Integrand size = 23, antiderivative size = 261 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {3}{256} (4 a-3 b) \left (8 a^2-14 a b+7 b^2\right ) x-\frac {\left (576 a^3-1744 a^2 b+1678 a b^2-525 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{1280 d}+\frac {\left (48 a^3-272 a^2 b+314 a b^2-105 b^3\right ) \cosh ^3(c+d x) \sinh (c+d x)}{640 d}+\frac {3 (2 a-3 b) \cosh ^5(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{80 d}+\frac {\cosh ^7(c+d x) \sinh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 d}-\frac {b \cosh ^3(c+d x) \sinh ^3(c+d x) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{160 d} \]
3/256*(4*a-3*b)*(8*a^2-14*a*b+7*b^2)*x-1/1280*(576*a^3-1744*a^2*b+1678*a*b ^2-525*b^3)*cosh(d*x+c)*sinh(d*x+c)/d+1/640*(48*a^3-272*a^2*b+314*a*b^2-10 5*b^3)*cosh(d*x+c)^3*sinh(d*x+c)/d+3/80*(2*a-3*b)*cosh(d*x+c)^5*sinh(d*x+c )^3*(a-(a-b)*tanh(d*x+c)^2)^2/d+1/10*cosh(d*x+c)^7*sinh(d*x+c)^3*(a-(a-b)* tanh(d*x+c)^2)^3/d-1/160*b*cosh(d*x+c)^3*sinh(d*x+c)^3*(a*(14*a-9*b)-(22*a -21*b)*(a-b)*tanh(d*x+c)^2)/d
Time = 1.66 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.62 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {120 (4 a-3 b) \left (8 a^2-14 a b+7 b^2\right ) (c+d x)-20 \left (128 a^3-360 a^2 b+336 a b^2-105 b^3\right ) \sinh (2 (c+d x))+40 \left (8 a^3-36 a^2 b+42 a b^2-15 b^3\right ) \sinh (4 (c+d x))+10 b \left (16 a^2-32 a b+15 b^2\right ) \sinh (6 (c+d x))+5 (6 a-5 b) b^2 \sinh (8 (c+d x))+2 b^3 \sinh (10 (c+d x))}{10240 d} \]
(120*(4*a - 3*b)*(8*a^2 - 14*a*b + 7*b^2)*(c + d*x) - 20*(128*a^3 - 360*a^ 2*b + 336*a*b^2 - 105*b^3)*Sinh[2*(c + d*x)] + 40*(8*a^3 - 36*a^2*b + 42*a *b^2 - 15*b^3)*Sinh[4*(c + d*x)] + 10*b*(16*a^2 - 32*a*b + 15*b^2)*Sinh[6* (c + d*x)] + 5*(6*a - 5*b)*b^2*Sinh[8*(c + d*x)] + 2*b^3*Sinh[10*(c + d*x) ])/(10240*d)
Time = 0.52 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.18, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 3666, 369, 27, 439, 439, 360, 25, 298, 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.
