\(\int \cos (c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx\) [29]

Optimal result

Integrand size = 29, antiderivative size = 185 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {7}{16} a^4 (8 A+7 B) x+\frac {4 a^4 (8 A+7 B) \sin (c+d x)}{5 d}+\frac {27 a^4 (8 A+7 B) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (8 A+7 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(6 A-B) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {B (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}-\frac {2 a^4 (8 A+7 B) \sin ^3(c+d x)}{15 d} \]

[Out]

7/16*a^4*(8*A+7*B)*x+4/5*a^4*(8*A+7*B)*sin(d*x+c)/d+27/80*a^4*(8*A+7*B)*cos(d*x+c)*sin(d*x+c)/d+1/40*a^4*(8*A+
7*B)*cos(d*x+c)^3*sin(d*x+c)/d+1/30*(6*A-B)*(a+a*cos(d*x+c))^4*sin(d*x+c)/d+1/6*B*(a+a*cos(d*x+c))^5*sin(d*x+c
)/a/d-2/15*a^4*(8*A+7*B)*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {3047, 3102, 2830, 2724, 2717, 2715, 8, 2713} \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=-\frac {2 a^4 (8 A+7 B) \sin ^3(c+d x)}{15 d}+\frac {4 a^4 (8 A+7 B) \sin (c+d x)}{5 d}+\frac {a^4 (8 A+7 B) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {27 a^4 (8 A+7 B) \sin (c+d x) \cos (c+d x)}{80 d}+\frac {7}{16} a^4 x (8 A+7 B)+\frac {(6 A-B) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac {B \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d} \]

[In]

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

[Out]

(7*a^4*(8*A + 7*B)*x)/16 + (4*a^4*(8*A + 7*B)*Sin[c + d*x])/(5*d) + (27*a^4*(8*A + 7*B)*Cos[c + d*x]*Sin[c + d
*x])/(80*d) + (a^4*(8*A + 7*B)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + ((6*A - B)*(a + a*Cos[c + d*x])^4*Sin[c +
d*x])/(30*d) + (B*(a + a*Cos[c + d*x])^5*Sin[c + d*x])/(6*a*d) - (2*a^4*(8*A + 7*B)*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 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (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) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \int (a+a \cos (c+d x))^4 \left (A \cos (c+d x)+B \cos ^2(c+d x)\right ) \, dx \\ & = \frac {B (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac {\int (a+a \cos (c+d x))^4 (5 a B+a (6 A-B) \cos (c+d x)) \, dx}{6 a} \\ & = \frac {(6 A-B) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {B (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac {1}{10} (8 A+7 B) \int (a+a \cos (c+d x))^4 \, dx \\ & = \frac {(6 A-B) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {B (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac {1}{10} (8 A+7 B) \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 (8 A+7 B) x+\frac {(6 A-B) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {B (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac {1}{10} \left (a^4 (8 A+7 B)\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (8 A+7 B)\right ) \int \cos (c+d x) \, dx+\frac {1}{5} \left (2 a^4 (8 A+7 B)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{5} \left (3 a^4 (8 A+7 B)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {1}{10} a^4 (8 A+7 B) x+\frac {2 a^4 (8 A+7 B) \sin (c+d x)}{5 d}+\frac {3 a^4 (8 A+7 B) \cos (c+d x) \sin (c+d x)}{10 d}+\frac {a^4 (8 A+7 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(6 A-B) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {B (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}+\frac {1}{40} \left (3 a^4 (8 A+7 B)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{10} \left (3 a^4 (8 A+7 B)\right ) \int 1 \, dx-\frac {\left (2 a^4 (8 A+7 B)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {2}{5} a^4 (8 A+7 B) x+\frac {4 a^4 (8 A+7 B) \sin (c+d x)}{5 d}+\frac {27 a^4 (8 A+7 B) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (8 A+7 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(6 A-B) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {B (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}-\frac {2 a^4 (8 A+7 B) \sin ^3(c+d x)}{15 d}+\frac {1}{80} \left (3 a^4 (8 A+7 B)\right ) \int 1 \, dx \\ & = \frac {7}{16} a^4 (8 A+7 B) x+\frac {4 a^4 (8 A+7 B) \sin (c+d x)}{5 d}+\frac {27 a^4 (8 A+7 B) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (8 A+7 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(6 A-B) (a+a \cos (c+d x))^4 \sin (c+d x)}{30 d}+\frac {B (a+a \cos (c+d x))^5 \sin (c+d x)}{6 a d}-\frac {2 a^4 (8 A+7 B) \sin ^3(c+d x)}{15 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.72 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {a^4 (2940 B c+3360 A d x+2940 B d x+120 (49 A+44 B) \sin (c+d x)+15 (128 A+127 B) \sin (2 (c+d x))+580 A \sin (3 (c+d x))+720 B \sin (3 (c+d x))+120 A \sin (4 (c+d x))+225 B \sin (4 (c+d x))+12 A \sin (5 (c+d x))+48 B \sin (5 (c+d x))+5 B \sin (6 (c+d x)))}{960 d} \]

