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

Optimal. Leaf size=345 \[ \frac {a b \left (6 a^2 C+126 A b^2+103 b^2 C\right ) \sin (c+d x) \cos ^3(c+d x)}{210 d}+\frac {\left (2 a^2 C+b^2 (7 A+6 C)\right ) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac {a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {1}{4} a b x \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right )+\frac {\left (35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right ) \sin (c+d x)}{105 d}+\frac {\left (4 a^4 C+3 a^2 b^2 (63 A+50 C)+4 b^4 (7 A+6 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{105 d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac {2 a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{21 d} \]

[Out]

1/4*a*b*(b^2*(6*A+5*C)+a^2*(8*A+6*C))*x+1/105*(35*a^4*(3*A+2*C)+84*a^2*b^2*(5*A+4*C)+8*b^4*(7*A+6*C))*sin(d*x+
c)/d+1/4*a*b*(b^2*(6*A+5*C)+a^2*(8*A+6*C))*cos(d*x+c)*sin(d*x+c)/d+1/105*(4*a^4*C+4*b^4*(7*A+6*C)+3*a^2*b^2*(6
3*A+50*C))*cos(d*x+c)^2*sin(d*x+c)/d+1/210*a*b*(126*A*b^2+6*C*a^2+103*C*b^2)*cos(d*x+c)^3*sin(d*x+c)/d+1/35*(2
*a^2*C+b^2*(7*A+6*C))*cos(d*x+c)^2*(a+b*cos(d*x+c))^2*sin(d*x+c)/d+2/21*a*C*cos(d*x+c)^2*(a+b*cos(d*x+c))^3*si
n(d*x+c)/d+1/7*C*cos(d*x+c)^2*(a+b*cos(d*x+c))^4*sin(d*x+c)/d

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Rubi [A]  time = 0.86, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3050, 3049, 3033, 3023, 2734} \[ \frac {\left (84 a^2 b^2 (5 A+4 C)+35 a^4 (3 A+2 C)+8 b^4 (7 A+6 C)\right ) \sin (c+d x)}{105 d}+\frac {\left (2 a^2 C+b^2 (7 A+6 C)\right ) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac {a b \left (6 a^2 C+126 A b^2+103 b^2 C\right ) \sin (c+d x) \cos ^3(c+d x)}{210 d}+\frac {\left (3 a^2 b^2 (63 A+50 C)+4 a^4 C+4 b^4 (7 A+6 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{105 d}+\frac {a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {1}{4} a b x \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac {2 a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{21 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*x)/4 + ((35*a^4*(3*A + 2*C) + 84*a^2*b^2*(5*A + 4*C) + 8*b^4*(7*A + 6
*C))*Sin[c + d*x])/(105*d) + (a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(4*d) + ((4*a
^4*C + 4*b^4*(7*A + 6*C) + 3*a^2*b^2*(63*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/(105*d) + (a*b*(126*A*b^2 + 6
*a^2*C + 103*b^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(210*d) + ((2*a^2*C + b^2*(7*A + 6*C))*Cos[c + d*x]^2*(a + b*
Cos[c + d*x])^2*Sin[c + d*x])/(35*d) + (2*a*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(21*d) + (C*
Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d)

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, 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 3033

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*d*Cos[e + f*x]*Sin[e + f*x]*(a + b
*Sin[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c*
(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e +
f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

Rule 3049

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/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3050

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/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n
*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*
x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0
] && NeQ[c, 0])))

Rubi steps

\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {1}{7} \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (a (7 A+2 C)+b (7 A+6 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {1}{42} \int \cos (c+d x) (a+b \cos (c+d x))^2 \left (2 a^2 (21 A+10 C)+4 a b (21 A+17 C) \cos (c+d x)+6 \left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {\left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {1}{210} \int \cos (c+d x) (a+b \cos (c+d x)) \left (2 a \left (6 b^2 (7 A+6 C)+a^2 (105 A+62 C)\right )+2 b \left (12 b^2 (7 A+6 C)+a^2 (315 A+244 C)\right ) \cos (c+d x)+4 a \left (126 A b^2+6 a^2 C+103 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {a b \left (126 A b^2+6 a^2 C+103 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac {\left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {1}{840} \int \cos (c+d x) \left (8 a^2 \left (6 b^2 (7 A+6 C)+a^2 (105 A+62 C)\right )+420 a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \cos (c+d x)+24 \left (4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {\left (4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{105 d}+\frac {a b \left (126 A b^2+6 a^2 C+103 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac {\left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {\int \cos (c+d x) \left (24 \left (35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right )+1260 a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \cos (c+d x)\right ) \, dx}{2520}\\ &=\frac {1}{4} a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) x+\frac {\left (35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right ) \sin (c+d x)}{105 d}+\frac {a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {\left (4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{105 d}+\frac {a b \left (126 A b^2+6 a^2 C+103 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac {\left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}\\ \end {align*}

