\left (\frac {3\,a^3}{8}+\frac {45\,a^2\,b}{8}+\frac {105\,a\,b^2}{8}+\frac {63\,b^3}{8}\right )+\frac {\frac {2\,\left (3\,a^2\,b+9\,a\,b^2+5\,b^3\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b+8\,a\,b^2+6\,b^3\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,a^2\,b+9\,a\,b^2+5\,b^3\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (3\,a^2\,b+12\,a\,b^2+10\,b^3\right )}{5\,d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}+\frac {\frac {2\,\left (3\,a^2\,b+8\,a\,b^2+6\,b^3\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b+9\,a\,b^2+5\,b^3\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,a^2\,b+12\,a\,b^2+10\,b^3\right )}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}+\frac {2\,\left (3\,a^2\,b+12\,a\,b^2+10\,b^3\right )}{5\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}\,{\left (a+b\right )}^3}{64\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,{\left (a+b\right )}^3}{64\,d}+\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,{\left (a+b\right )}^2\,\left (a+4\,b\right )}{8\,d}-\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^2\,\left (a+4\,b\right )}{8\,d} \]
((2*(12*a*b^2 + 3*a^2*b + 10*b^3))/(5*d) + (8*exp(2*c + 2*d*x)*(9*a*b^2 + 3*a^2*b + 5*b^3))/(5*d) + (12*exp(4*c + 4*d*x)*(8*a*b^2 + 3*a^2*b + 6*b^3) )/(5*d) + (8*exp(6*c + 6*d*x)*(9*a*b^2 + 3*a^2*b + 5*b^3))/(5*d) + (2*exp( 8*c + 8*d*x)*(12*a*b^2 + 3*a^2*b + 10*b^3))/(5*d))/(5*exp(2*c + 2*d*x) + 1 0*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*(9*a*b^2 + 3*a^2*b + 5*b^3))/(5*d) + (2*exp(2*c + 2*d* x)*(12*a*b^2 + 3*a^2*b + 10*b^3))/(5*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4 *d*x) + 1) + x*((105*a*b^2)/8 + (45*a^2*b)/8 + (3*a^3)/8 + (63*b^3)/8) + ( (2*(9*a*b^2 + 3*a^2*b + 5*b^3))/(5*d) + (6*exp(2*c + 2*d*x)*(8*a*b^2 + 3*a ^2*b + 6*b^3))/(5*d) + (6*exp(4*c + 4*d*x)*(9*a*b^2 + 3*a^2*b + 5*b^3))/(5 *d) + (2*exp(6*c + 6*d*x)*(12*a*b^2 + 3*a^2*b + 10*b^3))/(5*d))/(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1 ) + ((2*(8*a*b^2 + 3*a^2*b + 6*b^3))/(5*d) + (4*exp(2*c + 2*d*x)*(9*a*b^2 + 3*a^2*b + 5*b^3))/(5*d) + (2*exp(4*c + 4*d*x)*(12*a*b^2 + 3*a^2*b + 10*b ^3))/(5*d))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) + (2*(12*a*b^2 + 3*a^2*b + 10*b^3))/(5*d*(exp(2*c + 2*d*x) + 1)) - (exp (- 4*c - 4*d*x)*(a + b)^3)/(64*d) + (exp(4*c + 4*d*x)*(a + b)^3)/(64*d) + (exp(- 2*c - 2*d*x)*(a + b)^2*(a + 4*b))/(8*d) - (exp(2*c + 2*d*x)*(a + b) ^2*(a + 4*b))/(8*d)