| \(\displaystyle \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (i c+i d x)^4 \left (a-b \sin (i c+i d x)^2\right )^3dx\) |
| \(\Big \downarrow \) 3666 |
| \(\displaystyle \frac {\int \frac {\tanh ^4(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{\left (1-\tanh ^2(c+d x)\right )^6}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 \left (1-\tanh ^2(c+d x)\right )^5}-\frac {1}{10} \int \frac {3 \tanh ^2(c+d x) \left (a-3 (a-b) \tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{\left (1-\tanh ^2(c+d x)\right )^5}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 \left (1-\tanh ^2(c+d x)\right )^5}-\frac {3}{10} \int \frac {\tanh ^2(c+d x) \left (a-3 (a-b) \tanh ^2(c+d x)\right ) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{\left (1-\tanh ^2(c+d x)\right )^5}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 \left (1-\tanh ^2(c+d x)\right )^5}-\frac {3}{10} \left (\frac {1}{8} \int \frac {\tanh ^2(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right ) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right )^4}d\tanh (c+d x)-\frac {(2 a-3 b) \tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 \left (1-\tanh ^2(c+d x)\right )^4}\right )}{d}\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 \left (1-\tanh ^2(c+d x)\right )^5}-\frac {3}{10} \left (\frac {1}{8} \left (\frac {1}{6} \int \frac {\tanh ^2(c+d x) \left (3 a (14 a-9 b) (2 a-b)-(22 a-21 b) (6 a-5 b) (a-b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right )^3}d\tanh (c+d x)+\frac {b \tanh ^3(c+d x) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{6 \left (1-\tanh ^2(c+d x)\right )^3}\right )-\frac {(2 a-3 b) \tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 \left (1-\tanh ^2(c+d x)\right )^4}\right )}{d}\) |
| \(\Big \downarrow \) 360 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 \left (1-\tanh ^2(c+d x)\right )^5}-\frac {3}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (-\frac {1}{4} \int -\frac {48 a^3-272 b a^2+314 b^2 a-105 b^3+4 (22 a-21 b) (6 a-5 b) (a-b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)-\frac {\left (48 a^3-272 a^2 b+314 a b^2-105 b^3\right ) \tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b \tanh ^3(c+d x) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{6 \left (1-\tanh ^2(c+d x)\right )^3}\right )-\frac {(2 a-3 b) \tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 \left (1-\tanh ^2(c+d x)\right )^4}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 \left (1-\tanh ^2(c+d x)\right )^5}-\frac {3}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \int \frac {48 a^3-272 b a^2+314 b^2 a-105 b^3+4 (22 a-21 b) (6 a-5 b) (a-b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)-\frac {\left (48 a^3-272 a^2 b+314 a b^2-105 b^3\right ) \tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b \tanh ^3(c+d x) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{6 \left (1-\tanh ^2(c+d x)\right )^3}\right )-\frac {(2 a-3 b) \tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 \left (1-\tanh ^2(c+d x)\right )^4}\right )}{d}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 \left (1-\tanh ^2(c+d x)\right )^5}-\frac {3}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\left (576 a^3-1744 a^2 b+1678 a b^2-525 b^3\right ) \tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}-\frac {15}{2} (4 a-3 b) \left (8 a^2-14 a b+7 b^2\right ) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)\right )-\frac {\left (48 a^3-272 a^2 b+314 a b^2-105 b^3\right ) \tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b \tanh ^3(c+d x) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{6 \left (1-\tanh ^2(c+d x)\right )^3}\right )-\frac {(2 a-3 b) \tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 \left (1-\tanh ^2(c+d x)\right )^4}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^3}{10 \left (1-\tanh ^2(c+d x)\right )^5}-\frac {3}{10} \left (\frac {1}{8} \left (\frac {1}{6} \left (\frac {1}{4} \left (\frac {\left (576 a^3-1744 a^2 b+1678 a b^2-525 b^3\right ) \tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}-\frac {15}{2} (4 a-3 b) \left (8 a^2-14 a b+7 b^2\right ) \text {arctanh}(\tanh (c+d x))\right )-\frac {\left (48 a^3-272 a^2 b+314 a b^2-105 b^3\right ) \tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b \tanh ^3(c+d x) \left (a (14 a-9 b)-(22 a-21 b) (a-b) \tanh ^2(c+d x)\right )}{6 \left (1-\tanh ^2(c+d x)\right )^3}\right )-\frac {(2 a-3 b) \tanh ^3(c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{8 \left (1-\tanh ^2(c+d x)\right )^4}\right )}{d}\) |
((Tanh[c + d*x]^3*(a - (a - b)*Tanh[c + d*x]^2)^3)/(10*(1 - Tanh[c + d*x]^ 2)^5) - (3*(-1/8*((2*a - 3*b)*Tanh[c + d*x]^3*(a - (a - b)*Tanh[c + d*x]^2 )^2)/(1 - Tanh[c + d*x]^2)^4 + ((b*Tanh[c + d*x]^3*(a*(14*a - 9*b) - (22*a - 21*b)*(a - b)*Tanh[c + d*x]^2))/(6*(1 - Tanh[c + d*x]^2)^3) + (-1/4*((4 8*a^3 - 272*a^2*b + 314*a*b^2 - 105*b^3)*Tanh[c + d*x])/(1 - Tanh[c + d*x] ^2)^2 + ((-15*(4*a - 3*b)*(8*a^2 - 14*a*b + 7*b^2)*ArcTanh[Tanh[c + d*x]]) /2 + ((576*a^3 - 1744*a^2*b + 1678*a*b^2 - 525*b^3)*Tanh[c + d*x])/(2*(1 - Tanh[c + d*x]^2)))/4)/6)/8))/10)/d
3.1.19.3.1 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[((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[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : > Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1)) Int[(a + b*x^2)^(p + 1)*Expan dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & & (IntegerQ[p] || EqQ[m + 2*p + 1, 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[((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_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1)) , x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] & & IntegerQ[p]
Time = 2.48 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.59
| method | result | size |
| parallelrisch | \(\frac {\left (-2560 a^{3}+7200 a^{2} b -6720 a \,b^{2}+2100 b^{3}\right ) \sinh \left (2 d x +2 c \right )+\left (320 a^{3}-1440 a^{2} b +1680 a \,b^{2}-600 b^{3}\right ) \sinh \left (4 d x +4 c \right )+160 \left (a -\frac {5 b}{4}\right ) \left (a -\frac {3 b}{4}\right ) b \sinh \left (6 d x +6 c \right )+\left (30 a \,b^{2}-25 b^{3}\right ) \sinh \left (8 d x +8 c \right )+2 b^{3} \sinh \left (10 d x +10 c \right )+3840 d \left (a^{2}-\frac {7}{4} a b +\frac {7}{8} b^{2}\right ) x \left (a -\frac {3 b}{4}\right )}{10240 d}\) | \(153\) |
| derivativedivides | \(\frac {a^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{2} b \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+3 a \,b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{7}}{8}-\frac {7 \sinh \left (d x +c \right )^{5}}{48}+\frac {35 \sinh \left (d x +c \right )^{3}}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )+b^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{9}}{10}-\frac {9 \sinh \left (d x +c \right )^{7}}{80}+\frac {21 \sinh \left (d x +c \right )^{5}}{160}-\frac {21 \sinh \left (d x +c \right )^{3}}{128}+\frac {63 \sinh \left (d x +c \right )}{256}\right ) \cosh \left (d x +c \right )-\frac {63 d x}{256}-\frac {63 c}{256}\right )}{d}\) | \(222\) |