[In]

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

[Out]

(a^4*(2940*B*c + 3360*A*d*x + 2940*B*d*x + 120*(49*A + 44*B)*Sin[c + d*x] + 15*(128*A + 127*B)*Sin[2*(c + d*x)
] + 580*A*Sin[3*(c + d*x)] + 720*B*Sin[3*(c + d*x)] + 120*A*Sin[4*(c + d*x)] + 225*B*Sin[4*(c + d*x)] + 12*A*S
in[5*(c + d*x)] + 48*B*Sin[5*(c + d*x)] + 5*B*Sin[6*(c + d*x)]))/(960*d)

Maple [A] (verified)

Time = 4.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.60

method	result	size
parallelrisch	\(\frac {\left (\left (16 A +\frac {127 B}{8}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {29 A}{6}+6 B \right ) \sin \left (3 d x +3 c \right )+\left (A +\frac {15 B}{8}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {A}{10}+\frac {2 B}{5}\right ) \sin \left (5 d x +5 c \right )+\frac {B \sin \left (6 d x +6 c \right )}{24}+\left (49 A +44 B \right ) \sin \left (d x +c \right )+28 d x \left (A +\frac {7 B}{8}\right )\right ) a^{4}}{8 d}\)	\(111\)
risch	\(\frac {7 a^{4} x A}{2}+\frac {49 a^{4} B x}{16}+\frac {49 \sin \left (d x +c \right ) a^{4} A}{8 d}+\frac {11 \sin \left (d x +c \right ) B \,a^{4}}{2 d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{4}}{192 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} A}{80 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{4}}{20 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} A}{8 d}+\frac {15 \sin \left (4 d x +4 c \right ) B \,a^{4}}{64 d}+\frac {29 \sin \left (3 d x +3 c \right ) a^{4} A}{48 d}+\frac {3 \sin \left (3 d x +3 c \right ) B \,a^{4}}{4 d}+\frac {2 \sin \left (2 d x +2 c \right ) a^{4} A}{d}+\frac {127 \sin \left (2 d x +2 c \right ) B \,a^{4}}{64 d}\)	\(208\)
parts	\(\frac {\left (a^{4} A +4 B \,a^{4}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {B \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {\sin \left (d x +c \right ) a^{4} A}{d}\)	\(232\)
derivativedivides	\(\frac {\frac {a^{4} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+2 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{4} A \sin \left (d x +c \right )+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\)	\(306\)
default	\(\frac {\frac {a^{4} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+2 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{4} A \sin \left (d x +c \right )+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\)	\(306\)
norman	\(\frac {\frac {7 a^{4} \left (8 A +7 B \right ) x}{16}+\frac {281 a^{4} \left (8 A +7 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {231 a^{4} \left (8 A +7 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {119 a^{4} \left (8 A +7 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {7 a^{4} \left (8 A +7 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {21 a^{4} \left (8 A +7 B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {105 a^{4} \left (8 A +7 B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{4} \left (8 A +7 B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {105 a^{4} \left (8 A +7 B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{4} \left (8 A +7 B \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {7 a^{4} \left (8 A +7 B \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{4} \left (200 A +207 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{4} \left (1864 A +1471 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\)	\(329\)

