Integrand size = 23, antiderivative size = 144 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {1}{128} \left (48 a^2-16 a b+3 b^2\right ) x+\frac {\left (48 a^2-16 a b+3 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac {\left (48 a^2-16 a b+3 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{192 d}+\frac {(16 a-9 b) b \cosh ^5(c+d x) \sinh (c+d x)}{48 d}+\frac {b^2 \cosh ^7(c+d x) \sinh (c+d x)}{8 d} \] Output:
1/128*(48*a^2-16*a*b+3*b^2)*x+1/128*(48*a^2-16*a*b+3*b^2)*cosh(d*x+c)*sinh (d*x+c)/d+1/192*(48*a^2-16*a*b+3*b^2)*cosh(d*x+c)^3*sinh(d*x+c)/d+1/48*(16 *a-9*b)*b*cosh(d*x+c)^5*sinh(d*x+c)/d+1/8*b^2*cosh(d*x+c)^7*sinh(d*x+c)/d
Time = 1.60 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.68 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {24 \left (48 a^2-16 a b+3 b^2\right ) (c+d x)+96 a (8 a-b) \sinh (2 (c+d x))+24 \left (4 a^2+4 a b-b^2\right ) \sinh (4 (c+d x))+32 a b \sinh (6 (c+d x))+3 b^2 \sinh (8 (c+d x))}{3072 d} \] Input:
Integrate[Cosh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2,x]
Output:
(24*(48*a^2 - 16*a*b + 3*b^2)*(c + d*x) + 96*a*(8*a - b)*Sinh[2*(c + d*x)] + 24*(4*a^2 + 4*a*b - b^2)*Sinh[4*(c + d*x)] + 32*a*b*Sinh[6*(c + d*x)] + 3*b^2*Sinh[8*(c + d*x)])/(3072*d)
Time = 0.33 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 3670, 315, 25, 298, 215, 215, 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.
| \(\displaystyle \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (i c+i d x)^4 \left (a-b \sin (i c+i d x)^2\right )^2dx\) |
| \(\Big \downarrow \) 3670 |
| \(\displaystyle \frac {\int \frac {\left (a-(a-b) \tanh ^2(c+d x)\right )^2}{\left (1-\tanh ^2(c+d x)\right )^5}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\frac {b \tanh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 \left (1-\tanh ^2(c+d x)\right )^4}-\frac {1}{8} \int -\frac {a (8 a-b)-(8 a-3 b) (a-b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right )^4}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{8} \int \frac {a (8 a-b)-(8 a-3 b) (a-b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right )^4}d\tanh (c+d x)+\frac {b \tanh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 \left (1-\tanh ^2(c+d x)\right )^4}}{d}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (48 a^2-16 a b+3 b^2\right ) \int \frac {1}{\left (1-\tanh ^2(c+d x)\right )^3}d\tanh (c+d x)+\frac {b (10 a-3 b) \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}\right )+\frac {b \tanh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 \left (1-\tanh ^2(c+d x)\right )^4}}{d}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (48 a^2-16 a b+3 b^2\right ) \left (\frac {3}{4} \int \frac {1}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)+\frac {\tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b (10 a-3 b) \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}\right )+\frac {b \tanh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 \left (1-\tanh ^2(c+d x)\right )^4}}{d}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (48 a^2-16 a b+3 b^2\right ) \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)+\frac {\tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}\right )+\frac {\tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b (10 a-3 b) \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}\right )+\frac {b \tanh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 \left (1-\tanh ^2(c+d x)\right )^4}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{8} \left (\frac {1}{6} \left (48 a^2-16 a b+3 b^2\right ) \left (\frac {3}{4} \left (\frac {1}{2} \text {arctanh}(\tanh (c+d x))+\frac {\tanh (c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}\right )+\frac {\tanh (c+d x)}{4 \left (1-\tanh ^2(c+d x)\right )^2}\right )+\frac {b (10 a-3 b) \tanh (c+d x)}{6 \left (1-\tanh ^2(c+d x)\right )^3}\right )+\frac {b \tanh (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )}{8 \left (1-\tanh ^2(c+d x)\right )^4}}{d}\) |
Input:
Int[Cosh[c + d*x]^4*(a + b*Sinh[c + d*x]^2)^2,x]
Output:
((b*Tanh[c + d*x]*(a - (a - b)*Tanh[c + d*x]^2))/(8*(1 - Tanh[c + d*x]^2)^ 4) + (((10*a - 3*b)*b*Tanh[c + d*x])/(6*(1 - Tanh[c + d*x]^2)^3) + ((48*a^ 2 - 16*a*b + 3*b^2)*(Tanh[c + d*x]/(4*(1 - Tanh[c + d*x]^2)^2) + (3*(ArcTa nh[Tanh[c + d*x]]/2 + Tanh[c + d*x]/(2*(1 - Tanh[c + d*x]^2))))/4))/6)/8)/ d
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Su bst[Int[(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 = 0.14 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.19
