Integrand size = 20, antiderivative size = 237 \[ \int (c+d x)^3 (a+a \cosh (e+f x))^2 \, dx=\frac {3 a^2 c d^2 x}{4 f^2}+\frac {3 a^2 d^3 x^2}{8 f^2}+\frac {3 a^2 (c+d x)^4}{8 d}-\frac {12 a^2 d^3 \cosh (e+f x)}{f^4}-\frac {6 a^2 d (c+d x)^2 \cosh (e+f x)}{f^2}-\frac {3 a^2 d^3 \cosh ^2(e+f x)}{8 f^4}-\frac {3 a^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}+\frac {12 a^2 d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {2 a^2 (c+d x)^3 \sinh (e+f x)}{f}+\frac {3 a^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {a^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f} \]
3/4*a^2*c*d^2*x/f^2+3/8*a^2*d^3*x^2/f^2+3/8*a^2*(d*x+c)^4/d-12*a^2*d^3*cos h(f*x+e)/f^4-6*a^2*d*(d*x+c)^2*cosh(f*x+e)/f^2-3/8*a^2*d^3*cosh(f*x+e)^2/f ^4-3/4*a^2*d*(d*x+c)^2*cosh(f*x+e)^2/f^2+12*a^2*d^2*(d*x+c)*sinh(f*x+e)/f^ 3+2*a^2*(d*x+c)^3*sinh(f*x+e)/f+3/4*a^2*d^2*(d*x+c)*cosh(f*x+e)*sinh(f*x+e )/f^3+1/2*a^2*(d*x+c)^3*cosh(f*x+e)*sinh(f*x+e)/f
Time = 0.99 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.92 \[ \int (c+d x)^3 (a+a \cosh (e+f x))^2 \, dx=\frac {a^2 \left (-96 d \left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \cosh (e+f x)-3 d \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (1+2 f^2 x^2\right )\right ) \cosh (2 (e+f x))+2 f \left (3 f^3 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+16 (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (6+f^2 x^2\right )\right ) \sinh (e+f x)+(c+d x) \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (3+2 f^2 x^2\right )\right ) \sinh (2 (e+f x))\right )\right )}{16 f^4} \]
(a^2*(-96*d*(c^2*f^2 + 2*c*d*f^2*x + d^2*(2 + f^2*x^2))*Cosh[e + f*x] - 3* d*(2*c^2*f^2 + 4*c*d*f^2*x + d^2*(1 + 2*f^2*x^2))*Cosh[2*(e + f*x)] + 2*f* (3*f^3*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + 16*(c + d*x)*(c^2*f ^2 + 2*c*d*f^2*x + d^2*(6 + f^2*x^2))*Sinh[e + f*x] + (c + d*x)*(2*c^2*f^2 + 4*c*d*f^2*x + d^2*(3 + 2*f^2*x^2))*Sinh[2*(e + f*x)])))/(16*f^4)
Time = 0.51 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 3798, 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.
| \(\displaystyle \int (c+d x)^3 (a \cosh (e+f x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^3 \left (a+a \sin \left (i e+i f x+\frac {\pi }{2}\right )\right )^2dx\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \int \left (a^2 (c+d x)^3 \cosh ^2(e+f x)+2 a^2 (c+d x)^3 \cosh (e+f x)+a^2 (c+d x)^3\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {12 a^2 d^2 (c+d x) \sinh (e+f x)}{f^3}+\frac {3 a^2 d^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{4 f^3}-\frac {3 a^2 d (c+d x)^2 \cosh ^2(e+f x)}{4 f^2}-\frac {6 a^2 d (c+d x)^2 \cosh (e+f x)}{f^2}+\frac {2 a^2 (c+d x)^3 \sinh (e+f x)}{f}+\frac {a^2 (c+d x)^3 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac {3 a^2 d (c+d x)^2}{8 f^2}+\frac {3 a^2 (c+d x)^4}{8 d}-\frac {3 a^2 d^3 \cosh ^2(e+f x)}{8 f^4}-\frac {12 a^2 d^3 \cosh (e+f x)}{f^4}\) |
