Integrand size = 20, antiderivative size = 168 \[ \int (c+d x)^2 (a+a \cos (e+f x))^2 \, dx=-\frac {a^2 d^2 x}{4 f^2}+\frac {a^2 (c+d x)^3}{2 d}+\frac {4 a^2 d (c+d x) \cos (e+f x)}{f^2}+\frac {a^2 d (c+d x) \cos ^2(e+f x)}{2 f^2}-\frac {4 a^2 d^2 \sin (e+f x)}{f^3}+\frac {2 a^2 (c+d x)^2 \sin (e+f x)}{f}-\frac {a^2 d^2 \cos (e+f x) \sin (e+f x)}{4 f^3}+\frac {a^2 (c+d x)^2 \cos (e+f x) \sin (e+f x)}{2 f} \]
-1/4*a^2*d^2*x/f^2+1/2*a^2*(d*x+c)^3/d+4*a^2*d*(d*x+c)*cos(f*x+e)/f^2+1/2* a^2*d*(d*x+c)*cos(f*x+e)^2/f^2-4*a^2*d^2*sin(f*x+e)/f^3+2*a^2*(d*x+c)^2*si n(f*x+e)/f-1/4*a^2*d^2*cos(f*x+e)*sin(f*x+e)/f^3+1/2*a^2*(d*x+c)^2*cos(f*x +e)*sin(f*x+e)/f
Time = 0.78 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.15 \[ \int (c+d x)^2 (a+a \cos (e+f x))^2 \, dx=\frac {a^2 \left (12 c^2 f^3 x+12 c d f^3 x^2+4 d^2 f^3 x^3+32 d f (c+d x) \cos (e+f x)+2 d f (c+d x) \cos (2 (e+f x))-32 d^2 \sin (e+f x)+16 c^2 f^2 \sin (e+f x)+32 c d f^2 x \sin (e+f x)+16 d^2 f^2 x^2 \sin (e+f x)-d^2 \sin (2 (e+f x))+2 c^2 f^2 \sin (2 (e+f x))+4 c d f^2 x \sin (2 (e+f x))+2 d^2 f^2 x^2 \sin (2 (e+f x))\right )}{8 f^3} \]
(a^2*(12*c^2*f^3*x + 12*c*d*f^3*x^2 + 4*d^2*f^3*x^3 + 32*d*f*(c + d*x)*Cos [e + f*x] + 2*d*f*(c + d*x)*Cos[2*(e + f*x)] - 32*d^2*Sin[e + f*x] + 16*c^ 2*f^2*Sin[e + f*x] + 32*c*d*f^2*x*Sin[e + f*x] + 16*d^2*f^2*x^2*Sin[e + f* x] - d^2*Sin[2*(e + f*x)] + 2*c^2*f^2*Sin[2*(e + f*x)] + 4*c*d*f^2*x*Sin[2 *(e + f*x)] + 2*d^2*f^2*x^2*Sin[2*(e + f*x)]))/(8*f^3)
Time = 0.41 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, 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.
| \(\displaystyle \int (c+d x)^2 (a \cos (e+f x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^2 \left (a \sin \left (e+f x+\frac {\pi }{2}\right )+a\right )^2dx\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \int \left (a^2 (c+d x)^2 \cos ^2(e+f x)+2 a^2 (c+d x)^2 \cos (e+f x)+a^2 (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^2 d (c+d x) \cos ^2(e+f x)}{2 f^2}+\frac {4 a^2 d (c+d x) \cos (e+f x)}{f^2}+\frac {2 a^2 (c+d x)^2 \sin (e+f x)}{f}+\frac {a^2 (c+d x)^2 \sin (e+f x) \cos (e+f x)}{2 f}+\frac {a^2 (c+d x)^3}{2 d}-\frac {4 a^2 d^2 \sin (e+f x)}{f^3}-\frac {a^2 d^2 \sin (e+f x) \cos (e+f x)}{4 f^3}-\frac {a^2 d^2 x}{4 f^2}\) |
-1/4*(a^2*d^2*x)/f^2 + (a^2*(c + d*x)^3)/(2*d) + (4*a^2*d*(c + d*x)*Cos[e + f*x])/f^2 + (a^2*d*(c + d*x)*Cos[e + f*x]^2)/(2*f^2) - (4*a^2*d^2*Sin[e + f*x])/f^3 + (2*a^2*(c + d*x)^2*Sin[e + f*x])/f - (a^2*d^2*Cos[e + f*x]*S in[e + f*x])/(4*f^3) + (a^2*(c + d*x)^2*Cos[e + f*x]*Sin[e + f*x])/(2*f)
3.2.24.3.1 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 = 1.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.73
| method | result | size |
| parallelrisch | \(\frac {\left (\left (\left (d x +c \right )^{2} f^{2}-\frac {d^{2}}{2}\right ) \sin \left (2 f x +2 e \right )+d f \left (d x +c \right ) \cos \left (2 f x +2 e \right )+8 \left (\left (d x +c \right )^{2} f^{2}-2 d^{2}\right ) \sin \left (f x +e \right )+6 \left (\frac {8 d \left (d x +c \right ) \cos \left (f x +e \right )}{3}+x \left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) f^{2}-\frac {17 c d}{6}\right ) f \right ) a^{2}}{4 f^{3}}\) | \(122\) |