[In]

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

[Out]

1/8*((16*A+127/8*B)*sin(2*d*x+2*c)+(29/6*A+6*B)*sin(3*d*x+3*c)+(A+15/8*B)*sin(4*d*x+4*c)+(1/10*A+2/5*B)*sin(5*
d*x+5*c)+1/24*B*sin(6*d*x+6*c)+(49*A+44*B)*sin(d*x+c)+28*d*x*(A+7/8*B))*a^4/d

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.70 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {105 \, {\left (8 \, A + 7 \, B\right )} a^{4} d x + {\left (40 \, B a^{4} \cos \left (d x + c\right )^{5} + 48 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 10 \, {\left (24 \, A + 41 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 32 \, {\left (17 \, A + 18 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (8 \, A + 7 \, B\right )} a^{4} \cos \left (d x + c\right ) + 16 \, {\left (83 \, A + 72 \, B\right )} a^{4}\right )} \sin \left (d x + c\right )}{240 \, d} \]

[In]

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

[Out]

1/240*(105*(8*A + 7*B)*a^4*d*x + (40*B*a^4*cos(d*x + c)^5 + 48*(A + 4*B)*a^4*cos(d*x + c)^4 + 10*(24*A + 41*B)
*a^4*cos(d*x + c)^3 + 32*(17*A + 18*B)*a^4*cos(d*x + c)^2 + 105*(8*A + 7*B)*a^4*cos(d*x + c) + 16*(83*A + 72*B
)*a^4)*sin(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (170) = 340\).

Time = 0.43 (sec) , antiderivative size = 765, normalized size of antiderivative = 4.14 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\begin {cases} \frac {3 A a^{4} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 A a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + 2 A a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 A a^{4} x \cos ^{4}{\left (c + d x \right )}}{2} + 2 A a^{4} x \cos ^{2}{\left (c + d x \right )} + \frac {8 A a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 A a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 A a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {4 A a^{4} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {A a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 A a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {6 A a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 A a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {A a^{4} \sin {\left (c + d x \right )}}{d} + \frac {5 B a^{4} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 B a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 B a^{4} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {15 B a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {9 B a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {B a^{4} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {5 B a^{4} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {9 B a^{4} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {B a^{4} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {5 B a^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {32 B a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {5 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {16 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {9 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {8 B a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {11 B a^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {4 B a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {15 B a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac {4 B a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {B a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{4} \cos {\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((3*A*a**4*x*sin(c + d*x)**4/2 + 3*A*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2 + 2*A*a**4*x*sin(c + d*x)
**2 + 3*A*a**4*x*cos(c + d*x)**4/2 + 2*A*a**4*x*cos(c + d*x)**2 + 8*A*a**4*sin(c + d*x)**5/(15*d) + 4*A*a**4*s
in(c + d*x)**3*cos(c + d*x)**2/(3*d) + 3*A*a**4*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 4*A*a**4*sin(c + d*x)**3/
d + A*a**4*sin(c + d*x)*cos(c + d*x)**4/d + 5*A*a**4*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 6*A*a**4*sin(c + d*x
)*cos(c + d*x)**2/d + 2*A*a**4*sin(c + d*x)*cos(c + d*x)/d + A*a**4*sin(c + d*x)/d + 5*B*a**4*x*sin(c + d*x)**
6/16 + 15*B*a**4*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 9*B*a**4*x*sin(c + d*x)**4/4 + 15*B*a**4*x*sin(c + d*x
)**2*cos(c + d*x)**4/16 + 9*B*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2/2 + B*a**4*x*sin(c + d*x)**2/2 + 5*B*a**4
*x*cos(c + d*x)**6/16 + 9*B*a**4*x*cos(c + d*x)**4/4 + B*a**4*x*cos(c + d*x)**2/2 + 5*B*a**4*sin(c + d*x)**5*c
os(c + d*x)/(16*d) + 32*B*a**4*sin(c + d*x)**5/(15*d) + 5*B*a**4*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 16*B*
a**4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 9*B*a**4*sin(c + d*x)**3*cos(c + d*x)/(4*d) + 8*B*a**4*sin(c + d*
x)**3/(3*d) + 11*B*a**4*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 4*B*a**4*sin(c + d*x)*cos(c + d*x)**4/d + 15*B*a
**4*sin(c + d*x)*cos(c + d*x)**3/(4*d) + 4*B*a**4*sin(c + d*x)*cos(c + d*x)**2/d + B*a**4*sin(c + d*x)*cos(c +
d*x)/(2*d), Ne(d, 0)), (x*(A + B*cos(c))*(a*cos(c) + a)**4*cos(c), True))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.61 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 1920 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 120 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 960 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 960 \, A a^{4} \sin \left (d x + c\right )}{960 \, d} \]