| default | \(\frac {a^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{2} b \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+3 a \,b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{7}}{8}-\frac {7 \sinh \left (d x +c \right )^{5}}{48}+\frac {35 \sinh \left (d x +c \right )^{3}}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )+b^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{9}}{10}-\frac {9 \sinh \left (d x +c \right )^{7}}{80}+\frac {21 \sinh \left (d x +c \right )^{5}}{160}-\frac {21 \sinh \left (d x +c \right )^{3}}{128}+\frac {63 \sinh \left (d x +c \right )}{256}\right ) \cosh \left (d x +c \right )-\frac {63 d x}{256}-\frac {63 c}{256}\right )}{d}\) | \(222\) |
| parts | \(\frac {a^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {b^{3} \left (\left (\frac {\sinh \left (d x +c \right )^{9}}{10}-\frac {9 \sinh \left (d x +c \right )^{7}}{80}+\frac {21 \sinh \left (d x +c \right )^{5}}{160}-\frac {21 \sinh \left (d x +c \right )^{3}}{128}+\frac {63 \sinh \left (d x +c \right )}{256}\right ) \cosh \left (d x +c \right )-\frac {63 d x}{256}-\frac {63 c}{256}\right )}{d}+\frac {3 a \,b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{7}}{8}-\frac {7 \sinh \left (d x +c \right )^{5}}{48}+\frac {35 \sinh \left (d x +c \right )^{3}}{192}-\frac {35 \sinh \left (d x +c \right )}{128}\right ) \cosh \left (d x +c \right )+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {3 a^{2} b \left (\left (\frac {\sinh \left (d x +c \right )^{5}}{6}-\frac {5 \sinh \left (d x +c \right )^{3}}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d}\) | \(230\) |
| risch | \(\frac {105 a \,b^{2} x}{128}-\frac {15 a^{2} b x}{16}-\frac {63 b^{3} x}{256}+\frac {3 a^{3} x}{8}-\frac {b^{3} {\mathrm e}^{-10 d x -10 c}}{10240 d}+\frac {b^{3} {\mathrm e}^{10 d x +10 c}}{10240 d}-\frac {{\mathrm e}^{-4 d x -4 c} a^{3}}{64 d}+\frac {15 \,{\mathrm e}^{-4 d x -4 c} b^{3}}{512 d}-\frac {15 b^{3} {\mathrm e}^{-6 d x -6 c}}{2048 d}+\frac {5 b^{3} {\mathrm e}^{-8 d x -8 c}}{4096 d}-\frac {5 b^{3} {\mathrm e}^{8 d x +8 c}}{4096 d}+\frac {15 b^{3} {\mathrm e}^{6 d x +6 c}}{2048 d}+\frac {{\mathrm e}^{4 d x +4 c} a^{3}}{64 d}-\frac {15 \,{\mathrm e}^{4 d x +4 c} b^{3}}{512 d}-\frac {{\mathrm e}^{2 d x +2 c} a^{3}}{8 d}+\frac {105 \,{\mathrm e}^{2 d x +2 c} b^{3}}{1024 d}+\frac {{\mathrm e}^{-2 d x -2 c} a^{3}}{8 d}-\frac {105 \,{\mathrm e}^{-2 d x -2 c} b^{3}}{1024 d}-\frac {3 b^{2} {\mathrm e}^{-8 d x -8 c} a}{2048 d}-\frac {9 \,{\mathrm e}^{4 d x +4 c} a^{2} b}{128 d}+\frac {21 \,{\mathrm e}^{4 d x +4 c} a \,b^{2}}{256 d}+\frac {45 \,{\mathrm e}^{2 d x +2 c} a^{2} b}{128 d}-\frac {21 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{64 d}-\frac {45 \,{\mathrm e}^{-2 d x -2 c} a^{2} b}{128 d}+\frac {21 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{64 d}+\frac {9 \,{\mathrm e}^{-4 d x -4 c} a^{2} b}{128 d}-\frac {21 \,{\mathrm e}^{-4 d x -4 c} a \,b^{2}}{256 d}-\frac {b \,{\mathrm e}^{-6 d x -6 c} a^{2}}{128 d}+\frac {b^{2} {\mathrm e}^{-6 d x -6 c} a}{64 d}+\frac {3 b^{2} {\mathrm e}^{8 d x +8 c} a}{2048 d}+\frac {b \,{\mathrm e}^{6 d x +6 c} a^{2}}{128 d}-\frac {b^{2} {\mathrm e}^{6 d x +6 c} a}{64 d}\) | \(518\) |
1/10240*((-2560*a^3+7200*a^2*b-6720*a*b^2+2100*b^3)*sinh(2*d*x+2*c)+(320*a ^3-1440*a^2*b+1680*a*b^2-600*b^3)*sinh(4*d*x+4*c)+160*(a-5/4*b)*(a-3/4*b)* b*sinh(6*d*x+6*c)+(30*a*b^2-25*b^3)*sinh(8*d*x+8*c)+2*b^3*sinh(10*d*x+10*c )+3840*d*(a^2-7/4*a*b+7/8*b^2)*x*(a-3/4*b))/d