(3*a^2*d*(c + d*x)^2)/(8*f^2) + (3*a^2*(c + d*x)^4)/(8*d) - (12*a^2*d^3*Co sh[e + f*x])/f^4 - (6*a^2*d*(c + d*x)^2*Cosh[e + f*x])/f^2 - (3*a^2*d^3*Co sh[e + f*x]^2)/(8*f^4) - (3*a^2*d*(c + d*x)^2*Cosh[e + f*x]^2)/(4*f^2) + ( 12*a^2*d^2*(c + d*x)*Sinh[e + f*x])/f^3 + (2*a^2*(c + d*x)^3*Sinh[e + f*x] )/f + (3*a^2*d^2*(c + d*x)*Cosh[e + f*x]*Sinh[e + f*x])/(4*f^3) + (a^2*(c + d*x)^3*Cosh[e + f*x]*Sinh[e + f*x])/(2*f)
3.2.5.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ m, 0] || NeQ[a^2 - b^2, 0])
Time = 0.30 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.76
| method | result | size |
| parallelrisch | \(\frac {\left (\left (d x +c \right ) f \left (\left (d x +c \right )^{2} f^{2}+\frac {3 d^{2}}{2}\right ) \sinh \left (2 f x +2 e \right )-\frac {3 d \left (\left (d x +c \right )^{2} f^{2}+\frac {d^{2}}{2}\right ) \cosh \left (2 f x +2 e \right )}{2}+8 \left (d x +c \right ) \left (\left (d x +c \right )^{2} f^{2}+6 d^{2}\right ) f \sinh \left (f x +e \right )-24 d \left (\left (d x +c \right )^{2} f^{2}+2 d^{2}\right ) \cosh \left (f x +e \right )+\left (\frac {3}{2} d^{3} x^{4}+6 c^{3} x +6 d^{2} c \,x^{3}+9 d \,c^{2} x^{2}\right ) f^{4}-\frac {45 c^{2} d \,f^{2}}{2}-\frac {189 d^{3}}{4}\right ) a^{2}}{4 f^{4}}\) | \(180\) |
| risch | \(\frac {3 a^{2} d^{3} x^{4}}{8}+\frac {3 a^{2} d^{2} c \,x^{3}}{2}+\frac {9 a^{2} d \,c^{2} x^{2}}{4}+\frac {3 a^{2} c^{3} x}{2}+\frac {3 a^{2} c^{4}}{8 d}+\frac {a^{2} \left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}+12 c^{2} d \,f^{3} x -6 d^{3} f^{2} x^{2}+4 c^{3} f^{3}-12 c \,d^{2} f^{2} x -6 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f -3 d^{3}\right ) {\mathrm e}^{2 f x +2 e}}{32 f^{4}}+\frac {a^{2} \left (d^{3} x^{3} f^{3}+3 c \,d^{2} f^{3} x^{2}+3 c^{2} d \,f^{3} x -3 d^{3} f^{2} x^{2}+c^{3} f^{3}-6 c \,d^{2} f^{2} x -3 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f -6 d^{3}\right ) {\mathrm e}^{f x +e}}{f^{4}}-\frac {a^{2} \left (d^{3} x^{3} f^{3}+3 c \,d^{2} f^{3} x^{2}+3 c^{2} d \,f^{3} x +3 d^{3} f^{2} x^{2}+c^{3} f^{3}+6 c \,d^{2} f^{2} x +3 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +6 d^{3}\right ) {\mathrm e}^{-f x -e}}{f^{4}}-\frac {a^{2} \left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}+12 c^{2} d \,f^{3} x +6 d^{3} f^{2} x^{2}+4 c^{3} f^{3}+12 c \,d^{2} f^{2} x +6 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +3 d^{3}\right ) {\mathrm e}^{-2 f x -2 e}}{32 f^{4}}\) | \(481\) |
