\(\int (a+a \cos (c+d x))^4 (A+C \cos ^2(c+d x)) \, dx\) [30]

Optimal result

Integrand size = 25, antiderivative size = 179 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {7}{16} a^4 (10 A+7 C) x+\frac {4 a^4 (10 A+7 C) \sin (c+d x)}{5 d}+\frac {27 a^4 (10 A+7 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (10 A+7 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}-\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}-\frac {2 a^4 (10 A+7 C) \sin ^3(c+d x)}{15 d} \]

[Out]

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*(1
0*A+7*C)*cos(d*x+c)^3*sin(d*x+c)/d-1/30*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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3103, 2830, 2724, 2717, 2715, 8, 2713} \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 a^4 (10 A+7 C) \sin ^3(c+d x)}{15 d}+\frac {4 a^4 (10 A+7 C) \sin (c+d x)}{5 d}+\frac {a^4 (10 A+7 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {27 a^4 (10 A+7 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac {7}{16} a^4 x (10 A+7 C)+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}-\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d} \]

[In]

Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]

[Out]

(7*a^4*(10*A + 7*C)*x)/16 + (4*a^4*(10*A + 7*C)*Sin[c + d*x])/(5*d) + (27*a^4*(10*A + 7*C)*Cos[c + d*x]*Sin[c
+ d*x])/(80*d) + (a^4*(10*A + 7*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) - (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x
])/(30*d) + (C*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d) - (2*a^4*(10*A + 7*C)*Sin[c + d*x]^3)/(15*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2724

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTrig[(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]

Rule 2830

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] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S
in[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)]

Rule 3103

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] + Dist[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] &&  !Lt
Q[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {C (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac {\int (a+a \cos (c+d x))^4 (a (6 A+5 C)-a C \cos (c+d x)) \, dx}{6 a} \\ & = -\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 {1}{10} (10 A+7 C) \int (a+a \cos (c+d x))^4 \, dx \\ & = -\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 {1}{10} (10 A+7 C) \int \left (a^4+4 a^4 \cos (c+d x)+6 a^4 \cos ^2(c+d x)+4 a^4 \cos ^3(c+d x)+a^4 \cos ^4(c+d x)\right ) \, dx \\ & = \frac {1}{10} a^4 (10 A+7 C) x-\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 {1}{10} \left (a^4 (10 A+7 C)\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (10 A+7 C)\right ) \int \cos (c+d x) \, dx+\frac {1}{5} \left (2 a^4 (10 A+7 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{5} \left (3 a^4 (10 A+7 C)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {1}{10} a^4 (10 A+7 C) x+\frac {2 a^4 (10 A+7 C) \sin (c+d x)}{5 d}+\frac {3 a^4 (10 A+7 C) \cos (c+d x) \sin (c+d x)}{10 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 {1}{40} \left (3 a^4 (10 A+7 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{10} \left (3 a^4 (10 A+7 C)\right ) \int 1 \, dx-\frac {\left (2 a^4 (10 A+7 C)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {2}{5} 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}+\frac {1}{80} \left (3 a^4 (10 A+7 C)\right ) \int 1 \, 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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.66 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^4 (4200 A d x+2940 C d x+480 (14 A+11 C) \sin (c+d x)+15 (112 A+127 C) \sin (2 (c+d x))+320 A \sin (3 (c+d x))+720 C \sin (3 (c+d x))+30 A \sin (4 (c+d x))+225 C \sin (4 (c+d x))+48 C \sin (5 (c+d x))+5 C \sin (6 (c+d x)))}{960 d} \]

[In]

Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]

[Out]

(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*S
in[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)

Maple [A] (verified)

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	\(\frac {\frac {7 a^{4} \left (10 A +7 C \right ) x}{16}+\frac {281 a^{4} \left (10 A +7 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {231 a^{4} \left (10 A +7 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {119 a^{4} \left (10 A +7 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {7 a^{4} \left (10 A +7 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {21 a^{4} \left (10 A +7 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {105 a^{4} \left (10 A +7 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{4} \left (10 A +7 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {105 a^{4} \left (10 A +7 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{4} \left (10 A +7 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {7 a^{4} \left (10 A +7 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {3 a^{4} \left (62 A +69 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{4} \left (2138 A +1471 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\)	\(329\)

[In]

int((a+cos(d*x+c)*a)^4*(A+C*cos(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.70 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (10 \, A + 7 \, C\right )} a^{4} d x + {\left (40 \, C a^{4} \cos \left (d x + c\right )^{5} + 192 \, C a^{4} \cos \left (d x + c\right )^{4} + 10 \, {\left (6 \, A + 41 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 64 \, {\left (5 \, A + 9 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (54 \, A + 49 \, C\right )} a^{4} \cos \left (d x + c\right ) + 64 \, {\left (25 \, A + 18 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{240 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

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*co
s(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)*s
in(d*x + c))/d

Sympy [B] (verification not implemented)

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} \]

[In]

integrate((a+a*cos(d*x+c))**4*(A+C*cos(d*x+c)**2),x)

[Out]

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)*c
os(c + d*x)/(2*d), Ne(d, 0)), (x*(A + C*cos(c)**2)*(a*cos(c) + a)**4, True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.53 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1440 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 960 \, {\left (d x + c\right )} A a^{4} - 256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 3840 \, A a^{4} \sin \left (d x + c\right )}{960 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

-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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.88 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {C a^{4} \sin \left (5 \, d x + 5 \, c\right )}{20 \, d} + \frac {7}{16} \, {\left (10 \, A a^{4} + 7 \, C a^{4}\right )} x + \frac {{\left (2 \, A a^{4} + 15 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (4 \, A a^{4} + 9 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (112 \, A a^{4} + 127 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (14 \, A a^{4} + 11 \, C a^{4}\right )} \sin \left (d x + c\right )}{2 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.57 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.77 \[ \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (\frac {35\,A\,a^4}{4}+\frac {49\,C\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {595\,A\,a^4}{12}+\frac {833\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {231\,A\,a^4}{2}+\frac {1617\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {281\,A\,a^4}{2}+\frac {1967\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {1069\,A\,a^4}{12}+\frac {1471\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {93\,A\,a^4}{4}+\frac {207\,C\,a^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {7\,a^4\,\left (10\,A+7\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d}+\frac {7\,a^4\,\mathrm {atan}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (10\,A+7\,C\right )}{8\,\left (\frac {35\,A\,a^4}{4}+\frac {49\,C\,a^4}{8}\right )}\right )\,\left (10\,A+7\,C\right )}{8\,d} \]

[In]

int((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^4,x)

[Out]

(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 + (833*C*a^4)/24) + tan(c/2 + (d*x)/2)^7*((231*A*a^4)/2 + (1617*C*a^4)/20) +
tan(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*tan(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)