\[\frac {a^{2} \left (\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a b \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{5}}{6}-\frac {\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )}{6}-\frac {d x}{16}-\frac {c}{16}\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{3} \cosh \left (d x +c \right )^{5}}{8}-\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{5}}{16}+\frac {\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )}{16}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\]
Input:
int(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x)
Output:
1/d*(a^2*((1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c))*sinh(d*x+c)+3/8*d*x+3/8*c)+2 *a*b*(1/6*sinh(d*x+c)*cosh(d*x+c)^5-1/6*(1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c) )*sinh(d*x+c)-1/16*d*x-1/16*c)+b^2*(1/8*sinh(d*x+c)^3*cosh(d*x+c)^5-1/16*s inh(d*x+c)*cosh(d*x+c)^5+1/16*(1/4*cosh(d*x+c)^3+3/8*cosh(d*x+c))*sinh(d*x +c)+3/128*d*x+3/128*c))
Time = 0.09 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.47 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \, {\left (7 \, b^{2} \cosh \left (d x + c\right )^{3} + 8 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + {\left (21 \, b^{2} \cosh \left (d x + c\right )^{5} + 80 \, a b \cosh \left (d x + c\right )^{3} + 12 \, {\left (4 \, a^{2} + 4 \, a b - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} d x + 3 \, {\left (b^{2} \cosh \left (d x + c\right )^{7} + 8 \, a b \cosh \left (d x + c\right )^{5} + 4 \, {\left (4 \, a^{2} + 4 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{3} + 8 \, {\left (8 \, a^{2} - a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{384 \, d} \] Input:
integrate(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")
Output:
1/384*(3*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + 3*(7*b^2*cosh(d*x + c)^3 + 8* a*b*cosh(d*x + c))*sinh(d*x + c)^5 + (21*b^2*cosh(d*x + c)^5 + 80*a*b*cosh (d*x + c)^3 + 12*(4*a^2 + 4*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3* (48*a^2 - 16*a*b + 3*b^2)*d*x + 3*(b^2*cosh(d*x + c)^7 + 8*a*b*cosh(d*x + c)^5 + 4*(4*a^2 + 4*a*b - b^2)*cosh(d*x + c)^3 + 8*(8*a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (134) = 268\).
Time = 0.70 (sec) , antiderivative size = 481, normalized size of antiderivative = 3.34 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\begin {cases} \frac {3 a^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {3 a^{2} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {5 a^{2} \sinh {\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {a b x \sinh ^{6}{\left (c + d x \right )}}{8} - \frac {3 a b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac {3 a b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{8} - \frac {a b x \cosh ^{6}{\left (c + d x \right )}}{8} - \frac {a b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{8 d} + \frac {a b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac {a b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{8 d} + \frac {3 b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac {3 b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac {9 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac {3 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac {3 b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} - \frac {3 b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{128 d} + \frac {11 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac {11 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{128 d} - \frac {3 b^{2} \sinh {\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{2} \cosh ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cosh(d*x+c)**4*(a+b*sinh(d*x+c)**2)**2,x)
Output:
Piecewise((3*a**2*x*sinh(c + d*x)**4/8 - 3*a**2*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 3*a**2*x*cosh(c + d*x)**4/8 - 3*a**2*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) + 5*a**2*sinh(c + d*x)*cosh(c + d*x)**3/(8*d) + a*b*x*sinh(c + d*x)**6/8 - 3*a*b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/8 + 3*a*b*x*sinh( c + d*x)**2*cosh(c + d*x)**4/8 - a*b*x*cosh(c + d*x)**6/8 - a*b*sinh(c + d *x)**5*cosh(c + d*x)/(8*d) + a*b*sinh(c + d*x)**3*cosh(c + d*x)**3/(3*d) + a*b*sinh(c + d*x)*cosh(c + d*x)**5/(8*d) + 3*b**2*x*sinh(c + d*x)**8/128 - 3*b**2*x*sinh(c + d*x)**6*cosh(c + d*x)**2/32 + 9*b**2*x*sinh(c + d*x)** 4*cosh(c + d*x)**4/64 - 3*b**2*x*sinh(c + d*x)**2*cosh(c + d*x)**6/32 + 3* b**2*x*cosh(c + d*x)**8/128 - 3*b**2*sinh(c + d*x)**7*cosh(c + d*x)/(128*d ) + 11*b**2*sinh(c + d*x)**5*cosh(c + d*x)**3/(128*d) + 11*b**2*sinh(c + d *x)**3*cosh(c + d*x)**5/(128*d) - 3*b**2*sinh(c + d*x)*cosh(c + d*x)**7/(1 28*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**2*cosh(c)**4, True))
