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

Optimal. Leaf size=219 \[ -\frac {8 a^4 (14 A+11 C) \sin ^3(c+d x)}{105 d}+\frac {16 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac {a^4 (14 A+11 C) \sin (c+d x) \cos ^3(c+d x)}{70 d}+\frac {27 a^4 (14 A+11 C) \sin (c+d x) \cos (c+d x)}{140 d}+\frac {1}{4} a^4 x (14 A+11 C)+\frac {(21 A+4 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{105 d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{21 a d} \]

[Out]

1/4*a^4*(14*A+11*C)*x+16/35*a^4*(14*A+11*C)*sin(d*x+c)/d+27/140*a^4*(14*A+11*C)*cos(d*x+c)*sin(d*x+c)/d+1/70*a
^4*(14*A+11*C)*cos(d*x+c)^3*sin(d*x+c)/d+1/105*(21*A+4*C)*(a+a*cos(d*x+c))^4*sin(d*x+c)/d+1/7*C*cos(d*x+c)^2*(
a+a*cos(d*x+c))^4*sin(d*x+c)/d+2/21*C*(a+a*cos(d*x+c))^5*sin(d*x+c)/a/d-8/105*a^4*(14*A+11*C)*sin(d*x+c)^3/d

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Rubi [A]  time = 0.41, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {3046, 2968, 3023, 2751, 2645, 2637, 2635, 8, 2633} \[ -\frac {8 a^4 (14 A+11 C) \sin ^3(c+d x)}{105 d}+\frac {16 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac {a^4 (14 A+11 C) \sin (c+d x) \cos ^3(c+d x)}{70 d}+\frac {27 a^4 (14 A+11 C) \sin (c+d x) \cos (c+d x)}{140 d}+\frac {1}{4} a^4 x (14 A+11 C)+\frac {(21 A+4 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{105 d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{21 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(14*A + 11*C)*x)/4 + (16*a^4*(14*A + 11*C)*Sin[c + d*x])/(35*d) + (27*a^4*(14*A + 11*C)*Cos[c + d*x]*Sin[
c + d*x])/(140*d) + (a^4*(14*A + 11*C)*Cos[c + d*x]^3*Sin[c + d*x])/(70*d) + ((21*A + 4*C)*(a + a*Cos[c + d*x]
)^4*Sin[c + d*x])/(105*d) + (C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d) + (2*C*(a + a*Cos[c +
d*x])^5*Sin[c + d*x])/(21*a*d) - (8*a^4*(14*A + 11*C)*Sin[c + d*x]^3)/(105*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 2968

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 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]

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*
sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n
+ 1))/(d*f*(m + n + 2)), x] + Dist[1/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp
[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b
, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-
1)] && NeQ[m + n + 2, 0]

Rubi steps

\begin {align*} \int \cos (c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {\int \cos (c+d x) (a+a \cos (c+d x))^4 (a (7 A+2 C)+4 a C \cos (c+d x)) \, dx}{7 a}\\ &=\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {\int (a+a \cos (c+d x))^4 \left (a (7 A+2 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right ) \, dx}{7 a}\\ &=\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}+\frac {\int (a+a \cos (c+d x))^4 \left (20 a^2 C+2 a^2 (21 A+4 C) \cos (c+d x)\right ) \, dx}{42 a^2}\\ &=\frac {(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}+\frac {1}{35} (2 (14 A+11 C)) \int (a+a \cos (c+d x))^4 \, dx\\ &=\frac {(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}+\frac {1}{35} (2 (14 A+11 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 {2}{35} a^4 (14 A+11 C) x+\frac {(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}+\frac {1}{35} \left (2 a^4 (14 A+11 C)\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{35} \left (8 a^4 (14 A+11 C)\right ) \int \cos (c+d x) \, dx+\frac {1}{35} \left (8 a^4 (14 A+11 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{35} \left (12 a^4 (14 A+11 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {2}{35} a^4 (14 A+11 C) x+\frac {8 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac {6 a^4 (14 A+11 C) \cos (c+d x) \sin (c+d x)}{35 d}+\frac {a^4 (14 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}+\frac {1}{70} \left (3 a^4 (14 A+11 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{35} \left (6 a^4 (14 A+11 C)\right ) \int 1 \, dx-\frac {\left (8 a^4 (14 A+11 C)\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac {8}{35} a^4 (14 A+11 C) x+\frac {16 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac {27 a^4 (14 A+11 C) \cos (c+d x) \sin (c+d x)}{140 d}+\frac {a^4 (14 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}-\frac {8 a^4 (14 A+11 C) \sin ^3(c+d x)}{105 d}+\frac {1}{140} \left (3 a^4 (14 A+11 C)\right ) \int 1 \, dx\\ &=\frac {1}{4} a^4 (14 A+11 C) x+\frac {16 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac {27 a^4 (14 A+11 C) \cos (c+d x) \sin (c+d x)}{140 d}+\frac {a^4 (14 A+11 C) \cos ^3(c+d x) \sin (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{105 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {2 C (a+a \cos (c+d x))^5 \sin (c+d x)}{21 a d}-\frac {8 a^4 (14 A+11 C) \sin ^3(c+d x)}{105 d}\\ \end {align*}

