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

Optimal. Leaf size=273 \[ \frac {a^3 (a B+4 A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b^2 \sin (c+d x) \cos (c+d x) \left (-\left (a^2 (24 A-26 C)\right )+32 a b B+3 b^2 (4 A+3 C)\right )}{24 d}+\frac {b \sin (c+d x) \left (-\left (a^3 (12 A-19 C)\right )+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 d}+\frac {1}{8} x \left (8 a^4 C+32 a^3 b B+24 a^2 b^2 (2 A+C)+16 a b^3 B+b^4 (4 A+3 C)\right )-\frac {b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{12 d}-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d} \]

[Out]

1/8*(32*a^3*b*B+16*a*b^3*B+8*a^4*C+24*a^2*b^2*(2*A+C)+b^4*(4*A+3*C))*x+a^3*(4*A*b+B*a)*arctanh(sin(d*x+c))/d+1
/6*b*(34*a^2*b*B+4*b^3*B-a^3*(12*A-19*C)+8*a*b^2*(3*A+2*C))*sin(d*x+c)/d+1/24*b^2*(32*a*b*B-a^2*(24*A-26*C)+3*
b^2*(4*A+3*C))*cos(d*x+c)*sin(d*x+c)/d-1/12*b*(12*A*a-4*B*b-7*C*a)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d-1/4*b*(4*A-
C)*(a+b*cos(d*x+c))^3*sin(d*x+c)/d+A*(a+b*cos(d*x+c))^4*tan(d*x+c)/d

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Rubi [A]  time = 0.91, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3047, 3049, 3033, 3023, 2735, 3770} \[ \frac {b \sin (c+d x) \left (a^3 (-(12 A-19 C))+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 d}+\frac {b^2 \sin (c+d x) \cos (c+d x) \left (a^2 (-(24 A-26 C))+32 a b B+3 b^2 (4 A+3 C)\right )}{24 d}+\frac {1}{8} x \left (24 a^2 b^2 (2 A+C)+32 a^3 b B+8 a^4 C+16 a b^3 B+b^4 (4 A+3 C)\right )+\frac {a^3 (a B+4 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac {b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{12 d}-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((32*a^3*b*B + 16*a*b^3*B + 8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*x)/8 + (a^3*(4*A*b + a*B)*ArcTan
h[Sin[c + d*x]])/d + (b*(34*a^2*b*B + 4*b^3*B - a^3*(12*A - 19*C) + 8*a*b^2*(3*A + 2*C))*Sin[c + d*x])/(6*d) +
(b^2*(32*a*b*B - a^2*(24*A - 26*C) + 3*b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(24*d) - (b*(12*a*A - 4*b*
B - 7*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) - (b*(4*A - C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*
d) + (A*(a + b*Cos[c + d*x])^4*Tan[c + d*x])/d

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], 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 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\frac {A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}+\int (a+b \cos (c+d x))^3 \left (4 A b+a B+(b B+a C) \cos (c+d x)-b (4 A-C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {1}{4} \int (a+b \cos (c+d x))^2 \left (4 a (4 A b+a B)+\left (4 A b^2+8 a b B+4 a^2 C+3 b^2 C\right ) \cos (c+d x)-b (12 a A-4 b B-7 a C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {b (12 a A-4 b B-7 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {1}{12} \int (a+b \cos (c+d x)) \left (12 a^2 (4 A b+a B)+\left (36 a^2 b B+8 b^3 B+12 a^3 C+a b^2 (36 A+23 C)\right ) \cos (c+d x)+b \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac {b (12 a A-4 b B-7 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {1}{24} \int \left (24 a^3 (4 A b+a B)+3 \left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \cos (c+d x)+4 b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}+\frac {b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac {b (12 a A-4 b B-7 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {1}{24} \int \left (24 a^3 (4 A b+a B)+3 \left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {1}{8} \left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) x+\frac {b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}+\frac {b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac {b (12 a A-4 b B-7 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}+\left (a^3 (4 A b+a B)\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} \left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) x+\frac {a^3 (4 A b+a B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}+\frac {b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac {b (12 a A-4 b B-7 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}\\ \end {align*}