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Mathematica [A]  time = 0.85, size = 351, normalized size = 1.02 \[ \frac {560 a^4 C \sin (3 (c+d x))+13440 a^3 A b c+13440 a^3 A b d x+840 a^3 b C \sin (4 (c+d x))+10080 a^3 b c C+10080 a^3 b C d x+420 a b \left (16 a^2 (A+C)+b^2 (16 A+15 C)\right ) \sin (2 (c+d x))+3360 a^2 A b^2 \sin (3 (c+d x))+4200 a^2 b^2 C \sin (3 (c+d x))+504 a^2 b^2 C \sin (5 (c+d x))+105 \left (16 a^4 (4 A+3 C)+48 a^2 b^2 (6 A+5 C)+5 b^4 (8 A+7 C)\right ) \sin (c+d x)+840 a A b^3 \sin (4 (c+d x))+10080 a A b^3 c+10080 a A b^3 d x+1260 a b^3 C \sin (4 (c+d x))+140 a b^3 C \sin (6 (c+d x))+8400 a b^3 c C+8400 a b^3 C d x+700 A b^4 \sin (3 (c+d x))+84 A b^4 \sin (5 (c+d x))+735 b^4 C \sin (3 (c+d x))+147 b^4 C \sin (5 (c+d x))+15 b^4 C \sin (7 (c+d x))}{6720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13440*a^3*A*b*c + 10080*a*A*b^3*c + 10080*a^3*b*c*C + 8400*a*b^3*c*C + 13440*a^3*A*b*d*x + 10080*a*A*b^3*d*x
+ 10080*a^3*b*C*d*x + 8400*a*b^3*C*d*x + 105*(16*a^4*(4*A + 3*C) + 48*a^2*b^2*(6*A + 5*C) + 5*b^4*(8*A + 7*C))
*Sin[c + d*x] + 420*a*b*(16*a^2*(A + C) + b^2*(16*A + 15*C))*Sin[2*(c + d*x)] + 3360*a^2*A*b^2*Sin[3*(c + d*x)
] + 700*A*b^4*Sin[3*(c + d*x)] + 560*a^4*C*Sin[3*(c + d*x)] + 4200*a^2*b^2*C*Sin[3*(c + d*x)] + 735*b^4*C*Sin[
3*(c + d*x)] + 840*a*A*b^3*Sin[4*(c + d*x)] + 840*a^3*b*C*Sin[4*(c + d*x)] + 1260*a*b^3*C*Sin[4*(c + d*x)] + 8
4*A*b^4*Sin[5*(c + d*x)] + 504*a^2*b^2*C*Sin[5*(c + d*x)] + 147*b^4*C*Sin[5*(c + d*x)] + 140*a*b^3*C*Sin[6*(c
+ d*x)] + 15*b^4*C*Sin[7*(c + d*x)])/(6720*d)

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fricas [A]  time = 1.11, size = 251, normalized size = 0.73 \[ \frac {105 \, {\left (2 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} d x + {\left (60 \, C b^{4} \cos \left (d x + c\right )^{6} + 280 \, C a b^{3} \cos \left (d x + c\right )^{5} + 140 \, {\left (3 \, A + 2 \, C\right )} a^{4} + 336 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 32 \, {\left (7 \, A + 6 \, C\right )} b^{4} + 12 \, {\left (42 \, C a^{2} b^{2} + {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (6 \, C a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (35 \, C a^{4} + 42 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 4 \, {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (2 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{420 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*(105*(2*(4*A + 3*C)*a^3*b + (6*A + 5*C)*a*b^3)*d*x + (60*C*b^4*cos(d*x + c)^6 + 280*C*a*b^3*cos(d*x + c)
^5 + 140*(3*A + 2*C)*a^4 + 336*(5*A + 4*C)*a^2*b^2 + 32*(7*A + 6*C)*b^4 + 12*(42*C*a^2*b^2 + (7*A + 6*C)*b^4)*
cos(d*x + c)^4 + 70*(6*C*a^3*b + (6*A + 5*C)*a*b^3)*cos(d*x + c)^3 + 4*(35*C*a^4 + 42*(5*A + 4*C)*a^2*b^2 + 4*
(7*A + 6*C)*b^4)*cos(d*x + c)^2 + 105*(2*(4*A + 3*C)*a^3*b + (6*A + 5*C)*a*b^3)*cos(d*x + c))*sin(d*x + c))/d

