Integrand size = 23, antiderivative size = 122 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {1}{2} (a+b)^2 (a+7 b) x+\frac {(a+b)^3}{4 d (1-\tanh (c+d x))}+\frac {3 b (a+b)^2 \tanh (c+d x)}{d}+\frac {b^2 (3 a+2 b) \tanh ^3(c+d x)}{3 d}+\frac {b^3 \tanh ^5(c+d x)}{5 d}-\frac {(a+b)^3}{4 d (1+\tanh (c+d x))} \]
-1/2*(a+b)^2*(a+7*b)*x+1/4*(a+b)^3/d/(1-tanh(d*x+c))+3*b*(a+b)^2*tanh(d*x+ c)/d+1/3*b^2*(3*a+2*b)*tanh(d*x+c)^3/d+1/5*b^3*tanh(d*x+c)^5/d-1/4*(a+b)^3 /d/(tanh(d*x+c)+1)
Time = 3.99 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.78 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {-30 (a+b)^2 (a+7 b) (c+d x)+15 (a+b)^3 \sinh (2 (c+d x))+4 b \left (45 a^2+105 a b+58 b^2-b (15 a+16 b) \text {sech}^2(c+d x)+3 b^2 \text {sech}^4(c+d x)\right ) \tanh (c+d x)}{60 d} \]
(-30*(a + b)^2*(a + 7*b)*(c + d*x) + 15*(a + b)^3*Sinh[2*(c + d*x)] + 4*b* (45*a^2 + 105*a*b + 58*b^2 - b*(15*a + 16*b)*Sech[c + d*x]^2 + 3*b^2*Sech[ c + d*x]^4)*Tanh[c + d*x])/(60*d)
Time = 0.39 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.27, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 25, 4146, 369, 403, 25, 403, 25, 299, 219}
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.
| \(\displaystyle \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\sin (i c+i d x)^2 \left (a-b \tan (i c+i d x)^2\right )^3dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \sin (i c+i d x)^2 \left (a-b \tan (i c+i d x)^2\right )^3dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\tanh ^2(c+d x) \left (b \tanh ^2(c+d x)+a\right )^3}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {\frac {\tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{2 \left (1-\tanh ^2(c+d x)\right )}-\frac {1}{2} \int \frac {\left (b \tanh ^2(c+d x)+a\right )^2 \left (7 b \tanh ^2(c+d x)+a\right )}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{5} \int -\frac {\left (b \tanh ^2(c+d x)+a\right ) \left (b (33 a+35 b) \tanh ^2(c+d x)+a (5 a+7 b)\right )}{1-\tanh ^2(c+d x)}d\tanh (c+d x)+\frac {7}{5} b \tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^2\right )+\frac {\tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{2 \left (1-\tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {7}{5} b \tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^2-\frac {1}{5} \int \frac {\left (b \tanh ^2(c+d x)+a\right ) \left (b (33 a+35 b) \tanh ^2(c+d x)+a (5 a+7 b)\right )}{1-\tanh ^2(c+d x)}d\tanh (c+d x)\right )+\frac {\tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{2 \left (1-\tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \int -\frac {b \left (81 a^2+190 b a+105 b^2\right ) \tanh ^2(c+d x)+a \left (15 a^2+54 b a+35 b^2\right )}{1-\tanh ^2(c+d x)}d\tanh (c+d x)+\frac {1}{3} b (33 a+35 b) \tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )\right )+\frac {7}{5} b \tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^2\right )+\frac {\tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{2 \left (1-\tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} b (33 a+35 b) \tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )-\frac {1}{3} \int \frac {b \left (81 a^2+190 b a+105 b^2\right ) \tanh ^2(c+d x)+a \left (15 a^2+54 b a+35 b^2\right )}{1-\tanh ^2(c+d x)}d\tanh (c+d x)\right )+\frac {7}{5} b \tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^2\right )+\frac {\tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{2 \left (1-\tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \left (b \left (81 a^2+190 a b+105 b^2\right ) \tanh (c+d x)-15 (a+b)^2 (a+7 b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)\right )+\frac {1}{3} b (33 a+35 b) \tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )\right )+\frac {7}{5} b \tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^2\right )+\frac {\tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{2 \left (1-\tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{5} \left (\frac {1}{3} \left (b \left (81 a^2+190 a b+105 b^2\right ) \tanh (c+d x)-15 (a+b)^2 (a+7 b) \text {arctanh}(\tanh (c+d x))\right )+\frac {1}{3} b (33 a+35 b) \tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )\right )+\frac {7}{5} b \tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^2\right )+\frac {\tanh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3}{2 \left (1-\tanh ^2(c+d x)\right )}}{d}\) |