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Mathematica [A]  time = 3.05, size = 383, normalized size = 1.40 \[ \frac {b^2 \sec (c+d x) \left (3 \sin (3 (c+d x)) \left (48 a^2 C+32 a b B+8 A b^2+9 b^2 C\right )+b (8 (4 a C+b B) \sin (4 (c+d x))+3 b C \sin (5 (c+d x)))\right )+32 b \sin (c+d x) \left (24 a^3 C+36 a^2 b B+4 a b^2 (6 A+5 C)+5 b^3 B\right )+24 \left (8 a^4 B \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+8 a^4 c C+8 a^4 C d x-8 a^3 (a B+4 A b) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+32 a^3 A b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+32 a^3 b B c+32 a^3 b B d x+48 a^2 A b^2 c+48 a^2 A b^2 d x+24 a^2 b^2 c C+24 a^2 b^2 C d x+\tan (c+d x) \left (8 a^4 A+6 a^2 b^2 C+4 a b^3 B+b^4 (A+C)\right )+16 a b^3 B c+16 a b^3 B d x+4 A b^4 c+4 A b^4 d x+3 b^4 c C+3 b^4 C d x\right )}{192 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(32*b*(36*a^2*b*B + 5*b^3*B + 24*a^3*C + 4*a*b^2*(6*A + 5*C))*Sin[c + d*x] + b^2*Sec[c + d*x]*(3*(8*A*b^2 + 32
*a*b*B + 48*a^2*C + 9*b^2*C)*Sin[3*(c + d*x)] + b*(8*(b*B + 4*a*C)*Sin[4*(c + d*x)] + 3*b*C*Sin[5*(c + d*x)]))
+ 24*(48*a^2*A*b^2*c + 4*A*b^4*c + 32*a^3*b*B*c + 16*a*b^3*B*c + 8*a^4*c*C + 24*a^2*b^2*c*C + 3*b^4*c*C + 48*
a^2*A*b^2*d*x + 4*A*b^4*d*x + 32*a^3*b*B*d*x + 16*a*b^3*B*d*x + 8*a^4*C*d*x + 24*a^2*b^2*C*d*x + 3*b^4*C*d*x -
8*a^3*(4*A*b + a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 32*a^3*A*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x
)/2]] + 8*a^4*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (8*a^4*A + 4*a*b^3*B + 6*a^2*b^2*C + b^4*(A + C))*T
an[c + d*x]))/(192*d)

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fricas [A]  time = 0.48, size = 263, normalized size = 0.96 \[ \frac {3 \, {\left (8 \, C a^{4} + 32 \, B a^{3} b + 24 \, {\left (2 \, A + C\right )} a^{2} b^{2} + 16 \, B a b^{3} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} d x \cos \left (d x + c\right ) + 12 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, C b^{4} \cos \left (d x + c\right )^{4} + 24 \, A a^{4} + 8 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (24 \, C a^{2} b^{2} + 16 \, B a b^{3} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 16 \, {\left (6 \, C a^{3} b + 9 \, B a^{2} b^{2} + 2 \, {\left (3 \, A + 2 \, C\right )} a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*(8*C*a^4 + 32*B*a^3*b + 24*(2*A + C)*a^2*b^2 + 16*B*a*b^3 + (4*A + 3*C)*b^4)*d*x*cos(d*x + c) + 12*(B*
a^4 + 4*A*a^3*b)*cos(d*x + c)*log(sin(d*x + c) + 1) - 12*(B*a^4 + 4*A*a^3*b)*cos(d*x + c)*log(-sin(d*x + c) +
1) + (6*C*b^4*cos(d*x + c)^4 + 24*A*a^4 + 8*(4*C*a*b^3 + B*b^4)*cos(d*x + c)^3 + 3*(24*C*a^2*b^2 + 16*B*a*b^3
+ (4*A + 3*C)*b^4)*cos(d*x + c)^2 + 16*(6*C*a^3*b + 9*B*a^2*b^2 + 2*(3*A + 2*C)*a*b^3 + B*b^4)*cos(d*x + c))*s
in(d*x + c))/(d*cos(d*x + c))