[In]

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

[Out]

1/960*(64*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 - 1920*(sin(d*x + c)^3 - 3*sin(d*x +
c))*A*a^4 + 120*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 960*(2*d*x + 2*c + sin(2*d*x +
2*c))*A*a^4 + 256*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*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))*B*a^4 - 1280*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^4
+ 180*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 + 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*
a^4 + 960*A*a^4*sin(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.90 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {B a^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {7}{16} \, {\left (8 \, A a^{4} + 7 \, B a^{4}\right )} x + \frac {{\left (A a^{4} + 4 \, B a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (8 \, A a^{4} + 15 \, B a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (29 \, A a^{4} + 36 \, B a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (128 \, A a^{4} + 127 \, B a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (49 \, A a^{4} + 44 \, B a^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \]

[In]

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

[Out]

1/192*B*a^4*sin(6*d*x + 6*c)/d + 7/16*(8*A*a^4 + 7*B*a^4)*x + 1/80*(A*a^4 + 4*B*a^4)*sin(5*d*x + 5*c)/d + 1/64
*(8*A*a^4 + 15*B*a^4)*sin(4*d*x + 4*c)/d + 1/48*(29*A*a^4 + 36*B*a^4)*sin(3*d*x + 3*c)/d + 1/64*(128*A*a^4 + 1
27*B*a^4)*sin(2*d*x + 2*c)/d + 1/8*(49*A*a^4 + 44*B*a^4)*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 1.70 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.71 \[ \int \cos (c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {\left (7\,A\,a^4+\frac {49\,B\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {119\,A\,a^4}{3}+\frac {833\,B\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {462\,A\,a^4}{5}+\frac {1617\,B\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {562\,A\,a^4}{5}+\frac {1967\,B\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {233\,A\,a^4}{3}+\frac {1471\,B\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {207\,B\,a^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {7\,a^4\,\left (8\,A+7\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d}+\frac {7\,a^4\,\mathrm {atan}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,A+7\,B\right )}{8\,\left (7\,A\,a^4+\frac {49\,B\,a^4}{8}\right )}\right )\,\left (8\,A+7\,B\right )}{8\,d} \]

[In]

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

[Out]

(tan(c/2 + (d*x)/2)*(25*A*a^4 + (207*B*a^4)/8) + tan(c/2 + (d*x)/2)^11*(7*A*a^4 + (49*B*a^4)/8) + tan(c/2 + (d
*x)/2)^9*((119*A*a^4)/3 + (833*B*a^4)/24) + tan(c/2 + (d*x)/2)^3*((233*A*a^4)/3 + (1471*B*a^4)/24) + tan(c/2 +
(d*x)/2)^7*((462*A*a^4)/5 + (1617*B*a^4)/20) + tan(c/2 + (d*x)/2)^5*((562*A*a^4)/5 + (1967*B*a^4)/20))/(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*(8*A + 7*B)*(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)*(8*A + 7*B))/(8*(7*A*a^4 + (49*B*a^4)/8)))*(8*A + 7*B))/(8*d)