Time = 0.04 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.56 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {1}{64} \, a^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{2048} \, b^{2} {\left (\frac {{\left (8 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac {48 \, {\left (d x + c\right )}}{d} - \frac {8 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} + \frac {1}{192} \, a b {\left (\frac {{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} - \frac {24 \, {\left (d x + c\right )}}{d} + \frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \] Input:
integrate(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")
Output:
1/64*a^2*(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) - 1/2048*b^2*((8*e^(-4*d*x - 4*c) - 1)*e^(8*d* x + 8*c)/d - 48*(d*x + c)/d - (8*e^(-4*d*x - 4*c) - e^(-8*d*x - 8*c))/d) + 1/192*a*b*((3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + 1)*e^(6*d*x + 6*c)/ d - 24*(d*x + c)/d + (3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) - e^(-6*d*x - 6*c))/d)
Time = 0.14 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.33 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {1}{128} \, {\left (48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} x + \frac {b^{2} e^{\left (8 \, d x + 8 \, c\right )}}{2048 \, d} + \frac {a b e^{\left (6 \, d x + 6 \, c\right )}}{192 \, d} - \frac {a b e^{\left (-6 \, d x - 6 \, c\right )}}{192 \, d} - \frac {b^{2} e^{\left (-8 \, d x - 8 \, c\right )}}{2048 \, d} + \frac {{\left (4 \, a^{2} + 4 \, a b - b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{256 \, d} + \frac {{\left (8 \, a^{2} - a b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{64 \, d} - \frac {{\left (8 \, a^{2} - a b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{64 \, d} - \frac {{\left (4 \, a^{2} + 4 \, a b - b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{256 \, d} \] Input:
integrate(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")
Output:
1/128*(48*a^2 - 16*a*b + 3*b^2)*x + 1/2048*b^2*e^(8*d*x + 8*c)/d + 1/192*a *b*e^(6*d*x + 6*c)/d - 1/192*a*b*e^(-6*d*x - 6*c)/d - 1/2048*b^2*e^(-8*d*x - 8*c)/d + 1/256*(4*a^2 + 4*a*b - b^2)*e^(4*d*x + 4*c)/d + 1/64*(8*a^2 - a*b)*e^(2*d*x + 2*c)/d - 1/64*(8*a^2 - a*b)*e^(-2*d*x - 2*c)/d - 1/256*(4* a^2 + 4*a*b - b^2)*e^(-4*d*x - 4*c)/d
Time = 0.38 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.84 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {96\,a^2\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+12\,a^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )-3\,b^2\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+\frac {3\,b^2\,\mathrm {sinh}\left (8\,c+8\,d\,x\right )}{8}-12\,a\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+12\,a\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )+4\,a\,b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )+144\,a^2\,d\,x+9\,b^2\,d\,x-48\,a\,b\,d\,x}{384\,d} \] Input:
int(cosh(c + d*x)^4*(a + b*sinh(c + d*x)^2)^2,x)
Output:
(96*a^2*sinh(2*c + 2*d*x) + 12*a^2*sinh(4*c + 4*d*x) - 3*b^2*sinh(4*c + 4* d*x) + (3*b^2*sinh(8*c + 8*d*x))/8 - 12*a*b*sinh(2*c + 2*d*x) + 12*a*b*sin h(4*c + 4*d*x) + 4*a*b*sinh(6*c + 6*d*x) + 144*a^2*d*x + 9*b^2*d*x - 48*a* b*d*x)/(384*d)
Time = 0.17 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.82 \[ \int \cosh ^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {3 e^{16 d x +16 c} b^{2}+32 e^{14 d x +14 c} a b +96 e^{12 d x +12 c} a^{2}+96 e^{12 d x +12 c} a b -24 e^{12 d x +12 c} b^{2}+768 e^{10 d x +10 c} a^{2}-96 e^{10 d x +10 c} a b +2304 e^{8 d x +8 c} a^{2} d x -768 e^{8 d x +8 c} a b d x +144 e^{8 d x +8 c} b^{2} d x -768 e^{6 d x +6 c} a^{2}+96 e^{6 d x +6 c} a b -96 e^{4 d x +4 c} a^{2}-96 e^{4 d x +4 c} a b +24 e^{4 d x +4 c} b^{2}-32 e^{2 d x +2 c} a b -3 b^{2}}{6144 e^{8 d x +8 c} d} \] Input:
int(cosh(d*x+c)^4*(a+b*sinh(d*x+c)^2)^2,x)
Output:
(3*e**(16*c + 16*d*x)*b**2 + 32*e**(14*c + 14*d*x)*a*b + 96*e**(12*c + 12* d*x)*a**2 + 96*e**(12*c + 12*d*x)*a*b - 24*e**(12*c + 12*d*x)*b**2 + 768*e **(10*c + 10*d*x)*a**2 - 96*e**(10*c + 10*d*x)*a*b + 2304*e**(8*c + 8*d*x) *a**2*d*x - 768*e**(8*c + 8*d*x)*a*b*d*x + 144*e**(8*c + 8*d*x)*b**2*d*x - 768*e**(6*c + 6*d*x)*a**2 + 96*e**(6*c + 6*d*x)*a*b - 96*e**(4*c + 4*d*x) *a**2 - 96*e**(4*c + 4*d*x)*a*b + 24*e**(4*c + 4*d*x)*b**2 - 32*e**(2*c + 2*d*x)*a*b - 3*b**2)/(6144*e**(8*c + 8*d*x)*d)