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giac [B]  time = 0.37, size = 802, normalized size = 2.94 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(48*A*a^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) - 3*(8*C*a^4 + 32*B*a^3*b + 48*A*a^2*b^2 + 2
4*C*a^2*b^2 + 16*B*a*b^3 + 4*A*b^4 + 3*C*b^4)*(d*x + c) - 24*(B*a^4 + 4*A*a^3*b)*log(abs(tan(1/2*d*x + 1/2*c)
+ 1)) + 24*(B*a^4 + 4*A*a^3*b)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(96*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 144
*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 72*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 96*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 -
48*B*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 96*C*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 12*A*b^4*tan(1/2*d*x + 1/2*c)^7 + 24*B
*b^4*tan(1/2*d*x + 1/2*c)^7 - 15*C*b^4*tan(1/2*d*x + 1/2*c)^7 + 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 432*B*a^2
*b^2*tan(1/2*d*x + 1/2*c)^5 - 72*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 48*B*
a*b^3*tan(1/2*d*x + 1/2*c)^5 + 160*C*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 12*A*b^4*tan(1/2*d*x + 1/2*c)^5 + 40*B*b^4
*tan(1/2*d*x + 1/2*c)^5 + 9*C*b^4*tan(1/2*d*x + 1/2*c)^5 + 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 432*B*a^2*b^2*
tan(1/2*d*x + 1/2*c)^3 + 72*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 48*B*a*b^3
*tan(1/2*d*x + 1/2*c)^3 + 160*C*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 12*A*b^4*tan(1/2*d*x + 1/2*c)^3 + 40*B*b^4*tan(
1/2*d*x + 1/2*c)^3 - 9*C*b^4*tan(1/2*d*x + 1/2*c)^3 + 96*C*a^3*b*tan(1/2*d*x + 1/2*c) + 144*B*a^2*b^2*tan(1/2*
d*x + 1/2*c) + 72*C*a^2*b^2*tan(1/2*d*x + 1/2*c) + 96*A*a*b^3*tan(1/2*d*x + 1/2*c) + 48*B*a*b^3*tan(1/2*d*x +
1/2*c) + 96*C*a*b^3*tan(1/2*d*x + 1/2*c) + 12*A*b^4*tan(1/2*d*x + 1/2*c) + 24*B*b^4*tan(1/2*d*x + 1/2*c) + 15*
C*b^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d

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maple [A]  time = 0.37, size = 434, normalized size = 1.59 \[ a^{4} C x +\frac {A x \,b^{4}}{2}+\frac {3 C \,b^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 a^{3} b C \sin \left (d x +c \right )}{d}+\frac {6 a^{2} b^{2} B \sin \left (d x +c \right )}{d}+\frac {4 a A \,b^{3} \sin \left (d x +c \right )}{d}+\frac {8 C a \,b^{3} \sin \left (d x +c \right )}{3 d}+\frac {A \,b^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {A \,a^{4} \tan \left (d x +c \right )}{d}+\frac {a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{4} C c}{d}+3 C \,a^{2} b^{2} x +\frac {2 B \,b^{4} \sin \left (d x +c \right )}{3 d}+2 B x a \,b^{3}+4 B x \,a^{3} b +6 A x \,a^{2} b^{2}+\frac {A \,b^{4} c}{2 d}+\frac {3 C \,b^{4} c}{8 d}+\frac {2 B a \,b^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+\frac {3 C \,a^{2} b^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+\frac {4 C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a \,b^{3}}{3 d}+\frac {3 b^{4} C x}{8}+\frac {B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) b^{4}}{3 d}+\frac {C \,b^{4} \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {2 B a \,b^{3} c}{d}+\frac {4 B \,a^{3} b c}{d}+\frac {6 A \,a^{2} b^{2} c}{d}+\frac {3 C \,a^{2} b^{2} c}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^4*C*x+1/2*A*x*b^4+3/8/d*C*b^4*cos(d*x+c)*sin(d*x+c)+4/d*A*a^3*b*ln(sec(d*x+c)+tan(d*x+c))+4/d*a^3*b*C*sin(d*
x+c)+6/d*a^2*b^2*B*sin(d*x+c)+4/d*a*A*b^3*sin(d*x+c)+1/3/d*B*sin(d*x+c)*cos(d*x+c)^2*b^4+8/3/d*C*a*b^3*sin(d*x
+c)+1/2/d*A*b^4*cos(d*x+c)*sin(d*x+c)+1/d*A*a^4*tan(d*x+c)+1/d*a^4*B*ln(sec(d*x+c)+tan(d*x+c))+1/d*a^4*C*c+3*C
*a^2*b^2*x+2/3/d*B*b^4*sin(d*x+c)+2*B*x*a*b^3+4*B*x*a^3*b+6*A*x*a^2*b^2+1/2/d*A*b^4*c+3/8/d*C*b^4*c+2/d*B*a*b^
3*cos(d*x+c)*sin(d*x+c)+4/3/d*C*sin(d*x+c)*cos(d*x+c)^2*a*b^3+3/d*C*a^2*b^2*cos(d*x+c)*sin(d*x+c)+3/8*b^4*C*x+
2/d*B*a*b^3*c+4/d*B*a^3*b*c+6/d*A*a^2*b^2*c+3/d*C*a^2*b^2*c+1/4/d*C*b^4*sin(d*x+c)*cos(d*x+c)^3