((Tanh[c + d*x]*(a + b*Tanh[c + d*x]^2)^3)/(2*(1 - Tanh[c + d*x]^2)) + ((7 *b*Tanh[c + d*x]*(a + b*Tanh[c + d*x]^2)^2)/5 + ((-15*(a + b)^2*(a + 7*b)* ArcTanh[Tanh[c + d*x]] + b*(81*a^2 + 190*a*b + 105*b^2)*Tanh[c + d*x])/3 + (b*(33*a + 35*b)*Tanh[c + d*x]*(a + b*Tanh[c + d*x]^2))/3)/5)/2)/d
3.1.19.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
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[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
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 = 2.98 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.48
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )}-\frac {3 d x}{2}-\frac {3 c}{2}+\frac {3 \tanh \left (d x +c \right )}{2}\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{5}}{2 \cosh \left (d x +c \right )^{3}}-\frac {5 d x}{2}-\frac {5 c}{2}+\frac {5 \tanh \left (d x +c \right )}{2}+\frac {5 \tanh \left (d x +c \right )^{3}}{6}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{7}}{2 \cosh \left (d x +c \right )^{5}}-\frac {7 d x}{2}-\frac {7 c}{2}+\frac {7 \tanh \left (d x +c \right )}{2}+\frac {7 \tanh \left (d x +c \right )^{3}}{6}+\frac {7 \tanh \left (d x +c \right )^{5}}{10}\right )}{d}\) | \(180\) |
| default | \(\frac {a^{3} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )}-\frac {3 d x}{2}-\frac {3 c}{2}+\frac {3 \tanh \left (d x +c \right )}{2}\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{5}}{2 \cosh \left (d x +c \right )^{3}}-\frac {5 d x}{2}-\frac {5 c}{2}+\frac {5 \tanh \left (d x +c \right )}{2}+\frac {5 \tanh \left (d x +c \right )^{3}}{6}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{7}}{2 \cosh \left (d x +c \right )^{5}}-\frac {7 d x}{2}-\frac {7 c}{2}+\frac {7 \tanh \left (d x +c \right )}{2}+\frac {7 \tanh \left (d x +c \right )^{3}}{6}+\frac {7 \tanh \left (d x +c \right )^{5}}{10}\right )}{d}\) | \(180\) |
| risch | \(-\frac {a^{3} x}{2}-\frac {9 b \,a^{2} x}{2}-\frac {15 a \,b^{2} x}{2}-\frac {7 b^{3} x}{2}+\frac {{\mathrm e}^{2 d x +2 c} a^{3}}{8 d}+\frac {3 \,{\mathrm e}^{2 d x +2 c} a^{2} b}{8 d}+\frac {3 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{8 d}+\frac {{\mathrm e}^{2 d x +2 c} b^{3}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{3}}{8 d}-\frac {3 \,{\mathrm e}^{-2 d x -2 c} a^{2} b}{8 d}-\frac {3 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} b^{3}}{8 d}-\frac {2 b \left (45 a^{2} {\mathrm e}^{8 d x +8 c}+135 a b \,{\mathrm e}^{8 d x +8 c}+90 b^{2} {\mathrm e}^{8 d x +8 c}+180 a^{2} {\mathrm e}^{6 d x +6 c}+450 a b \,{\mathrm e}^{6 d x +6 c}+240 b^{2} {\mathrm e}^{6 d x +6 c}+270 a^{2} {\mathrm e}^{4 d x +4 c}+600 a b \,{\mathrm e}^{4 d x +4 c}+340 \,{\mathrm e}^{4 d x +4 c} b^{2}+180 a^{2} {\mathrm e}^{2 d x +2 c}+390 a b \,{\mathrm e}^{2 d x +2 c}+200 \,{\mathrm e}^{2 d x +2 c} b^{2}+45 a^{2}+105 a b +58 b^{2}\right )}{15 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{5}}\) | \(366\) |
1/d*(a^3*(1/2*cosh(d*x+c)*sinh(d*x+c)-1/2*d*x-1/2*c)+3*a^2*b*(1/2*sinh(d*x +c)^3/cosh(d*x+c)-3/2*d*x-3/2*c+3/2*tanh(d*x+c))+3*a*b^2*(1/2*sinh(d*x+c)^ 5/cosh(d*x+c)^3-5/2*d*x-5/2*c+5/2*tanh(d*x+c)+5/6*tanh(d*x+c)^3)+b^3*(1/2* sinh(d*x+c)^7/cosh(d*x+c)^5-7/2*d*x-7/2*c+7/2*tanh(d*x+c)+7/6*tanh(d*x+c)^ 3+7/10*tanh(d*x+c)^5))
Leaf count of result is larger than twice the leaf count of optimal. 725 vs. \(2 (110) = 220\).