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giac [A]  time = 0.46, size = 290, normalized size = 0.84 \[ \frac {C b^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {C a b^{3} \sin \left (6 \, d x + 6 \, c\right )}{48 \, d} + \frac {1}{4} \, {\left (8 \, A a^{3} b + 6 \, C a^{3} b + 6 \, A a b^{3} + 5 \, C a b^{3}\right )} x + \frac {{\left (24 \, C a^{2} b^{2} + 4 \, A b^{4} + 7 \, C b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (2 \, C a^{3} b + 2 \, A a b^{3} + 3 \, C a b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} + \frac {{\left (16 \, C a^{4} + 96 \, A a^{2} b^{2} + 120 \, C a^{2} b^{2} + 20 \, A b^{4} + 21 \, C b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (16 \, A a^{3} b + 16 \, C a^{3} b + 16 \, A a b^{3} + 15 \, C a b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{16 \, d} + \frac {{\left (64 \, A a^{4} + 48 \, C a^{4} + 288 \, A a^{2} b^{2} + 240 \, C a^{2} b^{2} + 40 \, A b^{4} + 35 \, C b^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/448*C*b^4*sin(7*d*x + 7*c)/d + 1/48*C*a*b^3*sin(6*d*x + 6*c)/d + 1/4*(8*A*a^3*b + 6*C*a^3*b + 6*A*a*b^3 + 5*
C*a*b^3)*x + 1/320*(24*C*a^2*b^2 + 4*A*b^4 + 7*C*b^4)*sin(5*d*x + 5*c)/d + 1/16*(2*C*a^3*b + 2*A*a*b^3 + 3*C*a
*b^3)*sin(4*d*x + 4*c)/d + 1/192*(16*C*a^4 + 96*A*a^2*b^2 + 120*C*a^2*b^2 + 20*A*b^4 + 21*C*b^4)*sin(3*d*x + 3
*c)/d + 1/16*(16*A*a^3*b + 16*C*a^3*b + 16*A*a*b^3 + 15*C*a*b^3)*sin(2*d*x + 2*c)/d + 1/64*(64*A*a^4 + 48*C*a^
4 + 288*A*a^2*b^2 + 240*C*a^2*b^2 + 40*A*b^4 + 35*C*b^4)*sin(d*x + c)/d

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maple [A]  time = 0.35, size = 332, normalized size = 0.96 \[ \frac {A \,a^{4} \sin \left (d x +c \right )+\frac {a^{4} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 A \,a^{3} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{3} b C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 A \,a^{2} b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {6 C \,a^{2} b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a A \,b^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 C a \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {A \,b^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {C \,b^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.59, size = 329, normalized size = 0.95 \[ -\frac {560 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b + 3360 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} - 672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{2} b^{2} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{3} - 112 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A b^{4} + 48 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C b^{4} - 1680 \, A a^{4} \sin \left (d x + c\right )}{1680 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.06, size = 798, normalized size = 2.31 \[ \frac {\left (2\,A\,a^4+2\,A\,b^4+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+12\,C\,a^2\,b^2-5\,A\,a\,b^3-4\,A\,a^3\,b-\frac {11\,C\,a\,b^3}{2}-5\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (12\,A\,a^4+\frac {20\,A\,b^4}{3}+\frac {28\,C\,a^4}{3}+4\,C\,b^4+56\,A\,a^2\,b^2+40\,C\,a^2\,b^2-12\,A\,a\,b^3-16\,A\,a^3\,b-\frac {14\,C\,a\,b^3}{3}-12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (30\,A\,a^4+\frac {226\,A\,b^4}{15}+\frac {58\,C\,a^4}{3}+\frac {86\,C\,b^4}{5}+116\,A\,a^2\,b^2+\frac {452\,C\,a^2\,b^2}{5}-9\,A\,a\,b^3-20\,A\,a^3\,b-\frac {85\,C\,a\,b^3}{6}-9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (40\,A\,a^4+\frac {104\,A\,b^4}{5}+24\,C\,a^4+\frac {424\,C\,b^4}{35}+144\,A\,a^2\,b^2+\frac {624\,C\,a^2\,b^2}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (30\,A\,a^4+\frac {226\,A\,b^4}{15}+\frac {58\,C\,a^4}{3}+\frac {86\,C\,b^4}{5}+116\,A\,a^2\,b^2+\frac {452\,C\,a^2\,b^2}{5}+9\,A\,a\,b^3+20\,A\,a^3\,b+\frac {85\,C\,a\,b^3}{6}+9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (12\,A\,a^4+\frac {20\,A\,b^4}{3}+\frac {28\,C\,a^4}{3}+4\,C\,b^4+56\,A\,a^2\,b^2+40\,C\,a^2\,b^2+12\,A\,a\,b^3+16\,A\,a^3\,b+\frac {14\,C\,a\,b^3}{3}+12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^4+2\,A\,b^4+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+12\,C\,a^2\,b^2+5\,A\,a\,b^3+4\,A\,a^3\,b+\frac {11\,C\,a\,b^3}{2}+5\,C\,a^3\,b\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\,b\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (8\,A\,a^2+6\,A\,b^2+6\,C\,a^2+5\,C\,b^2\right )}{2\,d}+\frac {a\,b\,\mathrm {atan}\left (\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,A\,a^2+6\,A\,b^2+6\,C\,a^2+5\,C\,b^2\right )}{2\,\left (3\,A\,a\,b^3+4\,A\,a^3\,b+\frac {5\,C\,a\,b^3}{2}+3\,C\,a^3\,b\right )}\right )\,\left (8\,A\,a^2+6\,A\,b^2+6\,C\,a^2+5\,C\,b^2\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 + b*cos(c + d*x))^4,x)