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maxima [A]  time = 0.35, size = 305, normalized size = 1.12 \[ \frac {96 \, {\left (d x + c\right )} C a^{4} + 384 \, {\left (d x + c\right )} B a^{3} b + 576 \, {\left (d x + c\right )} A a^{2} b^{2} + 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b^{2} + 96 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{3} - 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b^{3} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{4} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{4} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{4} + 48 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 192 \, A a^{3} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 384 \, C a^{3} b \sin \left (d x + c\right ) + 576 \, B a^{2} b^{2} \sin \left (d x + c\right ) + 384 \, A a b^{3} \sin \left (d x + c\right ) + 96 \, A a^{4} \tan \left (d x + c\right )}{96 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(96*(d*x + c)*C*a^4 + 384*(d*x + c)*B*a^3*b + 576*(d*x + c)*A*a^2*b^2 + 144*(2*d*x + 2*c + sin(2*d*x + 2*
c))*C*a^2*b^2 + 96*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a*b^3 - 128*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a*b^3 +
24*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*b^4 - 32*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*b^4 + 3*(12*d*x + 12*c + si
n(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*b^4 + 48*B*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 192*A*
a^3*b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 384*C*a^3*b*sin(d*x + c) + 576*B*a^2*b^2*sin(d*x + c)
+ 384*A*a*b^3*sin(d*x + c) + 96*A*a^4*tan(d*x + c))/d