| parts | \(\text {Expression too large to display}\) | \(853\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1071\) |
| default | \(\text {Expression too large to display}\) | \(1071\) |
1/4*((d*x+c)*f*((d*x+c)^2*f^2+3/2*d^2)*sinh(2*f*x+2*e)-3/2*d*((d*x+c)^2*f^ 2+1/2*d^2)*cosh(2*f*x+2*e)+8*(d*x+c)*((d*x+c)^2*f^2+6*d^2)*f*sinh(f*x+e)-2 4*d*((d*x+c)^2*f^2+2*d^2)*cosh(f*x+e)+(3/2*d^3*x^4+6*c^3*x+6*d^2*c*x^3+9*d *c^2*x^2)*f^4-45/2*c^2*d*f^2-189/4*d^3)*a^2/f^4
Time = 0.26 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.67 \[ \int (c+d x)^3 (a+a \cosh (e+f x))^2 \, dx=\frac {6 \, a^{2} d^{3} f^{4} x^{4} + 24 \, a^{2} c d^{2} f^{4} x^{3} + 36 \, a^{2} c^{2} d f^{4} x^{2} + 24 \, a^{2} c^{3} f^{4} x - 3 \, {\left (2 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c d^{2} f^{2} x + 2 \, a^{2} c^{2} d f^{2} + a^{2} d^{3}\right )} \cosh \left (f x + e\right )^{2} - 3 \, {\left (2 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c d^{2} f^{2} x + 2 \, a^{2} c^{2} d f^{2} + a^{2} d^{3}\right )} \sinh \left (f x + e\right )^{2} - 96 \, {\left (a^{2} d^{3} f^{2} x^{2} + 2 \, a^{2} c d^{2} f^{2} x + a^{2} c^{2} d f^{2} + 2 \, a^{2} d^{3}\right )} \cosh \left (f x + e\right ) + 4 \, {\left (8 \, a^{2} d^{3} f^{3} x^{3} + 24 \, a^{2} c d^{2} f^{3} x^{2} + 8 \, a^{2} c^{3} f^{3} + 48 \, a^{2} c d^{2} f + 24 \, {\left (a^{2} c^{2} d f^{3} + 2 \, a^{2} d^{3} f\right )} x + {\left (2 \, a^{2} d^{3} f^{3} x^{3} + 6 \, a^{2} c d^{2} f^{3} x^{2} + 2 \, a^{2} c^{3} f^{3} + 3 \, a^{2} c d^{2} f + 3 \, {\left (2 \, a^{2} c^{2} d f^{3} + a^{2} d^{3} f\right )} x\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )}{16 \, f^{4}} \]
1/16*(6*a^2*d^3*f^4*x^4 + 24*a^2*c*d^2*f^4*x^3 + 36*a^2*c^2*d*f^4*x^2 + 24 *a^2*c^3*f^4*x - 3*(2*a^2*d^3*f^2*x^2 + 4*a^2*c*d^2*f^2*x + 2*a^2*c^2*d*f^ 2 + a^2*d^3)*cosh(f*x + e)^2 - 3*(2*a^2*d^3*f^2*x^2 + 4*a^2*c*d^2*f^2*x + 2*a^2*c^2*d*f^2 + a^2*d^3)*sinh(f*x + e)^2 - 96*(a^2*d^3*f^2*x^2 + 2*a^2*c *d^2*f^2*x + a^2*c^2*d*f^2 + 2*a^2*d^3)*cosh(f*x + e) + 4*(8*a^2*d^3*f^3*x ^3 + 24*a^2*c*d^2*f^3*x^2 + 8*a^2*c^3*f^3 + 48*a^2*c*d^2*f + 24*(a^2*c^2*d *f^3 + 2*a^2*d^3*f)*x + (2*a^2*d^3*f^3*x^3 + 6*a^2*c*d^2*f^3*x^2 + 2*a^2*c ^3*f^3 + 3*a^2*c*d^2*f + 3*(2*a^2*c^2*d*f^3 + a^2*d^3*f)*x)*cosh(f*x + e)) *sinh(f*x + e))/f^4
Leaf count of result is larger than twice the leaf count of optimal. 779 vs. \(2 (243) = 486\).