| risch | \(\frac {a^{2} d^{2} x^{3}}{2}+\frac {3 a^{2} c d \,x^{2}}{2}+\frac {3 a^{2} c^{2} x}{2}+\frac {a^{2} c^{3}}{2 d}+\frac {4 a^{2} d \left (d x +c \right ) \cos \left (f x +e \right )}{f^{2}}+\frac {2 a^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}-2 d^{2}\right ) \sin \left (f x +e \right )}{f^{3}}+\frac {a^{2} d \left (d x +c \right ) \cos \left (2 f x +2 e \right )}{4 f^{2}}+\frac {a^{2} \left (2 d^{2} x^{2} f^{2}+4 c d \,f^{2} x +2 c^{2} f^{2}-d^{2}\right ) \sin \left (2 f x +2 e \right )}{8 f^{3}}\) | \(181\) |
| norman | \(\frac {\frac {8 a^{2} c d}{f^{2}}+a^{2} d^{2} x^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {a^{2} d^{2} x^{3}}{2}+\frac {6 a^{2} c d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f^{2}}+\frac {3 a^{2} c d \,x^{2}}{2}+\frac {a^{2} d^{2} x^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\frac {3 a^{2} \left (2 c^{2} f^{2}-5 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f^{3}}+\frac {a^{2} \left (6 c^{2} f^{2}+17 d^{2}\right ) x}{4 f^{2}}+\frac {a^{2} \left (10 c^{2} f^{2}-17 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 f^{3}}+3 a^{2} c d \,x^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {3 a^{2} c d \,x^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\frac {5 a^{2} d^{2} x^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {3 a^{2} d^{2} x^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {3 a^{2} \left (2 c^{2} f^{2}-d^{2}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f^{2}}+\frac {3 a^{2} \left (2 c^{2} f^{2}-5 d^{2}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f^{2}}+\frac {10 a^{2} c d x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {6 a^{2} c d x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}\) | \(400\) |
| parts | \(\frac {a^{2} \left (d x +c \right )^{3}}{3 d}+\frac {a^{2} \left (\frac {d^{2} \left (\left (f x +e \right )^{2} \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+\frac {\left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )}{2}-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{4}-\frac {f x}{4}-\frac {e}{4}-\frac {\left (f x +e \right )^{3}}{3}\right )}{f^{2}}+\frac {2 c d \left (\left (f x +e \right ) \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f}-\frac {2 d^{2} e \left (\left (f x +e \right ) \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f^{2}}+c^{2} \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 c d e \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {d^{2} e^{2} \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f^{2}}\right )}{f}+\frac {2 a^{2} \left (\frac {d^{2} \left (\left (f x +e \right )^{2} \sin \left (f x +e \right )-2 \sin \left (f x +e \right )+2 \left (f x +e \right ) \cos \left (f x +e \right )\right )}{f^{2}}+\frac {2 c d \left (\cos \left (f x +e \right )+\left (f x +e \right ) \sin \left (f x +e \right )\right )}{f}-\frac {2 d^{2} e \left (\cos \left (f x +e \right )+\left (f x +e \right ) \sin \left (f x +e \right )\right )}{f^{2}}+c^{2} \sin \left (f x +e \right )-\frac {2 c d e \sin \left (f x +e \right )}{f}+\frac {d^{2} e^{2} \sin \left (f x +e \right )}{f^{2}}\right )}{f}\) | \(454\) |