[Out]

(tan(c/2 + (d*x)/2)*(2*A*a^4 + 2*A*b^4 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2*b^2 + 12*C*a^2*b^2 + 5*A*a*b^3 + 4*A*a^3
*b + (11*C*a*b^3)/2 + 5*C*a^3*b) + tan(c/2 + (d*x)/2)^7*(40*A*a^4 + (104*A*b^4)/5 + 24*C*a^4 + (424*C*b^4)/35
+ 144*A*a^2*b^2 + (624*C*a^2*b^2)/5) + tan(c/2 + (d*x)/2)^13*(2*A*a^4 + 2*A*b^4 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2
*b^2 + 12*C*a^2*b^2 - 5*A*a*b^3 - 4*A*a^3*b - (11*C*a*b^3)/2 - 5*C*a^3*b) + tan(c/2 + (d*x)/2)^3*(12*A*a^4 + (
20*A*b^4)/3 + (28*C*a^4)/3 + 4*C*b^4 + 56*A*a^2*b^2 + 40*C*a^2*b^2 + 12*A*a*b^3 + 16*A*a^3*b + (14*C*a*b^3)/3
+ 12*C*a^3*b) + tan(c/2 + (d*x)/2)^11*(12*A*a^4 + (20*A*b^4)/3 + (28*C*a^4)/3 + 4*C*b^4 + 56*A*a^2*b^2 + 40*C*
a^2*b^2 - 12*A*a*b^3 - 16*A*a^3*b - (14*C*a*b^3)/3 - 12*C*a^3*b) + tan(c/2 + (d*x)/2)^5*(30*A*a^4 + (226*A*b^4
)/15 + (58*C*a^4)/3 + (86*C*b^4)/5 + 116*A*a^2*b^2 + (452*C*a^2*b^2)/5 + 9*A*a*b^3 + 20*A*a^3*b + (85*C*a*b^3)
/6 + 9*C*a^3*b) + tan(c/2 + (d*x)/2)^9*(30*A*a^4 + (226*A*b^4)/15 + (58*C*a^4)/3 + (86*C*b^4)/5 + 116*A*a^2*b^
2 + (452*C*a^2*b^2)/5 - 9*A*a*b^3 - 20*A*a^3*b - (85*C*a*b^3)/6 - 9*C*a^3*b))/(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*b*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*(8*A*a^2 + 6*A*b^2 +
6*C*a^2 + 5*C*b^2))/(2*d) + (a*b*atan((a*b*tan(c/2 + (d*x)/2)*(8*A*a^2 + 6*A*b^2 + 6*C*a^2 + 5*C*b^2))/(2*(3*
A*a*b^3 + 4*A*a^3*b + (5*C*a*b^3)/2 + 3*C*a^3*b)))*(8*A*a^2 + 6*A*b^2 + 6*C*a^2 + 5*C*b^2))/(2*d)