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mupad [B]  time = 5.71, size = 4781, normalized size = 17.51 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(((((A*b^4*1i)/2 + C*a^4*1i + (C*b^4*3i)/8 + A*a^2*b^2*6i + C*a^2*b^2*3i + B*a*b^3*2i + B*a^3*b*4i)*(16*A
*b^4 + 32*B*a^4 + 32*C*a^4 + 12*C*b^4 + 192*A*a^2*b^2 + 96*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 + 128*B*a^3*b)
+ tan(c/2 + (d*x)/2)*(8*A^2*b^8 + 32*B^2*a^8 + 32*C^2*a^8 + (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^
4 + 512*A^2*a^6*b^2 + 128*B^2*a^2*b^6 + 512*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312*C^2*a^4*b^4 +
192*C^2*a^6*b^2 + 12*A*C*b^8 + 64*A*B*a*b^7 + 256*A*B*a^7*b + 48*B*C*a*b^7 + 256*B*C*a^7*b + 896*A*B*a^3*b^5
+ 1536*A*B*a^5*b^3 + 240*A*C*a^2*b^6 + 1184*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B*C*a^5*b^3)
)*((A*b^4*1i)/2 + C*a^4*1i + (C*b^4*3i)/8 + A*a^2*b^2*6i + C*a^2*b^2*3i + B*a*b^3*2i + B*a^3*b*4i)*1i - (((A*b
^4*1i)/2 + C*a^4*1i + (C*b^4*3i)/8 + A*a^2*b^2*6i + C*a^2*b^2*3i + B*a*b^3*2i + B*a^3*b*4i)*(16*A*b^4 + 32*B*a
^4 + 32*C*a^4 + 12*C*b^4 + 192*A*a^2*b^2 + 96*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 + 128*B*a^3*b) - tan(c/2 +
(d*x)/2)*(8*A^2*b^8 + 32*B^2*a^8 + 32*C^2*a^8 + (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 512*A^2*a
^6*b^2 + 128*B^2*a^2*b^6 + 512*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312*C^2*a^4*b^4 + 192*C^2*a^6*
b^2 + 12*A*C*b^8 + 64*A*B*a*b^7 + 256*A*B*a^7*b + 48*B*C*a*b^7 + 256*B*C*a^7*b + 896*A*B*a^3*b^5 + 1536*A*B*a^
5*b^3 + 240*A*C*a^2*b^6 + 1184*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B*C*a^5*b^3))*((A*b^4*1i)
/2 + C*a^4*1i + (C*b^4*3i)/8 + A*a^2*b^2*6i + C*a^2*b^2*3i + B*a*b^3*2i + B*a^3*b*4i)*1i)/((((A*b^4*1i)/2 + C*
a^4*1i + (C*b^4*3i)/8 + A*a^2*b^2*6i + C*a^2*b^2*3i + B*a*b^3*2i + B*a^3*b*4i)*(16*A*b^4 + 32*B*a^4 + 32*C*a^4
+ 12*C*b^4 + 192*A*a^2*b^2 + 96*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 + 128*B*a^3*b) + tan(c/2 + (d*x)/2)*(8*A
^2*b^8 + 32*B^2*a^8 + 32*C^2*a^8 + (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 128*
B^2*a^2*b^6 + 512*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312*C^2*a^4*b^4 + 192*C^2*a^6*b^2 + 12*A*C*
b^8 + 64*A*B*a*b^7 + 256*A*B*a^7*b + 48*B*C*a*b^7 + 256*B*C*a^7*b + 896*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 240*A
*C*a^2*b^6 + 1184*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B*C*a^5*b^3))*((A*b^4*1i)/2 + C*a^4*1i
+ (C*b^4*3i)/8 + A*a^2*b^2*6i + C*a^2*b^2*3i + B*a*b^3*2i + B*a^3*b*4i) + (((A*b^4*1i)/2 + C*a^4*1i + (C*b^4*
3i)/8 + A*a^2*b^2*6i + C*a^2*b^2*3i + B*a*b^3*2i + B*a^3*b*4i)*(16*A*b^4 + 32*B*a^4 + 32*C*a^4 + 12*C*b^4 + 19
2*A*a^2*b^2 + 96*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 + 128*B*a^3*b) - tan(c/2 + (d*x)/2)*(8*A^2*b^8 + 32*B^2*
a^8 + 32*C^2*a^8 + (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 128*B^2*a^2*b^6 + 51
2*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312*C^2*a^4*b^4 + 192*C^2*a^6*b^2 + 12*A*C*b^8 + 64*A*B*a*b