| derivativedivides | \(\frac {a^{2} c^{2} \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 a^{2} c d e \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {2 a^{2} c d \left (\left (f x +e \right ) \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f}+\frac {a^{2} d^{2} e^{2} \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f^{2}}-\frac {2 a^{2} d^{2} e \left (\left (f x +e \right ) \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f^{2}}+\frac {a^{2} d^{2} \left (\left (f x +e \right )^{2} \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+\frac {\left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )}{2}-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{4}-\frac {f x}{4}-\frac {e}{4}-\frac {\left (f x +e \right )^{3}}{3}\right )}{f^{2}}+2 a^{2} c^{2} \sin \left (f x +e \right )-\frac {4 a^{2} c d e \sin \left (f x +e \right )}{f}+\frac {4 a^{2} c d \left (\cos \left (f x +e \right )+\left (f x +e \right ) \sin \left (f x +e \right )\right )}{f}+\frac {2 a^{2} d^{2} e^{2} \sin \left (f x +e \right )}{f^{2}}-\frac {4 a^{2} d^{2} e \left (\cos \left (f x +e \right )+\left (f x +e \right ) \sin \left (f x +e \right )\right )}{f^{2}}+\frac {2 a^{2} d^{2} \left (\left (f x +e \right )^{2} \sin \left (f x +e \right )-2 \sin \left (f x +e \right )+2 \left (f x +e \right ) \cos \left (f x +e \right )\right )}{f^{2}}+a^{2} c^{2} \left (f x +e \right )-\frac {2 a^{2} c d e \left (f x +e \right )}{f}+\frac {a^{2} c d \left (f x +e \right )^{2}}{f}+\frac {a^{2} d^{2} e^{2} \left (f x +e \right )}{f^{2}}-\frac {a^{2} d^{2} e \left (f x +e \right )^{2}}{f^{2}}+\frac {a^{2} d^{2} \left (f x +e \right )^{3}}{3 f^{2}}}{f}\) | \(564\) |
| default | \(\frac {a^{2} c^{2} \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 a^{2} c d e \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {2 a^{2} c d \left (\left (f x +e \right ) \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f}+\frac {a^{2} d^{2} e^{2} \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f^{2}}-\frac {2 a^{2} d^{2} e \left (\left (f x +e \right ) \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {\left (f x +e \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (f x +e \right )\right )}{4}\right )}{f^{2}}+\frac {a^{2} d^{2} \left (\left (f x +e \right )^{2} \left (\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+\frac {\left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )}{2}-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{4}-\frac {f x}{4}-\frac {e}{4}-\frac {\left (f x +e \right )^{3}}{3}\right )}{f^{2}}+2 a^{2} c^{2} \sin \left (f x +e \right )-\frac {4 a^{2} c d e \sin \left (f x +e \right )}{f}+\frac {4 a^{2} c d \left (\cos \left (f x +e \right )+\left (f x +e \right ) \sin \left (f x +e \right )\right )}{f}+\frac {2 a^{2} d^{2} e^{2} \sin \left (f x +e \right )}{f^{2}}-\frac {4 a^{2} d^{2} e \left (\cos \left (f x +e \right )+\left (f x +e \right ) \sin \left (f x +e \right )\right )}{f^{2}}+\frac {2 a^{2} d^{2} \left (\left (f x +e \right )^{2} \sin \left (f x +e \right )-2 \sin \left (f x +e \right )+2 \left (f x +e \right ) \cos \left (f x +e \right )\right )}{f^{2}}+a^{2} c^{2} \left (f x +e \right )-\frac {2 a^{2} c d e \left (f x +e \right )}{f}+\frac {a^{2} c d \left (f x +e \right )^{2}}{f}+\frac {a^{2} d^{2} e^{2} \left (f x +e \right )}{f^{2}}-\frac {a^{2} d^{2} e \left (f x +e \right )^{2}}{f^{2}}+\frac {a^{2} d^{2} \left (f x +e \right )^{3}}{3 f^{2}}}{f}\) | \(564\) |
1/4*(((d*x+c)^2*f^2-1/2*d^2)*sin(2*f*x+2*e)+d*f*(d*x+c)*cos(2*f*x+2*e)+8*( (d*x+c)^2*f^2-2*d^2)*sin(f*x+e)+6*(8/3*d*(d*x+c)*cos(f*x+e)+x*(1/3*x^2*d^2 +c*d*x+c^2)*f^2-17/6*c*d)*f)*a^2/f^3