Time = 0.30 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.56 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {5 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{9} + 10 \, {\left (6 \, b^{3} \cosh \left (d x + c\right )^{3} + {\left (6 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} + {\left (126 \, b^{3} \cosh \left (d x + c\right )^{5} + 70 \, {\left (6 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 15 \, {\left (16 \, a^{2} b - 32 \, a b^{2} + 15 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 10 \, {\left (6 \, b^{3} \cosh \left (d x + c\right )^{7} + 7 \, {\left (6 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (16 \, a^{2} b - 32 \, a b^{2} + 15 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \, {\left (8 \, a^{3} - 36 \, a^{2} b + 42 \, a b^{2} - 15 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 30 \, {\left (32 \, a^{3} - 80 \, a^{2} b + 70 \, a b^{2} - 21 \, b^{3}\right )} d x + 5 \, {\left (b^{3} \cosh \left (d x + c\right )^{9} + 2 \, {\left (6 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 3 \, {\left (16 \, a^{2} b - 32 \, a b^{2} + 15 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 8 \, {\left (8 \, a^{3} - 36 \, a^{2} b + 42 \, a b^{2} - 15 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 2 \, {\left (128 \, a^{3} - 360 \, a^{2} b + 336 \, a b^{2} - 105 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2560 \, d} \]
1/2560*(5*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + 10*(6*b^3*cosh(d*x + c)^3 + (6*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + (126*b^3*cosh(d*x + c)^ 5 + 70*(6*a*b^2 - 5*b^3)*cosh(d*x + c)^3 + 15*(16*a^2*b - 32*a*b^2 + 15*b^ 3)*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(6*b^3*cosh(d*x + c)^7 + 7*(6*a*b^2 - 5*b^3)*cosh(d*x + c)^5 + 5*(16*a^2*b - 32*a*b^2 + 15*b^3)*cosh(d*x + c) ^3 + 4*(8*a^3 - 36*a^2*b + 42*a*b^2 - 15*b^3)*cosh(d*x + c))*sinh(d*x + c) ^3 + 30*(32*a^3 - 80*a^2*b + 70*a*b^2 - 21*b^3)*d*x + 5*(b^3*cosh(d*x + c) ^9 + 2*(6*a*b^2 - 5*b^3)*cosh(d*x + c)^7 + 3*(16*a^2*b - 32*a*b^2 + 15*b^3 )*cosh(d*x + c)^5 + 8*(8*a^3 - 36*a^2*b + 42*a*b^2 - 15*b^3)*cosh(d*x + c) ^3 - 2*(128*a^3 - 360*a^2*b + 336*a*b^2 - 105*b^3)*cosh(d*x + c))*sinh(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (248) = 496\).
Time = 1.26 (sec) , antiderivative size = 777, normalized size of antiderivative = 2.98 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\begin {cases} \frac {3 a^{3} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{3} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac {5 a^{3} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} - \frac {3 a^{3} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {15 a^{2} b x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {45 a^{2} b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {45 a^{2} b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {15 a^{2} b x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {33 a^{2} b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 a^{2} b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{2 d} + \frac {15 a^{2} b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} + \frac {105 a b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac {105 a b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac {315 a b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac {105 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac {105 a b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac {279 a b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} - \frac {511 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac {385 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{128 d} - \frac {105 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} + \frac {63 b^{3} x \sinh ^{10}{\left (c + d x \right )}}{256} - \frac {315 b^{3} x \sinh ^{8}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{256} + \frac {315 b^{3} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{128} - \frac {315 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{128} + \frac {315 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{8}{\left (c + d x \right )}}{256} - \frac {63 b^{3} x \cosh ^{10}{\left (c + d x \right )}}{256} + \frac {193 b^{3} \sinh ^{9}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{256 d} - \frac {237 b^{3} \sinh ^{7}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac {21 b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{10 d} - \frac {147 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} + \frac {63 b^{3} \sinh {\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{256 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{3} \sinh ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((3*a**3*x*sinh(c + d*x)**4/8 - 3*a**3*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 3*a**3*x*cosh(c + d*x)**4/8 + 5*a**3*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) - 3*a**3*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) + 15*a**2*b*x* sinh(c + d*x)**6/16 - 45*a**2*b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 4 5*a**2*b*x*sinh(c + d*x)**2*cosh(c + d*x)**4/16 - 15*a**2*b*x*cosh(c + d*x )**6/16 + 33*a**2*b*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) - 5*a**2*b*sinh( c + d*x)**3*cosh(c + d*x)**3/(2*d) + 15*a**2*b*sinh(c + d*x)*cosh(c + d*x) **5/(16*d) + 105*a*b**2*x*sinh(c + d*x)**8/128 - 105*a*b**2*x*sinh(c + d*x )**6*cosh(c + d*x)**2/32 + 315*a*b**2*x*sinh(c + d*x)**4*cosh(c + d*x)**4/ 64 - 105*a*b**2*x*sinh(c + d*x)**2*cosh(c + d*x)**6/32 + 105*a*b**2*x*cosh (c + d*x)**8/128 + 279*a*b**2*sinh(c + d*x)**7*cosh(c + d*x)/(128*d) - 511 *a*b**2*sinh(c + d*x)**5*cosh(c + d*x)**3/(128*d) + 385*a*b**2*sinh(c + d* x)**3*cosh(c + d*x)**5/(128*d) - 105*a*b**2*sinh(c + d*x)*cosh(c + d*x)**7 /(128*d) + 63*b**3*x*sinh(c + d*x)**10/256 - 315*b**3*x*sinh(c + d*x)**8*c osh(c + d*x)**2/256 + 315*b**3*x*sinh(c + d*x)**6*cosh(c + d*x)**4/128 - 3 15*b**3*x*sinh(c + d*x)**4*cosh(c + d*x)**6/128 + 315*b**3*x*sinh(c + d*x) **2*cosh(c + d*x)**8/256 - 63*b**3*x*cosh(c + d*x)**10/256 + 193*b**3*sinh (c + d*x)**9*cosh(c + d*x)/(256*d) - 237*b**3*sinh(c + d*x)**7*cosh(c + d* x)**3/(128*d) + 21*b**3*sinh(c + d*x)**5*cosh(c + d*x)**5/(10*d) - 147*b** 3*sinh(c + d*x)**3*cosh(c + d*x)**7/(128*d) + 63*b**3*sinh(c + d*x)*cos...
Time = 0.20 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.55 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {1}{64} \, a^{3} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{20480} \, b^{3} {\left (\frac {{\left (25 \, e^{\left (-2 \, d x - 2 \, c\right )} - 150 \, e^{\left (-4 \, d x - 4 \, c\right )} + 600 \, e^{\left (-6 \, d x - 6 \, c\right )} - 2100 \, e^{\left (-8 \, d x - 8 \, c\right )} - 2\right )} e^{\left (10 \, d x + 10 \, c\right )}}{d} + \frac {5040 \, {\left (d x + c\right )}}{d} + \frac {2100 \, e^{\left (-2 \, d x - 2 \, c\right )} - 600 \, e^{\left (-4 \, d x - 4 \, c\right )} + 150 \, e^{\left (-6 \, d x - 6 \, c\right )} - 25 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d}\right )} - \frac {1}{2048} \, a b^{2} {\left (\frac {{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac {1680 \, {\left (d x + c\right )}}{d} - \frac {672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} - \frac {1}{128} \, a^{2} b {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \]
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) - 1/20480*b^3*((25*e^(-2*d*x - 2*c) - 150*e^(- 4*d*x - 4*c) + 600*e^(-6*d*x - 6*c) - 2100*e^(-8*d*x - 8*c) - 2)*e^(10*d*x + 10*c)/d + 5040*(d*x + c)/d + (2100*e^(-2*d*x - 2*c) - 600*e^(-4*d*x - 4 *c) + 150*e^(-6*d*x - 6*c) - 25*e^(-8*d*x - 8*c) + 2*e^(-10*d*x - 10*c))/d ) - 1/2048*a*b^2*((32*e^(-2*d*x - 2*c) - 168*e^(-4*d*x - 4*c) + 672*e^(-6* d*x - 6*c) - 3)*e^(8*d*x + 8*c)/d - 1680*(d*x + c)/d - (672*e^(-2*d*x - 2* c) - 168*e^(-4*d*x - 4*c) + 32*e^(-6*d*x - 6*c) - 3*e^(-8*d*x - 8*c))/d) - 1/128*a^2*b*((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)
