Integrand size = 25, antiderivative size = 179 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {7}{16} a^4 (10 A+7 C) x+\frac {4 a^4 (10 A+7 C) \sin (c+d x)}{5 d}+\frac {27 a^4 (10 A+7 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (10 A+7 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}-\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}-\frac {2 a^4 (10 A+7 C) \sin ^3(c+d x)}{15 d} \]
7/16*a^4*(10*A+7*C)*x+4/5*a^4*(10*A+7*C)*sin(d*x+c)/d+27/80*a^4*(10*A+7*C) *cos(d*x+c)*sin(d*x+c)/d+1/40*a^4*(10*A+7*C)*cos(d*x+c)^3*sin(d*x+c)/d-1/3 0*C*(a+a*cos(d*x+c))^4*sin(d*x+c)/d+1/6*C*(a+a*cos(d*x+c))^5*sin(d*x+c)/a/ d-2/15*a^4*(10*A+7*C)*sin(d*x+c)^3/d
Time = 0.56 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.66 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^4 (4200 A d x+2940 C d x+480 (14 A+11 C) \sin (c+d x)+15 (112 A+127 C) \sin (2 (c+d x))+320 A \sin (3 (c+d x))+720 C \sin (3 (c+d x))+30 A \sin (4 (c+d x))+225 C \sin (4 (c+d x))+48 C \sin (5 (c+d x))+5 C \sin (6 (c+d x)))}{960 d} \]
(a^4*(4200*A*d*x + 2940*C*d*x + 480*(14*A + 11*C)*Sin[c + d*x] + 15*(112*A + 127*C)*Sin[2*(c + d*x)] + 320*A*Sin[3*(c + d*x)] + 720*C*Sin[3*(c + d*x )] + 30*A*Sin[4*(c + d*x)] + 225*C*Sin[4*(c + d*x)] + 48*C*Sin[5*(c + d*x) ] + 5*C*Sin[6*(c + d*x)]))/(960*d)
Time = 0.53 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 3503, 3042, 3230, 3042, 3124, 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.
| \(\displaystyle \int (a \cos (c+d x)+a)^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3503 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^4 (a (6 A+5 C)-a C \cos (c+d x))dx}{6 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (6 A+5 C)-a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{6 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {3}{5} a (10 A+7 C) \int (\cos (c+d x) a+a)^4dx-\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} a (10 A+7 C) \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4dx-\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 3124 |
| \(\displaystyle \frac {\frac {3}{5} a (10 A+7 C) \int \left (\cos ^4(c+d x) a^4+4 \cos ^3(c+d x) a^4+6 \cos ^2(c+d x) a^4+4 \cos (c+d x) a^4+a^4\right )dx-\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3}{5} a (10 A+7 C) \left (-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {35 a^4 x}{8}\right )-\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}}{6 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}\) |
(C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d) + (-1/5*(a*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/d + (3*a*(10*A + 7*C)*((35*a^4*x)/8 + (8*a^4*Sin[c + d*x])/d + (27*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]^ 3*Sin[c + d*x])/(4*d) - (4*a^4*Sin[c + d*x]^3)/(3*d)))/5)/(6*a)
3.1.30.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTri g[(a + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^ (m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^ m*Simp[A*b*(m + 2) + b*C*(m + 1) - a*C*Sin[e + f*x], x], x], x] /; FreeQ[{a , b, e, f, A, C, m}, x] && !LtQ[m, -1]
Time = 7.90 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.59
| method | result | size |
| parallelrisch | \(\frac {a^{4} \left (\left (56 A +\frac {127 C}{2}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {32 A}{3}+24 C \right ) \sin \left (3 d x +3 c \right )+\left (A +\frac {15 C}{2}\right ) \sin \left (4 d x +4 c \right )+\frac {8 \sin \left (5 d x +5 c \right ) C}{5}+\frac {\sin \left (6 d x +6 c \right ) C}{6}+\left (224 A +176 C \right ) \sin \left (d x +c \right )+140 \left (A +\frac {7 C}{10}\right ) x d \right )}{32 d}\) | \(106\) |
| risch | \(\frac {35 a^{4} x A}{8}+\frac {49 a^{4} C x}{16}+\frac {7 \sin \left (d x +c \right ) a^{4} A}{d}+\frac {11 \sin \left (d x +c \right ) C \,a^{4}}{2 d}+\frac {\sin \left (6 d x +6 c \right ) C \,a^{4}}{192 d}+\frac {\sin \left (5 d x +5 c \right ) C \,a^{4}}{20 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} A}{32 d}+\frac {15 \sin \left (4 d x +4 c \right ) C \,a^{4}}{64 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} A}{3 d}+\frac {3 \sin \left (3 d x +3 c \right ) C \,a^{4}}{4 d}+\frac {7 \sin \left (2 d x +2 c \right ) a^{4} A}{4 d}+\frac {127 \sin \left (2 d x +2 c \right ) C \,a^{4}}{64 d}\) | \(190\) |
| parts | \(a^{4} x A +\frac {\left (a^{4} A +6 C \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (4 a^{4} A +4 C \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (6 a^{4} A +C \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {4 C \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {4 \sin \left (d x +c \right ) a^{4} A}{d}\) | \(230\) |
