3.318 \(\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=265 \[ -\frac {a^3 (133 A+119 B+108 C) \sin ^3(c+d x)}{105 d}+\frac {a^3 (133 A+119 B+108 C) \sin (c+d x)}{35 d}+\frac {a^3 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {(3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{15 d}+\frac {a^3 (26 A+23 B+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^3 x (26 A+23 B+21 C)+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{42 a d}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]

[Out]

1/16*a^3*(26*A+23*B+21*C)*x+1/35*a^3*(133*A+119*B+108*C)*sin(d*x+c)/d+1/16*a^3*(26*A+23*B+21*C)*cos(d*x+c)*sin
(d*x+c)/d+1/280*a^3*(154*A+147*B+129*C)*cos(d*x+c)^3*sin(d*x+c)/d+1/7*C*cos(d*x+c)^3*(a+a*cos(d*x+c))^3*sin(d*
x+c)/d+1/42*(7*B+3*C)*cos(d*x+c)^3*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/a/d+1/15*(3*A+4*B+3*C)*cos(d*x+c)^3*(a^3+
a^3*cos(d*x+c))*sin(d*x+c)/d-1/105*a^3*(133*A+119*B+108*C)*sin(d*x+c)^3/d

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Rubi [A]  time = 0.68, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {3045, 2976, 2968, 3023, 2748, 2635, 8, 2633} \[ -\frac {a^3 (133 A+119 B+108 C) \sin ^3(c+d x)}{105 d}+\frac {a^3 (133 A+119 B+108 C) \sin (c+d x)}{35 d}+\frac {a^3 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {(3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{15 d}+\frac {a^3 (26 A+23 B+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^3 x (26 A+23 B+21 C)+\frac {(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{42 a d}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(a^3*(26*A + 23*B + 21*C)*x)/16 + (a^3*(133*A + 119*B + 108*C)*Sin[c + d*x])/(35*d) + (a^3*(26*A + 23*B + 21*C
)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(154*A + 147*B + 129*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (C*C
os[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d) + ((7*B + 3*C)*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x]
)^2*Sin[c + d*x])/(42*a*d) + ((3*A + 4*B + 3*C)*Cos[c + d*x]^3*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d) -
(a^3*(133*A + 119*B + 108*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 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

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 2976

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*S
in[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&
NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 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 3045

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((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*(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)) + b*B*
d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, 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 ^2(c+d x) (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (a (7 A+3 C)+a (7 B+3 C) \cos (c+d x)) \, dx}{7 a}\\ &=\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (3 a^2 (14 A+7 B+9 C)+14 a^2 (3 A+4 B+3 C) \cos (c+d x)\right ) \, dx}{42 a}\\ &=\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^3 (112 A+91 B+87 C)+3 a^3 (154 A+147 B+129 C) \cos (c+d x)\right ) \, dx}{210 a}\\ &=\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {\int \cos ^2(c+d x) \left (3 a^4 (112 A+91 B+87 C)+\left (3 a^4 (112 A+91 B+87 C)+3 a^4 (154 A+147 B+129 C)\right ) \cos (c+d x)+3 a^4 (154 A+147 B+129 C) \cos ^2(c+d x)\right ) \, dx}{210 a}\\ &=\frac {a^3 (154 A+147 B+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {\int \cos ^2(c+d x) \left (105 a^4 (26 A+23 B+21 C)+24 a^4 (133 A+119 B+108 C) \cos (c+d x)\right ) \, dx}{840 a}\\ &=\frac {a^3 (154 A+147 B+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{8} \left (a^3 (26 A+23 B+21 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{35} \left (a^3 (133 A+119 B+108 C)\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac {a^3 (26 A+23 B+21 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (154 A+147 B+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{16} \left (a^3 (26 A+23 B+21 C)\right ) \int 1 \, dx-\frac {\left (a^3 (133 A+119 B+108 C)\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac {1}{16} a^3 (26 A+23 B+21 C) x+\frac {a^3 (133 A+119 B+108 C) \sin (c+d x)}{35 d}+\frac {a^3 (26 A+23 B+21 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (154 A+147 B+129 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 B+3 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^3(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^3 (133 A+119 B+108 C) \sin ^3(c+d x)}{105 d}\\ \end {align*}