Time = 0.46 (sec) , antiderivative size = 779, normalized size of antiderivative = 3.29 \[ \int (c+d x)^3 (a+a \cosh (e+f x))^2 \, dx=\begin {cases} - \frac {a^{2} c^{3} x \sinh ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c^{3} x \cosh ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{3} x + \frac {a^{2} c^{3} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} c^{3} \sinh {\left (e + f x \right )}}{f} - \frac {3 a^{2} c^{2} d x^{2} \sinh ^{2}{\left (e + f x \right )}}{4} + \frac {3 a^{2} c^{2} d x^{2} \cosh ^{2}{\left (e + f x \right )}}{4} + \frac {3 a^{2} c^{2} d x^{2}}{2} + \frac {3 a^{2} c^{2} d x \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} + \frac {6 a^{2} c^{2} d x \sinh {\left (e + f x \right )}}{f} - \frac {3 a^{2} c^{2} d \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {6 a^{2} c^{2} d \cosh {\left (e + f x \right )}}{f^{2}} - \frac {a^{2} c d^{2} x^{3} \sinh ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c d^{2} x^{3} \cosh ^{2}{\left (e + f x \right )}}{2} + a^{2} c d^{2} x^{3} + \frac {3 a^{2} c d^{2} x^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} + \frac {6 a^{2} c d^{2} x^{2} \sinh {\left (e + f x \right )}}{f} - \frac {3 a^{2} c d^{2} x \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {3 a^{2} c d^{2} x \cosh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {12 a^{2} c d^{2} x \cosh {\left (e + f x \right )}}{f^{2}} + \frac {3 a^{2} c d^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{4 f^{3}} + \frac {12 a^{2} c d^{2} \sinh {\left (e + f x \right )}}{f^{3}} - \frac {a^{2} d^{3} x^{4} \sinh ^{2}{\left (e + f x \right )}}{8} + \frac {a^{2} d^{3} x^{4} \cosh ^{2}{\left (e + f x \right )}}{8} + \frac {a^{2} d^{3} x^{4}}{4} + \frac {a^{2} d^{3} x^{3} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} d^{3} x^{3} \sinh {\left (e + f x \right )}}{f} - \frac {3 a^{2} d^{3} x^{2} \sinh ^{2}{\left (e + f x \right )}}{8 f^{2}} - \frac {3 a^{2} d^{3} x^{2} \cosh ^{2}{\left (e + f x \right )}}{8 f^{2}} - \frac {6 a^{2} d^{3} x^{2} \cosh {\left (e + f x \right )}}{f^{2}} + \frac {3 a^{2} d^{3} x \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{4 f^{3}} + \frac {12 a^{2} d^{3} x \sinh {\left (e + f x \right )}}{f^{3}} - \frac {3 a^{2} d^{3} \sinh ^{2}{\left (e + f x \right )}}{8 f^{4}} - \frac {12 a^{2} d^{3} \cosh {\left (e + f x \right )}}{f^{4}} & \text {for}\: f \neq 0 \\\left (a \cosh {\left (e \right )} + a\right )^{2} \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((-a**2*c**3*x*sinh(e + f*x)**2/2 + a**2*c**3*x*cosh(e + f*x)**2/ 2 + a**2*c**3*x + a**2*c**3*sinh(e + f*x)*cosh(e + f*x)/(2*f) + 2*a**2*c** 3*sinh(e + f*x)/f - 3*a**2*c**2*d*x**2*sinh(e + f*x)**2/4 + 3*a**2*c**2*d* x**2*cosh(e + f*x)**2/4 + 3*a**2*c**2*d*x**2/2 + 3*a**2*c**2*d*x*sinh(e + f*x)*cosh(e + f*x)/(2*f) + 6*a**2*c**2*d*x*sinh(e + f*x)/f - 3*a**2*c**2*d *sinh(e + f*x)**2/(4*f**2) - 6*a**2*c**2*d*cosh(e + f*x)/f**2 - a**2*c*d** 2*x**3*sinh(e + f*x)**2/2 + a**2*c*d**2*x**3*cosh(e + f*x)**2/2 + a**2*c*d **2*x**3 + 3*a**2*c*d**2*x**2*sinh(e + f*x)*cosh(e + f*x)/(2*f) + 6*a**2*c *d**2*x**2*sinh(e + f*x)/f - 3*a**2*c*d**2*x*sinh(e + f*x)**2/(4*f**2) - 3 *a**2*c*d**2*x*cosh(e + f*x)**2/(4*f**2) - 12*a**2*c*d**2*x*cosh(e + f*x)/ f**2 + 3*a**2*c*d**2*sinh(e + f*x)*cosh(e + f*x)/(4*f**3) + 12*a**2*c*d**2 *sinh(e + f*x)/f**3 - a**2*d**3*x**4*sinh(e + f*x)**2/8 + a**2*d**3*x**4*c osh(e + f*x)**2/8 + a**2*d**3*x**4/4 + a**2*d**3*x**3*sinh(e + f*x)*cosh(e + f*x)/(2*f) + 2*a**2*d**3*x**3*sinh(e + f*x)/f - 3*a**2*d**3*x**2*sinh(e + f*x)**2/(8*f**2) - 3*a**2*d**3*x**2*cosh(e + f*x)**2/(8*f**2) - 6*a**2* d**3*x**2*cosh(e + f*x)/f**2 + 3*a**2*d**3*x*sinh(e + f*x)*cosh(e + f*x)/( 4*f**3) + 12*a**2*d**3*x*sinh(e + f*x)/f**3 - 3*a**2*d**3*sinh(e + f*x)**2 /(8*f**4) - 12*a**2*d**3*cosh(e + f*x)/f**4, Ne(f, 0)), ((a*cosh(e) + a)** 2*(c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4), True))
Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (223) = 446\).