Time = 0.31 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.25 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {b^{3} e^{\left (10 \, d x + 10 \, c\right )}}{10240 \, d} - \frac {b^{3} e^{\left (-10 \, d x - 10 \, c\right )}}{10240 \, d} + \frac {3}{256} \, {\left (32 \, a^{3} - 80 \, a^{2} b + 70 \, a b^{2} - 21 \, b^{3}\right )} x + \frac {{\left (6 \, a b^{2} - 5 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )}}{4096 \, d} + \frac {{\left (16 \, a^{2} b - 32 \, a b^{2} + 15 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{2048 \, d} + \frac {{\left (8 \, a^{3} - 36 \, a^{2} b + 42 \, a b^{2} - 15 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{512 \, d} - \frac {{\left (128 \, a^{3} - 360 \, a^{2} b + 336 \, a b^{2} - 105 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{1024 \, d} + \frac {{\left (128 \, a^{3} - 360 \, a^{2} b + 336 \, a b^{2} - 105 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{1024 \, d} - \frac {{\left (8 \, a^{3} - 36 \, a^{2} b + 42 \, a b^{2} - 15 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{512 \, d} - \frac {{\left (16 \, a^{2} b - 32 \, a b^{2} + 15 \, b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{2048 \, d} - \frac {{\left (6 \, a b^{2} - 5 \, b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{4096 \, d} \]
1/10240*b^3*e^(10*d*x + 10*c)/d - 1/10240*b^3*e^(-10*d*x - 10*c)/d + 3/256 *(32*a^3 - 80*a^2*b + 70*a*b^2 - 21*b^3)*x + 1/4096*(6*a*b^2 - 5*b^3)*e^(8 *d*x + 8*c)/d + 1/2048*(16*a^2*b - 32*a*b^2 + 15*b^3)*e^(6*d*x + 6*c)/d + 1/512*(8*a^3 - 36*a^2*b + 42*a*b^2 - 15*b^3)*e^(4*d*x + 4*c)/d - 1/1024*(1 28*a^3 - 360*a^2*b + 336*a*b^2 - 105*b^3)*e^(2*d*x + 2*c)/d + 1/1024*(128* a^3 - 360*a^2*b + 336*a*b^2 - 105*b^3)*e^(-2*d*x - 2*c)/d - 1/512*(8*a^3 - 36*a^2*b + 42*a*b^2 - 15*b^3)*e^(-4*d*x - 4*c)/d - 1/2048*(16*a^2*b - 32* a*b^2 + 15*b^3)*e^(-6*d*x - 6*c)/d - 1/4096*(6*a*b^2 - 5*b^3)*e^(-8*d*x - 8*c)/d
Time = 1.95 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.92 \[ \int \sinh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {40\,a^3\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-320\,a^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {525\,b^3\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{2}-75\,b^3\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+\frac {75\,b^3\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}-\frac {25\,b^3\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}+\frac {b^3\,\mathrm {sinh}\left (10\,c+10\,d\,x\right )}{4}-840\,a\,b^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+900\,a^2\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+210\,a\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-180\,a^2\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-40\,a\,b^2\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+20\,a^2\,b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+\frac {15\,a\,b^2\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{4}+480\,a^3\,d\,x-315\,b^3\,d\,x+1050\,a\,b^2\,d\,x-1200\,a^2\,b\,d\,x}{1280\,d} \]
(40*a^3*sinh(4*c + 4*d*x) - 320*a^3*sinh(2*c + 2*d*x) + (525*b^3*sinh(2*c + 2*d*x))/2 - 75*b^3*sinh(4*c + 4*d*x) + (75*b^3*sinh(6*c + 6*d*x))/4 - (2 5*b^3*sinh(8*c + 8*d*x))/8 + (b^3*sinh(10*c + 10*d*x))/4 - 840*a*b^2*sinh( 2*c + 2*d*x) + 900*a^2*b*sinh(2*c + 2*d*x) + 210*a*b^2*sinh(4*c + 4*d*x) - 180*a^2*b*sinh(4*c + 4*d*x) - 40*a*b^2*sinh(6*c + 6*d*x) + 20*a^2*b*sinh( 6*c + 6*d*x) + (15*a*b^2*sinh(8*c + 8*d*x))/4 + 480*a^3*d*x - 315*b^3*d*x + 1050*a*b^2*d*x - 1200*a^2*b*d*x)/(1280*d)