^7 + 256*A*B*a^7*b + 48*B*C*a*b^7 + 256*B*C*a^7*b + 896*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 240*A*C*a^2*b^6 + 118
4*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B*C*a^5*b^3))*((A*b^4*1i)/2 + C*a^4*1i + (C*b^4*3i)/8
+ A*a^2*b^2*6i + C*a^2*b^2*3i + B*a*b^3*2i + B*a^3*b*4i) + 64*B*C^2*a^12 - 64*B^2*C*a^12 - 256*B^3*a^11*b + 64
*A^3*a^3*b^9 + 1536*A^3*a^5*b^7 - 512*A^3*a^6*b^6 + 9216*A^3*a^7*b^5 - 6144*A^3*a^8*b^4 + 256*B^3*a^6*b^6 + 10
24*B^3*a^8*b^4 - 128*B^3*a^9*b^3 + 1024*B^3*a^10*b^2 + 256*A*C^2*a^11*b + 512*B^2*C*a^11*b + 1152*A*B^2*a^5*b^
7 + 5888*A*B^2*a^7*b^5 - 1056*A*B^2*a^8*b^4 + 7168*A*B^2*a^9*b^3 - 2432*A*B^2*a^10*b^2 + 528*A^2*B*a^4*b^8 + 7
552*A^2*B*a^6*b^6 - 2304*A^2*B*a^7*b^5 + 14592*A^2*B*a^8*b^4 - 7168*A^2*B*a^9*b^3 + 36*A*C^2*a^3*b^9 + 576*A*C
^2*a^5*b^7 + 2496*A*C^2*a^7*b^5 + 1536*A*C^2*a^9*b^3 + 96*A^2*C*a^3*b^9 + 1920*A^2*C*a^5*b^7 - 384*A^2*C*a^6*b
^6 + 9472*A^2*C*a^7*b^5 - 3072*A^2*C*a^8*b^4 + 3072*A^2*C*a^9*b^3 - 1024*A^2*C*a^10*b^2 + 9*B*C^2*a^4*b^8 + 14
4*B*C^2*a^6*b^6 + 624*B*C^2*a^8*b^4 + 384*B*C^2*a^10*b^2 + 96*B^2*C*a^5*b^7 + 960*B^2*C*a^7*b^5 - 24*B^2*C*a^8
*b^4 + 1792*B^2*C*a^9*b^3 - 192*B^2*C*a^10*b^2 - 512*A*B*C*a^11*b + 408*A*B*C*a^4*b^8 + 4320*A*B*C*a^6*b^6 - 1
92*A*B*C*a^7*b^5 + 9536*A*B*C*a^8*b^4 - 1536*A*B*C*a^9*b^3 + 2816*A*B*C*a^10*b^2))*(A*b^4 + 2*C*a^4 + (3*C*b^4
)/4 + 12*A*a^2*b^2 + 6*C*a^2*b^2 + 4*B*a*b^3 + 8*B*a^3*b))/d + (tan(c/2 + (d*x)/2)*(2*A*a^4 + A*b^4 + 2*B*b^4
+ (5*C*b^4)/4 + 12*B*a^2*b^2 + 6*C*a^2*b^2 + 8*A*a*b^3 + 4*B*a*b^3 + 8*C*a*b^3 + 8*C*a^3*b) + tan(c/2 + (d*x)/
2)^3*(8*A*a^4 + (4*B*b^4)/3 - 2*C*b^4 + 24*B*a^2*b^2 + 16*A*a*b^3 + (16*C*a*b^3)/3 + 16*C*a^3*b) - tan(c/2 + (
d*x)/2)^7*((4*B*b^4)/3 - 8*A*a^4 + 2*C*b^4 + 24*B*a^2*b^2 + 16*A*a*b^3 + (16*C*a*b^3)/3 + 16*C*a^3*b) + tan(c/
2 + (d*x)/2)^9*(2*A*a^4 + A*b^4 - 2*B*b^4 + (5*C*b^4)/4 - 12*B*a^2*b^2 + 6*C*a^2*b^2 - 8*A*a*b^3 + 4*B*a*b^3 -
8*C*a*b^3 - 8*C*a^3*b) - tan(c/2 + (d*x)/2)^5*(2*A*b^4 - 12*A*a^4 - (3*C*b^4)/2 + 12*C*a^2*b^2 + 8*B*a*b^3))/
(d*(3*tan(c/2 + (d*x)/2)^2 + 2*tan(c/2 + (d*x)/2)^4 - 2*tan(c/2 + (d*x)/2)^6 - 3*tan(c/2 + (d*x)/2)^8 - tan(c/
2 + (d*x)/2)^10 + 1)) - (atan((((B*a^4 + 4*A*a^3*b)*(16*A*b^4 + 32*B*a^4 + 32*C*a^4 + 12*C*b^4 + 192*A*a^2*b^2
+ 96*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 + 128*B*a^3*b) + tan(c/2 + (d*x)/2)*(8*A^2*b^8 + 32*B^2*a^8 + 32*C^
2*a^8 + (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 128*B^2*a^2*b^6 + 512*B^2*a^4*b
^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312*C^2*a^4*b^4 + 192*C^2*a^6*b^2 + 12*A*C*b^8 + 64*A*B*a*b^7 + 256*A*
B*a^7*b + 48*B*C*a*b^7 + 256*B*C*a^7*b + 896*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 240*A*C*a^2*b^6 + 1184*A*C*a^4*b
^4 + 384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B*C*a^5*b^3))*(B*a^4 + 4*A*a^3*b)*1i - ((B*a^4 + 4*A*a^3*b)*(16*A
*b^4 + 32*B*a^4 + 32*C*a^4 + 12*C*b^4 + 192*A*a^2*b^2 + 96*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 + 128*B*a^3*b)