Time = 0.22 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.22 \[ \int (c+d x)^3 (a+a \cosh (e+f x))^2 \, dx=\frac {1}{4} \, a^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} + \frac {3}{16} \, {\left (4 \, x^{2} + \frac {{\left (2 \, f x e^{\left (2 \, e\right )} - e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{2}} - \frac {{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{2}}\right )} a^{2} c^{2} d + \frac {1}{16} \, {\left (8 \, x^{3} + \frac {3 \, {\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} - 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{3}} - \frac {3 \, {\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{3}}\right )} a^{2} c d^{2} + \frac {1}{32} \, {\left (4 \, x^{4} + \frac {{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} - 6 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 6 \, f x e^{\left (2 \, e\right )} - 3 \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{4}} - \frac {{\left (4 \, f^{3} x^{3} + 6 \, f^{2} x^{2} + 6 \, f x + 3\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{4}}\right )} a^{2} d^{3} + \frac {1}{8} \, a^{2} c^{3} {\left (4 \, x + \frac {e^{\left (2 \, f x + 2 \, e\right )}}{f} - \frac {e^{\left (-2 \, f x - 2 \, e\right )}}{f}\right )} + a^{2} c^{3} x + 3 \, a^{2} c^{2} d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} - \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + 3 \, a^{2} c d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} - \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + a^{2} d^{3} {\left (\frac {{\left (f^{3} x^{3} e^{e} - 3 \, f^{2} x^{2} e^{e} + 6 \, f x e^{e} - 6 \, e^{e}\right )} e^{\left (f x\right )}}{f^{4}} - \frac {{\left (f^{3} x^{3} + 3 \, f^{2} x^{2} + 6 \, f x + 6\right )} e^{\left (-f x - e\right )}}{f^{4}}\right )} + \frac {2 \, a^{2} c^{3} \sinh \left (f x + e\right )}{f} \]
1/4*a^2*d^3*x^4 + a^2*c*d^2*x^3 + 3/2*a^2*c^2*d*x^2 + 3/16*(4*x^2 + (2*f*x *e^(2*e) - e^(2*e))*e^(2*f*x)/f^2 - (2*f*x + 1)*e^(-2*f*x - 2*e)/f^2)*a^2* c^2*d + 1/16*(8*x^3 + 3*(2*f^2*x^2*e^(2*e) - 2*f*x*e^(2*e) + e^(2*e))*e^(2 *f*x)/f^3 - 3*(2*f^2*x^2 + 2*f*x + 1)*e^(-2*f*x - 2*e)/f^3)*a^2*c*d^2 + 1/ 32*(4*x^4 + (4*f^3*x^3*e^(2*e) - 6*f^2*x^2*e^(2*e) + 6*f*x*e^(2*e) - 3*e^( 2*e))*e^(2*f*x)/f^4 - (4*f^3*x^3 + 6*f^2*x^2 + 6*f*x + 3)*e^(-2*f*x - 2*e) /f^4)*a^2*d^3 + 1/8*a^2*c^3*(4*x + e^(2*f*x + 2*e)/f - e^(-2*f*x - 2*e)/f) + a^2*c^3*x + 3*a^2*c^2*d*((f*x*e^e - e^e)*e^(f*x)/f^2 - (f*x + 1)*e^(-f* x - e)/f^2) + 3*a^2*c*d^2*((f^2*x^2*e^e - 2*f*x*e^e + 2*e^e)*e^(f*x)/f^3 - (f^2*x^2 + 2*f*x + 2)*e^(-f*x - e)/f^3) + a^2*d^3*((f^3*x^3*e^e - 3*f^2*x ^2*e^e + 6*f*x*e^e - 6*e^e)*e^(f*x)/f^4 - (f^3*x^3 + 3*f^2*x^2 + 6*f*x + 6 )*e^(-f*x - e)/f^4) + 2*a^2*c^3*sinh(f*x + e)/f
Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (223) = 446\).