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Mathematica [A]  time = 1.06, size = 204, normalized size = 0.77 \[ \frac {a^3 (105 (184 A+168 B+155 C) \sin (c+d x)+105 (64 A+63 B+61 C) \sin (2 (c+d x))+2380 A \sin (3 (c+d x))+630 A \sin (4 (c+d x))+84 A \sin (5 (c+d x))+10920 A d x+2660 B \sin (3 (c+d x))+945 B \sin (4 (c+d x))+252 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+9660 B c+9660 B d x+2835 C \sin (3 (c+d x))+1155 C \sin (4 (c+d x))+399 C \sin (5 (c+d x))+105 C \sin (6 (c+d x))+15 C \sin (7 (c+d x))+5460 c C+8820 C d x)}{6720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(a^3*(9660*B*c + 5460*c*C + 10920*A*d*x + 9660*B*d*x + 8820*C*d*x + 105*(184*A + 168*B + 155*C)*Sin[c + d*x] +
105*(64*A + 63*B + 61*C)*Sin[2*(c + d*x)] + 2380*A*Sin[3*(c + d*x)] + 2660*B*Sin[3*(c + d*x)] + 2835*C*Sin[3*
(c + d*x)] + 630*A*Sin[4*(c + d*x)] + 945*B*Sin[4*(c + d*x)] + 1155*C*Sin[4*(c + d*x)] + 84*A*Sin[5*(c + d*x)]
+ 252*B*Sin[5*(c + d*x)] + 399*C*Sin[5*(c + d*x)] + 35*B*Sin[6*(c + d*x)] + 105*C*Sin[6*(c + d*x)] + 15*C*Sin
[7*(c + d*x)]))/(6720*d)

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fricas [A]  time = 0.45, size = 168, normalized size = 0.63 \[ \frac {105 \, {\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} d x + {\left (240 \, C a^{3} \cos \left (d x + c\right )^{6} + 280 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (7 \, A + 21 \, B + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 70 \, {\left (18 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (133 \, A + 119 \, B + 108 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 105 \, {\left (26 \, A + 23 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right ) + 32 \, {\left (133 \, A + 119 \, B + 108 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{1680 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

1/1680*(105*(26*A + 23*B + 21*C)*a^3*d*x + (240*C*a^3*cos(d*x + c)^6 + 280*(B + 3*C)*a^3*cos(d*x + c)^5 + 48*(
7*A + 21*B + 27*C)*a^3*cos(d*x + c)^4 + 70*(18*A + 23*B + 21*C)*a^3*cos(d*x + c)^3 + 16*(133*A + 119*B + 108*C
)*a^3*cos(d*x + c)^2 + 105*(26*A + 23*B + 21*C)*a^3*cos(d*x + c) + 32*(133*A + 119*B + 108*C)*a^3)*sin(d*x + c
))/d

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giac [A]  time = 1.46, size = 229, normalized size = 0.86 \[ \frac {C a^{3} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {1}{16} \, {\left (26 \, A a^{3} + 23 \, B a^{3} + 21 \, C a^{3}\right )} x + \frac {{\left (B a^{3} + 3 \, C a^{3}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (4 \, A a^{3} + 12 \, B a^{3} + 19 \, C a^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (6 \, A a^{3} + 9 \, B a^{3} + 11 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (68 \, A a^{3} + 76 \, B a^{3} + 81 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (64 \, A a^{3} + 63 \, B a^{3} + 61 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (184 \, A a^{3} + 168 \, B a^{3} + 155 \, C a^{3}\right )} \sin \left (d x + c\right )}{64 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