Time = 0.27 (sec) , antiderivative size = 725, normalized size of antiderivative = 5.94 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{7} - 4 \, {\left (90 \, a^{2} b + 210 \, a b^{2} + 116 \, b^{3} + 15 \, {\left (a^{3} + 9 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{5} - 20 \, {\left (90 \, a^{2} b + 210 \, a b^{2} + 116 \, b^{3} + 15 \, {\left (a^{3} + 9 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + {\left (75 \, a^{3} + 585 \, a^{2} b + 1065 \, a b^{2} + 539 \, b^{3} + 315 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} - 20 \, {\left (90 \, a^{2} b + 210 \, a b^{2} + 116 \, b^{3} + 15 \, {\left (a^{3} + 9 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{3} + 5 \, {\left (105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{4} + 27 \, a^{3} + 297 \, a^{2} b + 489 \, a b^{2} + 203 \, b^{3} + 2 \, {\left (75 \, a^{3} + 585 \, a^{2} b + 1065 \, a b^{2} + 539 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} - 20 \, {\left (2 \, {\left (90 \, a^{2} b + 210 \, a b^{2} + 116 \, b^{3} + 15 \, {\left (a^{3} + 9 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (90 \, a^{2} b + 210 \, a b^{2} + 116 \, b^{3} + 15 \, {\left (a^{3} + 9 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 40 \, {\left (90 \, a^{2} b + 210 \, a b^{2} + 116 \, b^{3} + 15 \, {\left (a^{3} + 9 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) + 5 \, {\left (21 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{6} + {\left (75 \, a^{3} + 585 \, a^{2} b + 1065 \, a b^{2} + 539 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 15 \, a^{3} + 189 \, a^{2} b + 285 \, a b^{2} + 175 \, b^{3} + 3 \, {\left (27 \, a^{3} + 297 \, a^{2} b + 489 \, a b^{2} + 203 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{120 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )\right )}} \]
1/120*(15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^7 - 4*(90*a^2*b + 210*a*b^2 + 116*b^3 + 15*(a^3 + 9*a^2*b + 15*a*b^2 + 7*b^3)*d*x)*cosh(d*x + c)^5 - 20*(90*a^2*b + 210*a*b^2 + 116*b^3 + 15*(a^3 + 9*a^2*b + 15*a*b^2 + 7*b^3)*d*x)*cosh(d*x + c)*sinh(d*x + c)^4 + (75*a^3 + 585*a^2*b + 1065* a*b^2 + 539*b^3 + 315*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sin h(d*x + c)^5 - 20*(90*a^2*b + 210*a*b^2 + 116*b^3 + 15*(a^3 + 9*a^2*b + 15 *a*b^2 + 7*b^3)*d*x)*cosh(d*x + c)^3 + 5*(105*(a^3 + 3*a^2*b + 3*a*b^2 + b ^3)*cosh(d*x + c)^4 + 27*a^3 + 297*a^2*b + 489*a*b^2 + 203*b^3 + 2*(75*a^3 + 585*a^2*b + 1065*a*b^2 + 539*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 - 20 *(2*(90*a^2*b + 210*a*b^2 + 116*b^3 + 15*(a^3 + 9*a^2*b + 15*a*b^2 + 7*b^3 )*d*x)*cosh(d*x + c)^3 + 3*(90*a^2*b + 210*a*b^2 + 116*b^3 + 15*(a^3 + 9*a ^2*b + 15*a*b^2 + 7*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^2 - 40*(90*a^2* b + 210*a*b^2 + 116*b^3 + 15*(a^3 + 9*a^2*b + 15*a*b^2 + 7*b^3)*d*x)*cosh( d*x + c) + 5*(21*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + (75*a^3 + 585*a^2*b + 1065*a*b^2 + 539*b^3)*cosh(d*x + c)^4 + 15*a^3 + 189*a^2*b + 285*a*b^2 + 175*b^3 + 3*(27*a^3 + 297*a^2*b + 489*a*b^2 + 203*b^3)*cosh( d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)*sinh(d*x + c)^4 + 5*d*cosh(d*x + c)^3 + 5*(2*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c) )*sinh(d*x + c)^2 + 10*d*cosh(d*x + c))
\[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \sinh ^{2}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (110) = 220\).
