3.330 \(\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=200 \[ -\frac {2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d}+\frac {4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac {a^4 (10 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {27 a^4 (10 A+8 B+7 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac {7}{16} a^4 x (10 A+8 B+7 C)+\frac {(6 B-C) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d} \]

[Out]

7/16*a^4*(10*A+8*B+7*C)*x+4/5*a^4*(10*A+8*B+7*C)*sin(d*x+c)/d+27/80*a^4*(10*A+8*B+7*C)*cos(d*x+c)*sin(d*x+c)/d
+1/40*a^4*(10*A+8*B+7*C)*cos(d*x+c)^3*sin(d*x+c)/d+1/30*(6*B-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+8*B+7*C)*sin(d*x+c)^3/d

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Rubi [A]  time = 0.28, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3023, 2751, 2645, 2637, 2635, 8, 2633} \[ -\frac {2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d}+\frac {4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac {a^4 (10 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {27 a^4 (10 A+8 B+7 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac {7}{16} a^4 x (10 A+8 B+7 C)+\frac {(6 B-C) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*a^4*(10*A + 8*B + 7*C)*x)/16 + (4*a^4*(10*A + 8*B + 7*C)*Sin[c + d*x])/(5*d) + (27*a^4*(10*A + 8*B + 7*C)*C
os[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(10*A + 8*B + 7*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + ((6*B - 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 + 8
*B + 7*C)*Sin[c + d*x]^3)/(15*d)

Rule 8

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

Rule 2633

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 2635

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] && Integer
Q[2*n]

Rule 2637

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

Rule 2645

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 2751

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*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)]

Rule 3023

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

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Mathematica [A]  time = 0.54, size = 163, normalized size = 0.82 \[ \frac {a^4 (120 (56 A+49 B+44 C) \sin (c+d x)+15 (112 A+128 B+127 C) \sin (2 (c+d x))+320 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+4200 A d x+580 B \sin (3 (c+d x))+120 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+3360 B d x+720 C \sin (3 (c+d x))+225 C \sin (4 (c+d x))+48 C \sin (5 (c+d x))+5 C \sin (6 (c+d x))+2940 C d x)}{960 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(4200*A*d*x + 3360*B*d*x + 2940*C*d*x + 120*(56*A + 49*B + 44*C)*Sin[c + d*x] + 15*(112*A + 128*B + 127*C
)*Sin[2*(c + d*x)] + 320*A*Sin[3*(c + d*x)] + 580*B*Sin[3*(c + d*x)] + 720*C*Sin[3*(c + d*x)] + 30*A*Sin[4*(c
+ d*x)] + 120*B*Sin[4*(c + d*x)] + 225*C*Sin[4*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 48*C*Sin[5*(c + d*x)] + 5*
C*Sin[6*(c + d*x)]))/(960*d)

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fricas [A]  time = 0.45, size = 145, normalized size = 0.72 \[ \frac {105 \, {\left (10 \, A + 8 \, B + 7 \, C\right )} a^{4} d x + {\left (40 \, C a^{4} \cos \left (d x + c\right )^{5} + 48 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 10 \, {\left (6 \, A + 24 \, B + 41 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 32 \, {\left (10 \, A + 17 \, B + 18 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (54 \, A + 56 \, B + 49 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \, {\left (100 \, A + 83 \, B + 72 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{240 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(105*(10*A + 8*B + 7*C)*a^4*d*x + (40*C*a^4*cos(d*x + c)^5 + 48*(B + 4*C)*a^4*cos(d*x + c)^4 + 10*(6*A +
24*B + 41*C)*a^4*cos(d*x + c)^3 + 32*(10*A + 17*B + 18*C)*a^4*cos(d*x + c)^2 + 15*(54*A + 56*B + 49*C)*a^4*co
s(d*x + c) + 16*(100*A + 83*B + 72*C)*a^4)*sin(d*x + c))/d

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giac [A]  time = 0.55, size = 196, normalized size = 0.98 \[ \frac {C a^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {7}{16} \, {\left (10 \, A a^{4} + 8 \, B a^{4} + 7 \, C a^{4}\right )} x + \frac {{\left (B a^{4} + 4 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (2 \, A a^{4} + 8 \, B a^{4} + 15 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (16 \, A a^{4} + 29 \, B a^{4} + 36 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (112 \, A a^{4} + 128 \, B a^{4} + 127 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (56 \, A a^{4} + 49 \, B a^{4} + 44 \, C a^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*C*a^4*sin(6*d*x + 6*c)/d + 7/16*(10*A*a^4 + 8*B*a^4 + 7*C*a^4)*x + 1/80*(B*a^4 + 4*C*a^4)*sin(5*d*x + 5*
c)/d + 1/64*(2*A*a^4 + 8*B*a^4 + 15*C*a^4)*sin(4*d*x + 4*c)/d + 1/48*(16*A*a^4 + 29*B*a^4 + 36*C*a^4)*sin(3*d*
x + 3*c)/d + 1/64*(112*A*a^4 + 128*B*a^4 + 127*C*a^4)*sin(2*d*x + 2*c)/d + 1/8*(56*A*a^4 + 49*B*a^4 + 44*C*a^4
)*sin(d*x + c)/d