- tan(c/2 + (d*x)/2)*(8*A^2*b^8 + 32*B^2*a^8 + 32*C^2*a^8 + (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^
4 + 512*A^2*a^6*b^2 + 128*B^2*a^2*b^6 + 512*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312*C^2*a^4*b^4 +
192*C^2*a^6*b^2 + 12*A*C*b^8 + 64*A*B*a*b^7 + 256*A*B*a^7*b + 48*B*C*a*b^7 + 256*B*C*a^7*b + 896*A*B*a^3*b^5
+ 1536*A*B*a^5*b^3 + 240*A*C*a^2*b^6 + 1184*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B*C*a^5*b^3)
)*(B*a^4 + 4*A*a^3*b)*1i)/(((B*a^4 + 4*A*a^3*b)*(16*A*b^4 + 32*B*a^4 + 32*C*a^4 + 12*C*b^4 + 192*A*a^2*b^2 + 9
6*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 + 128*B*a^3*b) + tan(c/2 + (d*x)/2)*(8*A^2*b^8 + 32*B^2*a^8 + 32*C^2*a^
8 + (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 128*B^2*a^2*b^6 + 512*B^2*a^4*b^4 +
512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312*C^2*a^4*b^4 + 192*C^2*a^6*b^2 + 12*A*C*b^8 + 64*A*B*a*b^7 + 256*A*B*a^
7*b + 48*B*C*a*b^7 + 256*B*C*a^7*b + 896*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 240*A*C*a^2*b^6 + 1184*A*C*a^4*b^4 +
384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B*C*a^5*b^3))*(B*a^4 + 4*A*a^3*b) + ((B*a^4 + 4*A*a^3*b)*(16*A*b^4 +
32*B*a^4 + 32*C*a^4 + 12*C*b^4 + 192*A*a^2*b^2 + 96*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 + 128*B*a^3*b) - tan(
c/2 + (d*x)/2)*(8*A^2*b^8 + 32*B^2*a^8 + 32*C^2*a^8 + (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 512
*A^2*a^6*b^2 + 128*B^2*a^2*b^6 + 512*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312*C^2*a^4*b^4 + 192*C^
2*a^6*b^2 + 12*A*C*b^8 + 64*A*B*a*b^7 + 256*A*B*a^7*b + 48*B*C*a*b^7 + 256*B*C*a^7*b + 896*A*B*a^3*b^5 + 1536*
A*B*a^5*b^3 + 240*A*C*a^2*b^6 + 1184*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B*C*a^5*b^3))*(B*a^
4 + 4*A*a^3*b) + 64*B*C^2*a^12 - 64*B^2*C*a^12 - 256*B^3*a^11*b + 64*A^3*a^3*b^9 + 1536*A^3*a^5*b^7 - 512*A^3*
a^6*b^6 + 9216*A^3*a^7*b^5 - 6144*A^3*a^8*b^4 + 256*B^3*a^6*b^6 + 1024*B^3*a^8*b^4 - 128*B^3*a^9*b^3 + 1024*B^
3*a^10*b^2 + 256*A*C^2*a^11*b + 512*B^2*C*a^11*b + 1152*A*B^2*a^5*b^7 + 5888*A*B^2*a^7*b^5 - 1056*A*B^2*a^8*b^
4 + 7168*A*B^2*a^9*b^3 - 2432*A*B^2*a^10*b^2 + 528*A^2*B*a^4*b^8 + 7552*A^2*B*a^6*b^6 - 2304*A^2*B*a^7*b^5 + 1
4592*A^2*B*a^8*b^4 - 7168*A^2*B*a^9*b^3 + 36*A*C^2*a^3*b^9 + 576*A*C^2*a^5*b^7 + 2496*A*C^2*a^7*b^5 + 1536*A*C
^2*a^9*b^3 + 96*A^2*C*a^3*b^9 + 1920*A^2*C*a^5*b^7 - 384*A^2*C*a^6*b^6 + 9472*A^2*C*a^7*b^5 - 3072*A^2*C*a^8*b
^4 + 3072*A^2*C*a^9*b^3 - 1024*A^2*C*a^10*b^2 + 9*B*C^2*a^4*b^8 + 144*B*C^2*a^6*b^6 + 624*B*C^2*a^8*b^4 + 384*
B*C^2*a^10*b^2 + 96*B^2*C*a^5*b^7 + 960*B^2*C*a^7*b^5 - 24*B^2*C*a^8*b^4 + 1792*B^2*C*a^9*b^3 - 192*B^2*C*a^10
*b^2 - 512*A*B*C*a^11*b + 408*A*B*C*a^4*b^8 + 4320*A*B*C*a^6*b^6 - 192*A*B*C*a^7*b^5 + 9536*A*B*C*a^8*b^4 - 15
36*A*B*C*a^9*b^3 + 2816*A*B*C*a^10*b^2))*(B*a^4*2i + A*a^3*b*8i))/d

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**2,x)

[Out]

Timed out

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