Time = 0.30 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.26 \[ \int (c+d x)^2 (a+a \cos (e+f x))^2 \, dx=\frac {2 \, a^{2} d^{2} f^{3} x^{3} + 6 \, a^{2} c d f^{3} x^{2} + 2 \, {\left (a^{2} d^{2} f x + a^{2} c d f\right )} \cos \left (f x + e\right )^{2} + {\left (6 \, a^{2} c^{2} f^{3} - a^{2} d^{2} f\right )} x + 16 \, {\left (a^{2} d^{2} f x + a^{2} c d f\right )} \cos \left (f x + e\right ) + {\left (8 \, a^{2} d^{2} f^{2} x^{2} + 16 \, a^{2} c d f^{2} x + 8 \, a^{2} c^{2} f^{2} - 16 \, a^{2} d^{2} + {\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} - a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, f^{3}} \]
1/4*(2*a^2*d^2*f^3*x^3 + 6*a^2*c*d*f^3*x^2 + 2*(a^2*d^2*f*x + a^2*c*d*f)*c os(f*x + e)^2 + (6*a^2*c^2*f^3 - a^2*d^2*f)*x + 16*(a^2*d^2*f*x + a^2*c*d* f)*cos(f*x + e) + (8*a^2*d^2*f^2*x^2 + 16*a^2*c*d*f^2*x + 8*a^2*c^2*f^2 - 16*a^2*d^2 + (2*a^2*d^2*f^2*x^2 + 4*a^2*c*d*f^2*x + 2*a^2*c^2*f^2 - a^2*d^ 2)*cos(f*x + e))*sin(f*x + e))/f^3
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (163) = 326\).
Time = 0.33 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.71 \[ \int (c+d x)^2 (a+a \cos (e+f x))^2 \, dx=\begin {cases} \frac {a^{2} c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{2} x + \frac {a^{2} c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} c^{2} \sin {\left (e + f x \right )}}{f} + \frac {a^{2} c d x^{2} \sin ^{2}{\left (e + f x \right )}}{2} + \frac {a^{2} c d x^{2} \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c d x^{2} + \frac {a^{2} c d x \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {4 a^{2} c d x \sin {\left (e + f x \right )}}{f} - \frac {a^{2} c d \sin ^{2}{\left (e + f x \right )}}{2 f^{2}} + \frac {4 a^{2} c d \cos {\left (e + f x \right )}}{f^{2}} + \frac {a^{2} d^{2} x^{3} \sin ^{2}{\left (e + f x \right )}}{6} + \frac {a^{2} d^{2} x^{3} \cos ^{2}{\left (e + f x \right )}}{6} + \frac {a^{2} d^{2} x^{3}}{3} + \frac {a^{2} d^{2} x^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} d^{2} x^{2} \sin {\left (e + f x \right )}}{f} - \frac {a^{2} d^{2} x \sin ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {a^{2} d^{2} x \cos ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {4 a^{2} d^{2} x \cos {\left (e + f x \right )}}{f^{2}} - \frac {a^{2} d^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f^{3}} - \frac {4 a^{2} d^{2} \sin {\left (e + f x \right )}}{f^{3}} & \text {for}\: f \neq 0 \\\left (a \cos {\left (e \right )} + a\right )^{2} \left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a**2*c**2*x*sin(e + f*x)**2/2 + a**2*c**2*x*cos(e + f*x)**2/2 + a**2*c**2*x + a**2*c**2*sin(e + f*x)*cos(e + f*x)/(2*f) + 2*a**2*c**2*sin (e + f*x)/f + a**2*c*d*x**2*sin(e + f*x)**2/2 + a**2*c*d*x**2*cos(e + f*x) **2/2 + a**2*c*d*x**2 + a**2*c*d*x*sin(e + f*x)*cos(e + f*x)/f + 4*a**2*c* d*x*sin(e + f*x)/f - a**2*c*d*sin(e + f*x)**2/(2*f**2) + 4*a**2*c*d*cos(e + f*x)/f**2 + a**2*d**2*x**3*sin(e + f*x)**2/6 + a**2*d**2*x**3*cos(e + f* x)**2/6 + a**2*d**2*x**3/3 + a**2*d**2*x**2*sin(e + f*x)*cos(e + f*x)/(2*f ) + 2*a**2*d**2*x**2*sin(e + f*x)/f - a**2*d**2*x*sin(e + f*x)**2/(4*f**2) + a**2*d**2*x*cos(e + f*x)**2/(4*f**2) + 4*a**2*d**2*x*cos(e + f*x)/f**2 - a**2*d**2*sin(e + f*x)*cos(e + f*x)/(4*f**3) - 4*a**2*d**2*sin(e + f*x)/ f**3, Ne(f, 0)), ((a*cos(e) + a)**2*(c**2*x + c*d*x**2 + d**2*x**3/3), Tru e))
Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (158) = 316\).