1/448*C*a^3*sin(7*d*x + 7*c)/d + 1/16*(26*A*a^3 + 23*B*a^3 + 21*C*a^3)*x + 1/192*(B*a^3 + 3*C*a^3)*sin(6*d*x +
6*c)/d + 1/320*(4*A*a^3 + 12*B*a^3 + 19*C*a^3)*sin(5*d*x + 5*c)/d + 1/64*(6*A*a^3 + 9*B*a^3 + 11*C*a^3)*sin(4
*d*x + 4*c)/d + 1/192*(68*A*a^3 + 76*B*a^3 + 81*C*a^3)*sin(3*d*x + 3*c)/d + 1/64*(64*A*a^3 + 63*B*a^3 + 61*C*a
^3)*sin(2*d*x + 2*c)/d + 1/64*(184*A*a^3 + 168*B*a^3 + 155*C*a^3)*sin(d*x + c)/d

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maple [A]  time = 0.37, size = 427, normalized size = 1.61 \[ \frac {\frac {A \,a^{3} \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^{3} B \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 {C \,a^{3} \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}+3 A \,a^{3} \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 {3 a^{3} 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}+3 C \,a^{3} \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 )+A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 a^{3} 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 )+\frac {3 C \,a^{3} \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^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{3} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,a^{3} \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} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)

[Out]

1/d*(1/5*A*a^3*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+a^3*B*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*c
os(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+1/7*C*a^3*(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)+3*A*a^3*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+3/5*a^3*B*(8/3+cos(d*x+c)^4+4/3*cos(d
*x+c)^2)*sin(d*x+c)+3*C*a^3*(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)+A
*a^3*(2+cos(d*x+c)^2)*sin(d*x+c)+3*a^3*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)+3/5*C*a^
3*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+A*a^3*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+1/3*a^3*B*(2+
cos(d*x+c)^2)*sin(d*x+c)+C*a^3*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c))

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maxima [A]  time = 0.39, size = 425, normalized size = 1.60 \[ \frac {448 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} - 6720 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 630 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 1344 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 630 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C a^{3} + 1344 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} - 105 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3}}{6720 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

1/6720*(448*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^3 - 6720*(sin(d*x + c)^3 - 3*sin(d*x
+ c))*A*a^3 + 630*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 + 1680*(2*d*x + 2*c + sin(2*d*
x + 2*c))*A*a^3 + 1344*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^3 - 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))*B*a^3 - 2240*(sin(d*x + c)^3 - 3*sin(d*x + c))*
B*a^3 + 630*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^3 - 192*(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^3 + 1344*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x
+ c))*C*a^3 - 105*(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^3 + 2
10*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^3)/d

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mupad [B]  time = 2.91, size = 410, normalized size = 1.55 \[ \frac {\left (\frac {13\,A\,a^3}{4}+\frac {23\,B\,a^3}{8}+\frac {21\,C\,a^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {65\,A\,a^3}{3}+\frac {115\,B\,a^3}{6}+\frac {35\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3679\,A\,a^3}{60}+\frac {6509\,B\,a^3}{120}+\frac {1981\,C\,a^3}{40}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {464\,A\,a^3}{5}+\frac {432\,B\,a^3}{5}+\frac {2608\,C\,a^3}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {5089\,A\,a^3}{60}+\frac {2993\,B\,a^3}{40}+\frac {3011\,C\,a^3}{40}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {143\,A\,a^3}{3}+\frac {79\,B\,a^3}{2}+\frac {61\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,A\,a^3}{4}+\frac {105\,B\,a^3}{8}+\frac {107\,C\,a^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (26\,A+23\,B+21\,C\right )}{8\,\left (\frac {13\,A\,a^3}{4}+\frac {23\,B\,a^3}{8}+\frac {21\,C\,a^3}{8}\right )}\right )\,\left (26\,A+23\,B+21\,C\right )}{8\,d}-\frac {a^3\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (26\,A+23\,B+21\,C\right )}{8\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^2*(a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)

[Out]