Time = 0.29 (sec) , antiderivative size = 577, normalized size of antiderivative = 2.43 \[ \int (c+d x)^3 (a+a \cosh (e+f x))^2 \, dx=\frac {3}{8} \, a^{2} d^{3} x^{4} + \frac {3}{2} \, a^{2} c d^{2} x^{3} + \frac {9}{4} \, a^{2} c^{2} d x^{2} + \frac {3}{2} \, a^{2} c^{3} x + \frac {{\left (4 \, a^{2} d^{3} f^{3} x^{3} + 12 \, a^{2} c d^{2} f^{3} x^{2} + 12 \, a^{2} c^{2} d f^{3} x - 6 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c^{3} f^{3} - 12 \, a^{2} c d^{2} f^{2} x - 6 \, a^{2} c^{2} d f^{2} + 6 \, a^{2} d^{3} f x + 6 \, a^{2} c d^{2} f - 3 \, a^{2} d^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{32 \, f^{4}} + \frac {{\left (a^{2} d^{3} f^{3} x^{3} + 3 \, a^{2} c d^{2} f^{3} x^{2} + 3 \, a^{2} c^{2} d f^{3} x - 3 \, a^{2} d^{3} f^{2} x^{2} + a^{2} c^{3} f^{3} - 6 \, a^{2} c d^{2} f^{2} x - 3 \, a^{2} c^{2} d f^{2} + 6 \, a^{2} d^{3} f x + 6 \, a^{2} c d^{2} f - 6 \, a^{2} d^{3}\right )} e^{\left (f x + e\right )}}{f^{4}} - \frac {{\left (a^{2} d^{3} f^{3} x^{3} + 3 \, a^{2} c d^{2} f^{3} x^{2} + 3 \, a^{2} c^{2} d f^{3} x + 3 \, a^{2} d^{3} f^{2} x^{2} + a^{2} c^{3} f^{3} + 6 \, a^{2} c d^{2} f^{2} x + 3 \, a^{2} c^{2} d f^{2} + 6 \, a^{2} d^{3} f x + 6 \, a^{2} c d^{2} f + 6 \, a^{2} d^{3}\right )} e^{\left (-f x - e\right )}}{f^{4}} - \frac {{\left (4 \, a^{2} d^{3} f^{3} x^{3} + 12 \, a^{2} c d^{2} f^{3} x^{2} + 12 \, a^{2} c^{2} d f^{3} x + 6 \, a^{2} d^{3} f^{2} x^{2} + 4 \, a^{2} c^{3} f^{3} + 12 \, a^{2} c d^{2} f^{2} x + 6 \, a^{2} c^{2} d f^{2} + 6 \, a^{2} d^{3} f x + 6 \, a^{2} c d^{2} f + 3 \, a^{2} d^{3}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{32 \, f^{4}} \]
3/8*a^2*d^3*x^4 + 3/2*a^2*c*d^2*x^3 + 9/4*a^2*c^2*d*x^2 + 3/2*a^2*c^3*x + 1/32*(4*a^2*d^3*f^3*x^3 + 12*a^2*c*d^2*f^3*x^2 + 12*a^2*c^2*d*f^3*x - 6*a^ 2*d^3*f^2*x^2 + 4*a^2*c^3*f^3 - 12*a^2*c*d^2*f^2*x - 6*a^2*c^2*d*f^2 + 6*a ^2*d^3*f*x + 6*a^2*c*d^2*f - 3*a^2*d^3)*e^(2*f*x + 2*e)/f^4 + (a^2*d^3*f^3 *x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x - 3*a^2*d^3*f^2*x^2 + a^2*c ^3*f^3 - 6*a^2*c*d^2*f^2*x - 3*a^2*c^2*d*f^2 + 6*a^2*d^3*f*x + 6*a^2*c*d^2 *f - 6*a^2*d^3)*e^(f*x + e)/f^4 - (a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + 3*a^2*d^3*f^2*x^2 + a^2*c^3*f^3 + 6*a^2*c*d^2*f^2*x + 3*a^2*c^2*d*f^2 + 6*a^2*d^3*f*x + 6*a^2*c*d^2*f + 6*a^2*d^3)*e^(-f*x - e) /f^4 - 1/32*(4*a^2*d^3*f^3*x^3 + 12*a^2*c*d^2*f^3*x^2 + 12*a^2*c^2*d*f^3*x + 6*a^2*d^3*f^2*x^2 + 4*a^2*c^3*f^3 + 12*a^2*c*d^2*f^2*x + 6*a^2*c^2*d*f^ 2 + 6*a^2*d^3*f*x + 6*a^2*c*d^2*f + 3*a^2*d^3)*e^(-2*f*x - 2*e)/f^4