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maple [B]  time = 0.31, size = 416, normalized size = 2.08 \[ \frac {a^{4} C \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{4} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 a^{4} C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 a^{4} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {4 a^{4} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+6 A \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 A \,a^{4} \sin \left (d x +c \right )+a^{4} B \sin \left (d x +c \right )+A \,a^{4} \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^4*C*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+1/5*a^4*B*(8/3+cos
(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+4/5*a^4*C*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+A*a^4*(1/4*(co
s(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4*a^4*B*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8
*d*x+3/8*c)+6*a^4*C*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4/3*A*a^4*(2+cos(d*x+c)^2)*si
n(d*x+c)+2*a^4*B*(2+cos(d*x+c)^2)*sin(d*x+c)+4/3*a^4*C*(2+cos(d*x+c)^2)*sin(d*x+c)+6*A*a^4*(1/2*cos(d*x+c)*sin
(d*x+c)+1/2*d*x+1/2*c)+4*a^4*B*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+a^4*C*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*
d*x+1/2*c)+4*A*a^4*sin(d*x+c)+a^4*B*sin(d*x+c)+A*a^4*(d*x+c))

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maxima [B]  time = 0.34, size = 400, normalized size = 2.00 \[ -\frac {1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1440 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 960 \, {\left (d x + c\right )} A a^{4} - 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} + 1920 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 120 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 960 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 3840 \, A a^{4} \sin \left (d x + c\right ) - 960 \, B a^{4} \sin \left (d x + c\right )}{960 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+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 - 64*(3*sin(d*x + c)^5 - 10*sin(
d*x + c)^3 + 15*sin(d*x + c))*B*a^4 + 1920*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^4 - 120*(12*d*x + 12*c + sin(
4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 - 960*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*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) - 960*B
*a^4*sin(d*x + c))/d

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mupad [B]  time = 4.24, size = 334, normalized size = 1.67 \[ \frac {\left (\frac {35\,A\,a^4}{4}+7\,B\,a^4+\frac {49\,C\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {595\,A\,a^4}{12}+\frac {119\,B\,a^4}{3}+\frac {833\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {231\,A\,a^4}{2}+\frac {462\,B\,a^4}{5}+\frac {1617\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {281\,A\,a^4}{2}+\frac {562\,B\,a^4}{5}+\frac {1967\,C\,a^4}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {1069\,A\,a^4}{12}+\frac {233\,B\,a^4}{3}+\frac {1471\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {93\,A\,a^4}{4}+25\,B\,a^4+\frac {207\,C\,a^4}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atan}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (10\,A+8\,B+7\,C\right )}{8\,\left (\frac {35\,A\,a^4}{4}+7\,B\,a^4+\frac {49\,C\,a^4}{8}\right )}\right )\,\left (10\,A+8\,B+7\,C\right )}{8\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(c/2 + (d*x)/2)^11*((35*A*a^4)/4 + 7*B*a^4 + (49*C*a^4)/8) + tan(c/2 + (d*x)/2)^9*((595*A*a^4)/12 + (119*B
*a^4)/3 + (833*C*a^4)/24) + tan(c/2 + (d*x)/2)^7*((231*A*a^4)/2 + (462*B*a^4)/5 + (1617*C*a^4)/20) + tan(c/2 +
(d*x)/2)^3*((1069*A*a^4)/12 + (233*B*a^4)/3 + (1471*C*a^4)/24) + tan(c/2 + (d*x)/2)^5*((281*A*a^4)/2 + (562*B
*a^4)/5 + (1967*C*a^4)/20) + tan(c/2 + (d*x)/2)*((93*A*a^4)/4 + 25*B*a^4 + (207*C*a^4)/8))/(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)^1
0 + tan(c/2 + (d*x)/2)^12 + 1)) + (7*a^4*atan((7*a^4*tan(c/2 + (d*x)/2)*(10*A + 8*B + 7*C))/(8*((35*A*a^4)/4 +
7*B*a^4 + (49*C*a^4)/8)))*(10*A + 8*B + 7*C))/(8*d)

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sympy [A]  time = 5.34, size = 1005, normalized size = 5.02 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))**4*(A+B*cos(d*x+c)+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 + 3*B*a**4*x*sin(c +
d*x)**4/2 + 3*B*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2 + 2*B*a**4*x*sin(c + d*x)**2 + 3*B*a**4*x*cos(c + d*x)*
*4/2 + 2*B*a**4*x*cos(c + d*x)**2 + 8*B*a**4*sin(c + d*x)**5/(15*d) + 4*B*a**4*sin(c + d*x)**3*cos(c + d*x)**2
/(3*d) + 3*B*a**4*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 4*B*a**4*sin(c + d*x)**3/d + B*a**4*sin(c + d*x)*cos(c
+ d*x)**4/d + 5*B*a**4*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 6*B*a**4*sin(c + d*x)*cos(c + d*x)**2/d + 2*B*a**4
*sin(c + d*x)*cos(c + d*x)/d + B*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*
cos(c) + a)**4*(A + B*cos(c) + C*cos(c)**2), True))

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