(tan(c/2 + (d*x)/2)^13*((13*A*a^3)/4 + (23*B*a^3)/8 + (21*C*a^3)/8) + tan(c/2 + (d*x)/2)^11*((65*A*a^3)/3 + (1
15*B*a^3)/6 + (35*C*a^3)/2) + tan(c/2 + (d*x)/2)^3*((143*A*a^3)/3 + (79*B*a^3)/2 + (61*C*a^3)/2) + tan(c/2 + (
d*x)/2)^7*((464*A*a^3)/5 + (432*B*a^3)/5 + (2608*C*a^3)/35) + tan(c/2 + (d*x)/2)^5*((5089*A*a^3)/60 + (2993*B*
a^3)/40 + (3011*C*a^3)/40) + tan(c/2 + (d*x)/2)^9*((3679*A*a^3)/60 + (6509*B*a^3)/120 + (1981*C*a^3)/40) + tan
(c/2 + (d*x)/2)*((51*A*a^3)/4 + (105*B*a^3)/8 + (107*C*a^3)/8))/(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)^1
2 + tan(c/2 + (d*x)/2)^14 + 1)) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(26*A + 23*B + 21*C))/(8*((13*A*a^3)/4 + (
23*B*a^3)/8 + (21*C*a^3)/8)))*(26*A + 23*B + 21*C))/(8*d) - (a^3*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*(26*A +
23*B + 21*C))/(8*d)

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sympy [A]  time = 8.44, size = 1149, normalized size = 4.34 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2*(a+a*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)

[Out]

Piecewise((9*A*a**3*x*sin(c + d*x)**4/8 + 9*A*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + A*a**3*x*sin(c + d*x)
**2/2 + 9*A*a**3*x*cos(c + d*x)**4/8 + A*a**3*x*cos(c + d*x)**2/2 + 8*A*a**3*sin(c + d*x)**5/(15*d) + 4*A*a**3
*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 9*A*a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 2*A*a**3*sin(c + d*x)**
3/d + A*a**3*sin(c + d*x)*cos(c + d*x)**4/d + 15*A*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 3*A*a**3*sin(c +
d*x)*cos(c + d*x)**2/d + A*a**3*sin(c + d*x)*cos(c + d*x)/(2*d) + 5*B*a**3*x*sin(c + d*x)**6/16 + 15*B*a**3*x*
sin(c + d*x)**4*cos(c + d*x)**2/16 + 9*B*a**3*x*sin(c + d*x)**4/8 + 15*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**
4/16 + 9*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 5*B*a**3*x*cos(c + d*x)**6/16 + 9*B*a**3*x*cos(c + d*x)*
*4/8 + 5*B*a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 8*B*a**3*sin(c + d*x)**5/(5*d) + 5*B*a**3*sin(c + d*x)**
3*cos(c + d*x)**3/(6*d) + 4*B*a**3*sin(c + d*x)**3*cos(c + d*x)**2/d + 9*B*a**3*sin(c + d*x)**3*cos(c + d*x)/(
8*d) + 2*B*a**3*sin(c + d*x)**3/(3*d) + 11*B*a**3*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 3*B*a**3*sin(c + d*x)*
cos(c + d*x)**4/d + 15*B*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) + B*a**3*sin(c + d*x)*cos(c + d*x)**2/d + 15*
C*a**3*x*sin(c + d*x)**6/16 + 45*C*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*C*a**3*x*sin(c + d*x)**4/8 +
45*C*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*C*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 15*C*a**3*x*co
s(c + d*x)**6/16 + 3*C*a**3*x*cos(c + d*x)**4/8 + 16*C*a**3*sin(c + d*x)**7/(35*d) + 8*C*a**3*sin(c + d*x)**5*
cos(c + d*x)**2/(5*d) + 15*C*a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 8*C*a**3*sin(c + d*x)**5/(5*d) + 2*C*a
**3*sin(c + d*x)**3*cos(c + d*x)**4/d + 5*C*a**3*sin(c + d*x)**3*cos(c + d*x)**3/(2*d) + 4*C*a**3*sin(c + d*x)
**3*cos(c + d*x)**2/d + 3*C*a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d) + C*a**3*sin(c + d*x)*cos(c + d*x)**6/d +
33*C*a**3*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 3*C*a**3*sin(c + d*x)*cos(c + d*x)**4/d + 5*C*a**3*sin(c + d*x
)*cos(c + d*x)**3/(8*d), Ne(d, 0)), (x*(a*cos(c) + a)**3*(A + B*cos(c) + C*cos(c)**2)*cos(c)**2, True))

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