Time = 0.21 (sec) , antiderivative size = 377, normalized size of antiderivative = 3.09 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {1}{8} \, a^{3} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{120} \, b^{3} {\left (\frac {420 \, {\left (d x + c\right )}}{d} + \frac {15 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {1003 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3350 \, e^{\left (-4 \, d x - 4 \, c\right )} + 5590 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3915 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1455 \, e^{\left (-10 \, d x - 10 \, c\right )} + 15}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 5 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 10 \, e^{\left (-8 \, d x - 8 \, c\right )} + 5 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )}\right )}}\right )} - \frac {1}{8} \, a b^{2} {\left (\frac {60 \, {\left (d x + c\right )}}{d} + \frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {121 \, e^{\left (-2 \, d x - 2 \, c\right )} + 201 \, e^{\left (-4 \, d x - 4 \, c\right )} + 147 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )}\right )}}\right )} - \frac {3}{8} \, a^{2} b {\left (\frac {12 \, {\left (d x + c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )}\right )}}\right )} \]
-1/8*a^3*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/120*b^3*(420*( d*x + c)/d + 15*e^(-2*d*x - 2*c)/d - (1003*e^(-2*d*x - 2*c) + 3350*e^(-4*d *x - 4*c) + 5590*e^(-6*d*x - 6*c) + 3915*e^(-8*d*x - 8*c) + 1455*e^(-10*d* x - 10*c) + 15)/(d*(e^(-2*d*x - 2*c) + 5*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 10*e^(-8*d*x - 8*c) + 5*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c)))) - 1/8*a*b^2*(60*(d*x + c)/d + 3*e^(-2*d*x - 2*c)/d - (121*e^(-2*d*x - 2*c ) + 201*e^(-4*d*x - 4*c) + 147*e^(-6*d*x - 6*c) + 3)/(d*(e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + 3*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c)))) - 3/8*a^2* b*(12*(d*x + c)/d + e^(-2*d*x - 2*c)/d - (17*e^(-2*d*x - 2*c) + 1)/(d*(e^( -2*d*x - 2*c) + e^(-4*d*x - 4*c))))
Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (110) = 220\).
Time = 0.48 (sec) , antiderivative size = 393, normalized size of antiderivative = 3.22 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {15 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 45 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 15 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 60 \, {\left (a^{3} + 9 \, a^{2} b + 15 \, a b^{2} + 7 \, b^{3}\right )} {\left (d x + c\right )} + 15 \, {\left (2 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 18 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 30 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 14 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - \frac {16 \, {\left (45 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 135 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 90 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 180 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 450 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 240 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 270 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 600 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 340 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 180 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 390 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 200 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, a^{2} b + 105 \, a b^{2} + 58 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{120 \, d} \]