| derivativedivides | \(\frac {a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+C \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {4 C \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+6 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \sin \left (d x +c \right )+\frac {4 C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{4} A \left (d x +c \right )+C \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(284\) |
| default | \(\frac {a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+C \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {4 C \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+6 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \sin \left (d x +c \right )+\frac {4 C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{4} A \left (d x +c \right )+C \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(284\) |
| norman | \(\frac {\frac {7 a^{4} \left (10 A +7 C \right ) x}{16}+\frac {281 a^{4} \left (10 A +7 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {231 a^{4} \left (10 A +7 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {119 a^{4} \left (10 A +7 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {7 a^{4} \left (10 A +7 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {21 a^{4} \left (10 A +7 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {105 a^{4} \left (10 A +7 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{4} \left (10 A +7 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {105 a^{4} \left (10 A +7 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{4} \left (10 A +7 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {7 a^{4} \left (10 A +7 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {3 a^{4} \left (62 A +69 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{4} \left (2138 A +1471 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(329\) |
1/32*a^4*((56*A+127/2*C)*sin(2*d*x+2*c)+(32/3*A+24*C)*sin(3*d*x+3*c)+(A+15 /2*C)*sin(4*d*x+4*c)+8/5*sin(5*d*x+5*c)*C+1/6*sin(6*d*x+6*c)*C+(224*A+176* C)*sin(d*x+c)+140*(A+7/10*C)*x*d)/d
Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.70 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (10 \, A + 7 \, C\right )} a^{4} d x + {\left (40 \, C a^{4} \cos \left (d x + c\right )^{5} + 192 \, C a^{4} \cos \left (d x + c\right )^{4} + 10 \, {\left (6 \, A + 41 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 64 \, {\left (5 \, A + 9 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (54 \, A + 49 \, C\right )} a^{4} \cos \left (d x + c\right ) + 64 \, {\left (25 \, A + 18 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{240 \, d} \]
1/240*(105*(10*A + 7*C)*a^4*d*x + (40*C*a^4*cos(d*x + c)^5 + 192*C*a^4*cos (d*x + c)^4 + 10*(6*A + 41*C)*a^4*cos(d*x + c)^3 + 64*(5*A + 9*C)*a^4*cos( d*x + c)^2 + 15*(54*A + 49*C)*a^4*cos(d*x + c) + 64*(25*A + 18*C)*a^4)*sin (d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 707 vs. \(2 (167) = 334\).
Time = 0.41 (sec) , antiderivative size = 707, normalized size of antiderivative = 3.95 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {3 A a^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + 3 A a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 A a^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + 3 A a^{4} x \cos ^{2}{\left (c + d x \right )} + A a^{4} x + \frac {3 A a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {8 A a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 A a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {4 A a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 A a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {4 A a^{4} \sin {\left (c + d x \right )}}{d} + \frac {5 C a^{4} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 C a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 C a^{4} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {15 C a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {9 C a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {C a^{4} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {5 C a^{4} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {9 C a^{4} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {C a^{4} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {5 C a^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {32 C a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {5 C a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {16 C a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {9 C a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {8 C a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {11 C a^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {4 C a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {15 C a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac {4 C a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{4} & \text {otherwise} \end {cases} \]