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Mathematica [A]  time = 0.59, size = 145, normalized size = 0.66 \[ \frac {a^4 (105 (392 A+323 C) \sin (c+d x)+420 (32 A+31 C) \sin (2 (c+d x))+4060 A \sin (3 (c+d x))+840 A \sin (4 (c+d x))+84 A \sin (5 (c+d x))+23520 A d x+5495 C \sin (3 (c+d x))+2100 C \sin (4 (c+d x))+651 C \sin (5 (c+d x))+140 C \sin (6 (c+d x))+15 C \sin (7 (c+d x))+11760 c C+18480 C d x)}{6720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(11760*c*C + 23520*A*d*x + 18480*C*d*x + 105*(392*A + 323*C)*Sin[c + d*x] + 420*(32*A + 31*C)*Sin[2*(c +
d*x)] + 4060*A*Sin[3*(c + d*x)] + 5495*C*Sin[3*(c + d*x)] + 840*A*Sin[4*(c + d*x)] + 2100*C*Sin[4*(c + d*x)] +
84*A*Sin[5*(c + d*x)] + 651*C*Sin[5*(c + d*x)] + 140*C*Sin[6*(c + d*x)] + 15*C*Sin[7*(c + d*x)]))/(6720*d)

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fricas [A]  time = 0.57, size = 146, normalized size = 0.67 \[ \frac {105 \, {\left (14 \, A + 11 \, C\right )} a^{4} d x + {\left (60 \, C a^{4} \cos \left (d x + c\right )^{6} + 280 \, C a^{4} \cos \left (d x + c\right )^{5} + 12 \, {\left (7 \, A + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (6 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 4 \, {\left (238 \, A + 227 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (14 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right ) + 4 \, {\left (581 \, A + 454 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(105*(14*A + 11*C)*a^4*d*x + (60*C*a^4*cos(d*x + c)^6 + 280*C*a^4*cos(d*x + c)^5 + 12*(7*A + 48*C)*a^4*c
os(d*x + c)^4 + 70*(6*A + 11*C)*a^4*cos(d*x + c)^3 + 4*(238*A + 227*C)*a^4*cos(d*x + c)^2 + 105*(14*A + 11*C)*
a^4*cos(d*x + c) + 4*(581*A + 454*C)*a^4)*sin(d*x + c))/d

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giac [A]  time = 1.53, size = 185, normalized size = 0.84 \[ \frac {C a^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {C a^{4} \sin \left (6 \, d x + 6 \, c\right )}{48 \, d} + \frac {1}{4} \, {\left (14 \, A a^{4} + 11 \, C a^{4}\right )} x + \frac {{\left (4 \, A a^{4} + 31 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (2 \, A a^{4} + 5 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} + \frac {{\left (116 \, A a^{4} + 157 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (32 \, A a^{4} + 31 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{16 \, d} + \frac {{\left (392 \, A a^{4} + 323 \, C a^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/448*C*a^4*sin(7*d*x + 7*c)/d + 1/48*C*a^4*sin(6*d*x + 6*c)/d + 1/4*(14*A*a^4 + 11*C*a^4)*x + 1/320*(4*A*a^4
+ 31*C*a^4)*sin(5*d*x + 5*c)/d + 1/16*(2*A*a^4 + 5*C*a^4)*sin(4*d*x + 4*c)/d + 1/192*(116*A*a^4 + 157*C*a^4)*s
in(3*d*x + 3*c)/d + 1/16*(32*A*a^4 + 31*C*a^4)*sin(2*d*x + 2*c)/d + 1/64*(392*A*a^4 + 323*C*a^4)*sin(d*x + c)/
d

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maple [A]  time = 0.33, size = 322, normalized size = 1.47 \[ \frac {\frac {A \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {a^{4} C \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+4 A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} C \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+2 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {6 a^{4} C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 A \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \,a^{4} \sin \left (d x +c \right )+\frac {a^{4} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/5*A*a^4*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+1/7*a^4*C*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/
5*cos(d*x+c)^2)*sin(d*x+c)+4*A*a^4*(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*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)+2*A*a^4*(2+cos(d*x+c)^2)*sin(d*x+c)
+6/5*a^4*C*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+4*A*a^4*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+4*
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)+A*a^4*sin(d*x+c)+1/3*a^4*C*(2+cos(d*x+c)^2)
*sin(d*x+c))