1/120*(15*a^3*e^(2*d*x + 2*c) + 45*a^2*b*e^(2*d*x + 2*c) + 45*a*b^2*e^(2*d *x + 2*c) + 15*b^3*e^(2*d*x + 2*c) - 60*(a^3 + 9*a^2*b + 15*a*b^2 + 7*b^3) *(d*x + c) + 15*(2*a^3*e^(2*d*x + 2*c) + 18*a^2*b*e^(2*d*x + 2*c) + 30*a*b ^2*e^(2*d*x + 2*c) + 14*b^3*e^(2*d*x + 2*c) - a^3 - 3*a^2*b - 3*a*b^2 - b^ 3)*e^(-2*d*x - 2*c) - 16*(45*a^2*b*e^(8*d*x + 8*c) + 135*a*b^2*e^(8*d*x + 8*c) + 90*b^3*e^(8*d*x + 8*c) + 180*a^2*b*e^(6*d*x + 6*c) + 450*a*b^2*e^(6 *d*x + 6*c) + 240*b^3*e^(6*d*x + 6*c) + 270*a^2*b*e^(4*d*x + 4*c) + 600*a* b^2*e^(4*d*x + 4*c) + 340*b^3*e^(4*d*x + 4*c) + 180*a^2*b*e^(2*d*x + 2*c) + 390*a*b^2*e^(2*d*x + 2*c) + 200*b^3*e^(2*d*x + 2*c) + 45*a^2*b + 105*a*b ^2 + 58*b^3)/(e^(2*d*x + 2*c) + 1)^5)/d
Time = 0.33 (sec) , antiderivative size = 668, normalized size of antiderivative = 5.48 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^3}{8\,d}-\frac {\frac {2\,\left (3\,a^2\,b+6\,a\,b^2+2\,b^3\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2\,b+3\,a\,b^2+2\,b^3\right )}{5\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {2\,\left (3\,a^2\,b+6\,a\,b^2+2\,b^3\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a^2\,b+3\,a\,b^2+2\,b^3\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,a^2\,b+6\,a\,b^2+2\,b^3\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (9\,a^2\,b+15\,a\,b^2+10\,b^3\right )}{5\,d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {\frac {2\,\left (9\,a^2\,b+15\,a\,b^2+10\,b^3\right )}{15\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^2\,b+3\,a\,b^2+2\,b^3\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b+6\,a\,b^2+2\,b^3\right )}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {6\,\left (a^2\,b+3\,a\,b^2+2\,b^3\right )}{5\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,{\left (a+b\right )}^3}{8\,d}-\frac {\frac {6\,\left (a^2\,b+3\,a\,b^2+2\,b^3\right )}{5\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b+6\,a\,b^2+2\,b^3\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a^2\,b+3\,a\,b^2+2\,b^3\right )}{5\,d}+\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (3\,a^2\,b+6\,a\,b^2+2\,b^3\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (9\,a^2\,b+15\,a\,b^2+10\,b^3\right )}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1}-\frac {x\,{\left (a+b\right )}^2\,\left (a+7\,b\right )}{2} \]
(exp(2*c + 2*d*x)*(a + b)^3)/(8*d) - ((2*(6*a*b^2 + 3*a^2*b + 2*b^3))/(5*d ) + (6*exp(2*c + 2*d*x)*(3*a*b^2 + a^2*b + 2*b^3))/(5*d))/(2*exp(2*c + 2*d *x) + exp(4*c + 4*d*x) + 1) - ((2*(6*a*b^2 + 3*a^2*b + 2*b^3))/(5*d) + (6* exp(6*c + 6*d*x)*(3*a*b^2 + a^2*b + 2*b^3))/(5*d) + (6*exp(4*c + 4*d*x)*(6 *a*b^2 + 3*a^2*b + 2*b^3))/(5*d) + (2*exp(2*c + 2*d*x)*(15*a*b^2 + 9*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*(15*a*b^2 + 9*a^2*b + 10*b^3))/(15*d) + (6*exp(4*c + 4*d*x)*(3*a*b^2 + a^2*b + 2*b^3))/(5*d) + (4*exp(2*c + 2*d *x)*(6*a*b^2 + 3*a^2*b + 2*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) - (6*(3*a*b^2 + a^2*b + 2*b^3))/(5*d*(exp(2 *c + 2*d*x) + 1)) - (exp(- 2*c - 2*d*x)*(a + b)^3)/(8*d) - ((6*(3*a*b^2 + a^2*b + 2*b^3))/(5*d) + (8*exp(2*c + 2*d*x)*(6*a*b^2 + 3*a^2*b + 2*b^3))/( 5*d) + (6*exp(8*c + 8*d*x)*(3*a*b^2 + a^2*b + 2*b^3))/(5*d) + (8*exp(6*c + 6*d*x)*(6*a*b^2 + 3*a^2*b + 2*b^3))/(5*d) + (4*exp(4*c + 4*d*x)*(15*a*b^2 + 9*a^2*b + 10*b^3))/(5*d))/(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 1 0*exp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1) - (x*(a + b)^2*(a + 7*b))/2