Piecewise((3*A*a**4*x*sin(c + d*x)**4/8 + 3*A*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*A*a**4*x*sin(c + d*x)**2 + 3*A*a**4*x*cos(c + d*x)**4/8 + 3*A*a**4*x*cos(c + d*x)**2 + A*a**4*x + 3*A*a**4*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 8*A*a**4*sin(c + d*x)**3/(3*d) + 5*A*a**4*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 4*A*a**4*sin(c + d*x)*cos(c + d*x)**2/d + 3*A*a**4*sin(c + d*x)*cos(c + d*x)/d + 4*A*a**4*sin(c + d*x)/d + 5*C*a**4*x*sin(c + d*x) **6/16 + 15*C*a**4*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 9*C*a**4*x*sin(c + d*x)**4/4 + 15*C*a**4*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 9*C*a**4*x *sin(c + d*x)**2*cos(c + d*x)**2/2 + C*a**4*x*sin(c + d*x)**2/2 + 5*C*a**4 *x*cos(c + d*x)**6/16 + 9*C*a**4*x*cos(c + d*x)**4/4 + C*a**4*x*cos(c + d* x)**2/2 + 5*C*a**4*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 32*C*a**4*sin(c + d*x)**5/(15*d) + 5*C*a**4*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 16*C*a* *4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 9*C*a**4*sin(c + d*x)**3*cos(c + d*x)/(4*d) + 8*C*a**4*sin(c + d*x)**3/(3*d) + 11*C*a**4*sin(c + d*x)*cos (c + d*x)**5/(16*d) + 4*C*a**4*sin(c + d*x)*cos(c + d*x)**4/d + 15*C*a**4* sin(c + d*x)*cos(c + d*x)**3/(4*d) + 4*C*a**4*sin(c + d*x)*cos(c + d*x)**2 /d + C*a**4*sin(c + d*x)*cos(c + d*x)/(2*d), Ne(d, 0)), (x*(A + C*cos(c)** 2)*(a*cos(c) + a)**4, True))
Time = 0.23 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.53 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1440 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 960 \, {\left (d x + c\right )} A a^{4} - 256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 3840 \, A a^{4} \sin \left (d x + c\right )}{960 \, d} \]
-1/960*(1280*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 - 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 - 1440*(2*d*x + 2*c + sin(2* d*x + 2*c))*A*a^4 - 960*(d*x + c)*A*a^4 - 256*(3*sin(d*x + c)^5 - 10*sin(d *x + c)^3 + 15*sin(d*x + c))*C*a^4 + 5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60 *c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*C*a^4 + 1280*(sin(d*x + c)^ 3 - 3*sin(d*x + c))*C*a^4 - 180*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin( 2*d*x + 2*c))*C*a^4 - 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^4 - 3840*A* a^4*sin(d*x + c))/d
Time = 0.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.88 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {C a^{4} \sin \left (5 \, d x + 5 \, c\right )}{20 \, d} + \frac {7}{16} \, {\left (10 \, A a^{4} + 7 \, C a^{4}\right )} x + \frac {{\left (2 \, A a^{4} + 15 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (4 \, A a^{4} + 9 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (112 \, A a^{4} + 127 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (14 \, A a^{4} + 11 \, C a^{4}\right )} \sin \left (d x + c\right )}{2 \, d} \]
1/192*C*a^4*sin(6*d*x + 6*c)/d + 1/20*C*a^4*sin(5*d*x + 5*c)/d + 7/16*(10* A*a^4 + 7*C*a^4)*x + 1/64*(2*A*a^4 + 15*C*a^4)*sin(4*d*x + 4*c)/d + 1/12*( 4*A*a^4 + 9*C*a^4)*sin(3*d*x + 3*c)/d + 1/64*(112*A*a^4 + 127*C*a^4)*sin(2 *d*x + 2*c)/d + 1/2*(14*A*a^4 + 11*C*a^4)*sin(d*x + c)/d
Time = 1.57 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.77 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (\frac {35\,A\,a^4}{4}+\frac {49\,C\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {595\,A\,a^4}{12}+\frac {833\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {231\,A\,a^4}{2}+\frac {1617\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {281\,A\,a^4}{2}+\frac {1967\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {1069\,A\,a^4}{12}+\frac {1471\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {93\,A\,a^4}{4}+\frac {207\,C\,a^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {7\,a^4\,\left (10\,A+7\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d}+\frac {7\,a^4\,\mathrm {atan}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (10\,A+7\,C\right )}{8\,\left (\frac {35\,A\,a^4}{4}+\frac {49\,C\,a^4}{8}\right )}\right )\,\left (10\,A+7\,C\right )}{8\,d} \]
(tan(c/2 + (d*x)/2)*((93*A*a^4)/4 + (207*C*a^4)/8) + tan(c/2 + (d*x)/2)^11 *((35*A*a^4)/4 + (49*C*a^4)/8) + tan(c/2 + (d*x)/2)^9*((595*A*a^4)/12 + (8 33*C*a^4)/24) + tan(c/2 + (d*x)/2)^7*((231*A*a^4)/2 + (1617*C*a^4)/20) + t an(c/2 + (d*x)/2)^5*((281*A*a^4)/2 + (1967*C*a^4)/20) + tan(c/2 + (d*x)/2) ^3*((1069*A*a^4)/12 + (1471*C*a^4)/24))/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*ta n(c/2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 + 6 *tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) - (7*a^4*(10*A + 7*C) *(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(8*d) + (7*a^4*atan((7*a^4*tan(c/2 + (d*x)/2)*(10*A + 7*C))/(8*((35*A*a^4)/4 + (49*C*a^4)/8)))*(10*A + 7*C))/ (8*d)