Time = 0.32 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.94 \[ \int (c+d x)^2 (a+a \cos (e+f x))^2 \, dx=\frac {6 \, {\left (2 \, f x + 2 \, e + \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} + 24 \, {\left (f x + e\right )} a^{2} c^{2} + \frac {8 \, {\left (f x + e\right )}^{3} a^{2} d^{2}}{f^{2}} - \frac {24 \, {\left (f x + e\right )}^{2} a^{2} d^{2} e}{f^{2}} + \frac {6 \, {\left (2 \, f x + 2 \, e + \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{2} e^{2}}{f^{2}} + \frac {24 \, {\left (f x + e\right )} a^{2} d^{2} e^{2}}{f^{2}} + \frac {24 \, {\left (f x + e\right )}^{2} a^{2} c d}{f} - \frac {12 \, {\left (2 \, f x + 2 \, e + \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d e}{f} - \frac {48 \, {\left (f x + e\right )} a^{2} c d e}{f} + 48 \, a^{2} c^{2} \sin \left (f x + e\right ) + \frac {48 \, a^{2} d^{2} e^{2} \sin \left (f x + e\right )}{f^{2}} - \frac {96 \, a^{2} c d e \sin \left (f x + e\right )}{f} - \frac {6 \, {\left (2 \, {\left (f x + e\right )}^{2} + 2 \, {\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right ) + \cos \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{2} e}{f^{2}} - \frac {96 \, {\left ({\left (f x + e\right )} \sin \left (f x + e\right ) + \cos \left (f x + e\right )\right )} a^{2} d^{2} e}{f^{2}} + \frac {6 \, {\left (2 \, {\left (f x + e\right )}^{2} + 2 \, {\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right ) + \cos \left (2 \, f x + 2 \, e\right )\right )} a^{2} c d}{f} + \frac {96 \, {\left ({\left (f x + e\right )} \sin \left (f x + e\right ) + \cos \left (f x + e\right )\right )} a^{2} c d}{f} + \frac {{\left (4 \, {\left (f x + e\right )}^{3} + 6 \, {\left (f x + e\right )} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (2 \, {\left (f x + e\right )}^{2} - 1\right )} \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d^{2}}{f^{2}} + \frac {48 \, {\left (2 \, {\left (f x + e\right )} \cos \left (f x + e\right ) + {\left ({\left (f x + e\right )}^{2} - 2\right )} \sin \left (f x + e\right )\right )} a^{2} d^{2}}{f^{2}}}{24 \, f} \]
1/24*(6*(2*f*x + 2*e + sin(2*f*x + 2*e))*a^2*c^2 + 24*(f*x + e)*a^2*c^2 + 8*(f*x + e)^3*a^2*d^2/f^2 - 24*(f*x + e)^2*a^2*d^2*e/f^2 + 6*(2*f*x + 2*e + sin(2*f*x + 2*e))*a^2*d^2*e^2/f^2 + 24*(f*x + e)*a^2*d^2*e^2/f^2 + 24*(f *x + e)^2*a^2*c*d/f - 12*(2*f*x + 2*e + sin(2*f*x + 2*e))*a^2*c*d*e/f - 48 *(f*x + e)*a^2*c*d*e/f + 48*a^2*c^2*sin(f*x + e) + 48*a^2*d^2*e^2*sin(f*x + e)/f^2 - 96*a^2*c*d*e*sin(f*x + e)/f - 6*(2*(f*x + e)^2 + 2*(f*x + e)*si n(2*f*x + 2*e) + cos(2*f*x + 2*e))*a^2*d^2*e/f^2 - 96*((f*x + e)*sin(f*x + e) + cos(f*x + e))*a^2*d^2*e/f^2 + 6*(2*(f*x + e)^2 + 2*(f*x + e)*sin(2*f *x + 2*e) + cos(2*f*x + 2*e))*a^2*c*d/f + 96*((f*x + e)*sin(f*x + e) + cos (f*x + e))*a^2*c*d/f + (4*(f*x + e)^3 + 6*(f*x + e)*cos(2*f*x + 2*e) + 3*( 2*(f*x + e)^2 - 1)*sin(2*f*x + 2*e))*a^2*d^2/f^2 + 48*(2*(f*x + e)*cos(f*x + e) + ((f*x + e)^2 - 2)*sin(f*x + e))*a^2*d^2/f^2)/f