Time = 3.47 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.91 \[ \int (c+d x)^3 (a+a \cosh (e+f x))^2 \, dx=\frac {16\,a^2\,c^3\,f^3\,\mathrm {sinh}\left (e+f\,x\right )-\frac {3\,a^2\,d^3\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )}{2}-96\,a^2\,d^3\,\mathrm {cosh}\left (e+f\,x\right )+12\,a^2\,c^3\,f^4\,x+2\,a^2\,c^3\,f^3\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+3\,a^2\,d^3\,f^4\,x^4+96\,a^2\,c\,d^2\,f\,\mathrm {sinh}\left (e+f\,x\right )+96\,a^2\,d^3\,f\,x\,\mathrm {sinh}\left (e+f\,x\right )-3\,a^2\,d^3\,f^2\,x^2\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )+2\,a^2\,d^3\,f^3\,x^3\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-48\,a^2\,c^2\,d\,f^2\,\mathrm {cosh}\left (e+f\,x\right )+3\,a^2\,c\,d^2\,f\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+3\,a^2\,d^3\,f\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-3\,a^2\,c^2\,d\,f^2\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )+18\,a^2\,c^2\,d\,f^4\,x^2+12\,a^2\,c\,d^2\,f^4\,x^3-48\,a^2\,d^3\,f^2\,x^2\,\mathrm {cosh}\left (e+f\,x\right )+16\,a^2\,d^3\,f^3\,x^3\,\mathrm {sinh}\left (e+f\,x\right )-6\,a^2\,c\,d^2\,f^2\,x\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )+6\,a^2\,c^2\,d\,f^3\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )+48\,a^2\,c\,d^2\,f^3\,x^2\,\mathrm {sinh}\left (e+f\,x\right )+6\,a^2\,c\,d^2\,f^3\,x^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )-96\,a^2\,c\,d^2\,f^2\,x\,\mathrm {cosh}\left (e+f\,x\right )+48\,a^2\,c^2\,d\,f^3\,x\,\mathrm {sinh}\left (e+f\,x\right )}{8\,f^4} \]
(16*a^2*c^3*f^3*sinh(e + f*x) - (3*a^2*d^3*cosh(2*e + 2*f*x))/2 - 96*a^2*d ^3*cosh(e + f*x) + 12*a^2*c^3*f^4*x + 2*a^2*c^3*f^3*sinh(2*e + 2*f*x) + 3* a^2*d^3*f^4*x^4 + 96*a^2*c*d^2*f*sinh(e + f*x) + 96*a^2*d^3*f*x*sinh(e + f *x) - 3*a^2*d^3*f^2*x^2*cosh(2*e + 2*f*x) + 2*a^2*d^3*f^3*x^3*sinh(2*e + 2 *f*x) - 48*a^2*c^2*d*f^2*cosh(e + f*x) + 3*a^2*c*d^2*f*sinh(2*e + 2*f*x) + 3*a^2*d^3*f*x*sinh(2*e + 2*f*x) - 3*a^2*c^2*d*f^2*cosh(2*e + 2*f*x) + 18* a^2*c^2*d*f^4*x^2 + 12*a^2*c*d^2*f^4*x^3 - 48*a^2*d^3*f^2*x^2*cosh(e + f*x ) + 16*a^2*d^3*f^3*x^3*sinh(e + f*x) - 6*a^2*c*d^2*f^2*x*cosh(2*e + 2*f*x) + 6*a^2*c^2*d*f^3*x*sinh(2*e + 2*f*x) + 48*a^2*c*d^2*f^3*x^2*sinh(e + f*x ) + 6*a^2*c*d^2*f^3*x^2*sinh(2*e + 2*f*x) - 96*a^2*c*d^2*f^2*x*cosh(e + f* x) + 48*a^2*c^2*d*f^3*x*sinh(e + f*x))/(8*f^4)