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sympy [A]  time = 8.04, size = 850, normalized size = 2.46 \[ \begin {cases} \frac {A a^{4} \sin {\left (c + d x \right )}}{d} + 2 A a^{3} b x \sin ^{2}{\left (c + d x \right )} + 2 A a^{3} b x \cos ^{2}{\left (c + d x \right )} + \frac {2 A a^{3} b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {4 A a^{2} b^{2} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {6 A a^{2} b^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 A a b^{3} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 A a b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \frac {3 A a b^{3} x \cos ^{4}{\left (c + d x \right )}}{2} + \frac {3 A a b^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {5 A a b^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {8 A b^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 A b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {A b^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{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 ^{2}{\left (c + d x \right )}}{d} + \frac {3 C a^{3} b x \sin ^{4}{\left (c + d x \right )}}{2} + 3 C a^{3} b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \frac {3 C a^{3} b x \cos ^{4}{\left (c + d x \right )}}{2} + \frac {3 C a^{3} b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {5 C a^{3} b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {16 C a^{2} b^{2} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {8 C a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {6 C a^{2} b^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C a b^{3} x \sin ^{6}{\left (c + d x \right )}}{4} + \frac {15 C a b^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {15 C a b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4} + \frac {5 C a b^{3} x \cos ^{6}{\left (c + d x \right )}}{4} + \frac {5 C a b^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {10 C a b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {11 C a b^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{4 d} + \frac {16 C b^{4} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 C b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 C b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {C b^{4} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a + b \cos {\relax (c )}\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+b*cos(d*x+c))**4*(A+C*cos(d*x+c)**2),x)

[Out]

Piecewise((A*a**4*sin(c + d*x)/d + 2*A*a**3*b*x*sin(c + d*x)**2 + 2*A*a**3*b*x*cos(c + d*x)**2 + 2*A*a**3*b*si
n(c + d*x)*cos(c + d*x)/d + 4*A*a**2*b**2*sin(c + d*x)**3/d + 6*A*a**2*b**2*sin(c + d*x)*cos(c + d*x)**2/d + 3
*A*a*b**3*x*sin(c + d*x)**4/2 + 3*A*a*b**3*x*sin(c + d*x)**2*cos(c + d*x)**2 + 3*A*a*b**3*x*cos(c + d*x)**4/2
+ 3*A*a*b**3*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 5*A*a*b**3*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 8*A*b**4*sin
(c + d*x)**5/(15*d) + 4*A*b**4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + A*b**4*sin(c + d*x)*cos(c + d*x)**4/d +
2*C*a**4*sin(c + d*x)**3/(3*d) + C*a**4*sin(c + d*x)*cos(c + d*x)**2/d + 3*C*a**3*b*x*sin(c + d*x)**4/2 + 3*C
*a**3*b*x*sin(c + d*x)**2*cos(c + d*x)**2 + 3*C*a**3*b*x*cos(c + d*x)**4/2 + 3*C*a**3*b*sin(c + d*x)**3*cos(c
+ d*x)/(2*d) + 5*C*a**3*b*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 16*C*a**2*b**2*sin(c + d*x)**5/(5*d) + 8*C*a**2
*b**2*sin(c + d*x)**3*cos(c + d*x)**2/d + 6*C*a**2*b**2*sin(c + d*x)*cos(c + d*x)**4/d + 5*C*a*b**3*x*sin(c +
d*x)**6/4 + 15*C*a*b**3*x*sin(c + d*x)**4*cos(c + d*x)**2/4 + 15*C*a*b**3*x*sin(c + d*x)**2*cos(c + d*x)**4/4
+ 5*C*a*b**3*x*cos(c + d*x)**6/4 + 5*C*a*b**3*sin(c + d*x)**5*cos(c + d*x)/(4*d) + 10*C*a*b**3*sin(c + d*x)**3
*cos(c + d*x)**3/(3*d) + 11*C*a*b**3*sin(c + d*x)*cos(c + d*x)**5/(4*d) + 16*C*b**4*sin(c + d*x)**7/(35*d) + 8
*C*b**4*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2*C*b**4*sin(c + d*x)**3*cos(c + d*x)**4/d + C*b**4*sin(c + d*
x)*cos(c + d*x)**6/d, Ne(d, 0)), (x*(A + C*cos(c)**2)*(a + b*cos(c))**4*cos(c), True))

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