Time = 0.28 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.21 \[ \int (c+d x)^2 (a+a \cos (e+f x))^2 \, dx=\frac {1}{2} \, a^{2} d^{2} x^{3} + \frac {3}{2} \, a^{2} c d x^{2} + \frac {3}{2} \, a^{2} c^{2} x + \frac {{\left (a^{2} d^{2} f x + a^{2} c d f\right )} \cos \left (2 \, f x + 2 \, e\right )}{4 \, f^{3}} + \frac {4 \, {\left (a^{2} d^{2} f x + a^{2} c d f\right )} \cos \left (f x + e\right )}{f^{3}} + \frac {{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} - a^{2} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{8 \, f^{3}} + \frac {2 \, {\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2} - 2 \, a^{2} d^{2}\right )} \sin \left (f x + e\right )}{f^{3}} \]
1/2*a^2*d^2*x^3 + 3/2*a^2*c*d*x^2 + 3/2*a^2*c^2*x + 1/4*(a^2*d^2*f*x + a^2 *c*d*f)*cos(2*f*x + 2*e)/f^3 + 4*(a^2*d^2*f*x + a^2*c*d*f)*cos(f*x + e)/f^ 3 + 1/8*(2*a^2*d^2*f^2*x^2 + 4*a^2*c*d*f^2*x + 2*a^2*c^2*f^2 - a^2*d^2)*si n(2*f*x + 2*e)/f^3 + 2*(a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^2 - 2*a^2*d^2)*sin(f*x + e)/f^3
Time = 15.20 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.52 \[ \int (c+d x)^2 (a+a \cos (e+f x))^2 \, dx=\frac {8\,a^2\,c^2\,f^2\,\sin \left (e+f\,x\right )-\frac {a^2\,d^2\,\sin \left (2\,e+2\,f\,x\right )}{2}-16\,a^2\,d^2\,\sin \left (e+f\,x\right )+6\,a^2\,c^2\,f^3\,x+a^2\,c^2\,f^2\,\sin \left (2\,e+2\,f\,x\right )+2\,a^2\,d^2\,f^3\,x^3+a^2\,c\,d\,f\,\cos \left (2\,e+2\,f\,x\right )+16\,a^2\,d^2\,f\,x\,\cos \left (e+f\,x\right )+a^2\,d^2\,f^2\,x^2\,\sin \left (2\,e+2\,f\,x\right )+6\,a^2\,c\,d\,f^3\,x^2+a^2\,d^2\,f\,x\,\cos \left (2\,e+2\,f\,x\right )+16\,a^2\,c\,d\,f\,\cos \left (e+f\,x\right )+8\,a^2\,d^2\,f^2\,x^2\,\sin \left (e+f\,x\right )+16\,a^2\,c\,d\,f^2\,x\,\sin \left (e+f\,x\right )+2\,a^2\,c\,d\,f^2\,x\,\sin \left (2\,e+2\,f\,x\right )}{4\,f^3} \]
(8*a^2*c^2*f^2*sin(e + f*x) - (a^2*d^2*sin(2*e + 2*f*x))/2 - 16*a^2*d^2*si n(e + f*x) + 6*a^2*c^2*f^3*x + a^2*c^2*f^2*sin(2*e + 2*f*x) + 2*a^2*d^2*f^ 3*x^3 + a^2*c*d*f*cos(2*e + 2*f*x) + 16*a^2*d^2*f*x*cos(e + f*x) + a^2*d^2 *f^2*x^2*sin(2*e + 2*f*x) + 6*a^2*c*d*f^3*x^2 + a^2*d^2*f*x*cos(2*e + 2*f* x) + 16*a^2*c*d*f*cos(e + f*x) + 8*a^2*d^2*f^2*x^2*sin(e + f*x) + 16*a^2*c *d*f^2*x*sin(e + f*x) + 2*a^2*c*d*f^2*x*sin(2*e + 2*f*x))/(4*f^3)