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maxima [A]  time = 0.35, size = 319, normalized size = 1.46 \[ \frac {112 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 3360 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 48 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C a^{4} + 672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 560 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 1680 \, A a^{4} \sin \left (d x + c\right )}{1680 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(112*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 - 3360*(sin(d*x + c)^3 - 3*sin(d*x
+ c))*A*a^4 + 210*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 1680*(2*d*x + 2*c + sin(2*d*
x + 2*c))*A*a^4 - 48*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*C*a^4 + 672*
(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^4 - 35*(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 - 560*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 + 210*(12*d*x + 12
*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4 + 1680*A*a^4*sin(d*x + c))/d

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mupad [B]  time = 2.30, size = 353, normalized size = 1.61 \[ \frac {\left (7\,A\,a^4+\frac {11\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {140\,A\,a^4}{3}+\frac {110\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1981\,A\,a^4}{15}+\frac {3113\,C\,a^4}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {1024\,A\,a^4}{5}+\frac {5632\,C\,a^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {2851\,A\,a^4}{15}+\frac {1501\,C\,a^4}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {308\,A\,a^4}{3}+70\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {53\,C\,a^4}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^4\,\left (14\,A+11\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{2\,d}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (14\,A+11\,C\right )}{2\,\left (7\,A\,a^4+\frac {11\,C\,a^4}{2}\right )}\right )\,\left (14\,A+11\,C\right )}{2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(tan(c/2 + (d*x)/2)*(25*A*a^4 + (53*C*a^4)/2) + tan(c/2 + (d*x)/2)^13*(7*A*a^4 + (11*C*a^4)/2) + tan(c/2 + (d*
x)/2)^11*((140*A*a^4)/3 + (110*C*a^4)/3) + tan(c/2 + (d*x)/2)^3*((308*A*a^4)/3 + 70*C*a^4) + tan(c/2 + (d*x)/2
)^5*((2851*A*a^4)/15 + (1501*C*a^4)/10) + tan(c/2 + (d*x)/2)^9*((1981*A*a^4)/15 + (3113*C*a^4)/30) + tan(c/2 +
(d*x)/2)^7*((1024*A*a^4)/5 + (5632*C*a^4)/35))/(d*(7*tan(c/2 + (d*x)/2)^2 + 21*tan(c/2 + (d*x)/2)^4 + 35*tan(
c/2 + (d*x)/2)^6 + 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 + 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d
*x)/2)^14 + 1)) - (a^4*(14*A + 11*C)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(2*d) + (a^4*atan((a^4*tan(c/2 + (d
*x)/2)*(14*A + 11*C))/(2*(7*A*a^4 + (11*C*a^4)/2)))*(14*A + 11*C))/(2*d)

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sympy [A]  time = 8.21, size = 799, normalized size = 3.65 \[ \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 C a^{4} x \sin ^{6}{\left (c + d x \right )}}{4} + \frac {15 C a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C a^{4} x \sin ^{4}{\left (c + d x \right )}}{2} + \frac {15 C a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4} + 3 C a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \frac {5 C a^{4} x \cos ^{6}{\left (c + d x \right )}}{4} + \frac {3 C a^{4} x \cos ^{4}{\left (c + d x \right )}}{2} + \frac {16 C a^{4} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 C a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {5 C a^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {16 C a^{4} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {2 C a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {10 C a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {8 C a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 C a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C a^{4} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} + \frac {11 C a^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{4 d} + \frac {6 C a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {C a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a \cos {\relax (c )} + a\right )^{4} \cos {\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(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)**4/2 + 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*C*a**4*x*sin(c + d*x)**
6/4 + 15*C*a**4*x*sin(c + d*x)**4*cos(c + d*x)**2/4 + 3*C*a**4*x*sin(c + d*x)**4/2 + 15*C*a**4*x*sin(c + d*x)*
*2*cos(c + d*x)**4/4 + 3*C*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2 + 5*C*a**4*x*cos(c + d*x)**6/4 + 3*C*a**4*x*
cos(c + d*x)**4/2 + 16*C*a**4*sin(c + d*x)**7/(35*d) + 8*C*a**4*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 5*C*a*
*4*sin(c + d*x)**5*cos(c + d*x)/(4*d) + 16*C*a**4*sin(c + d*x)**5/(5*d) + 2*C*a**4*sin(c + d*x)**3*cos(c + d*x
)**4/d + 10*C*a**4*sin(c + d*x)**3*cos(c + d*x)**3/(3*d) + 8*C*a**4*sin(c + d*x)**3*cos(c + d*x)**2/d + 3*C*a*
*4*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 2*C*a**4*sin(c + d*x)**3/(3*d) + C*a**4*sin(c + d*x)*cos(c + d*x)**6/d
+ 11*C*a**4*sin(c + d*x)*cos(c + d*x)**5/(4*d) + 6*C*a**4*sin(c + d*x)*cos(c + d*x)**4/d + 5*C*a**4*sin(c + d
*x)*cos(c + d*x)**3/(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*cos(c), True))

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