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

Optimal. Leaf size=303 \[ -\frac {b^2 \sin (c+d x) \cos (c+d x) \left (a^2 (4 A+6 C)+18 a b B+3 b^2 (6 A-C)\right )}{6 d}+\frac {\tan (c+d x) \left (a^2 (4 A+6 C)+15 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{6 d}+\frac {1}{2} b^2 x \left (12 a^2 C+8 a b B+2 A b^2+b^2 C\right )-\frac {b \sin (c+d x) \left (4 a^3 (2 A+3 C)+39 a^2 b B+4 a b^2 (11 A-6 C)-6 b^3 B\right )}{6 d}+\frac {a \left (a^3 B+4 a^2 b (A+2 C)+12 a b^2 B+8 A b^3\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {(3 a B+4 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^4}{3 d} \]

[Out]

1/2*b^2*(2*A*b^2+8*B*a*b+12*C*a^2+C*b^2)*x+1/2*a*(8*A*b^3+a^3*B+12*a*b^2*B+4*a^2*b*(A+2*C))*arctanh(sin(d*x+c)
)/d-1/6*b*(39*a^2*b*B-6*b^3*B+4*a*b^2*(11*A-6*C)+4*a^3*(2*A+3*C))*sin(d*x+c)/d-1/6*b^2*(18*a*b*B+3*b^2*(6*A-C)
+a^2*(4*A+6*C))*cos(d*x+c)*sin(d*x+c)/d+1/6*(12*A*b^2+15*a*b*B+a^2*(4*A+6*C))*(a+b*cos(d*x+c))^2*tan(d*x+c)/d+
1/6*(4*A*b+3*B*a)*(a+b*cos(d*x+c))^3*sec(d*x+c)*tan(d*x+c)/d+1/3*A*(a+b*cos(d*x+c))^4*sec(d*x+c)^2*tan(d*x+c)/
d

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Rubi [A]  time = 1.08, antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3047, 3033, 3023, 2735, 3770} \[ -\frac {b \sin (c+d x) \left (4 a^3 (2 A+3 C)+39 a^2 b B+4 a b^2 (11 A-6 C)-6 b^3 B\right )}{6 d}+\frac {a \left (4 a^2 b (A+2 C)+a^3 B+12 a b^2 B+8 A b^3\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {b^2 \sin (c+d x) \cos (c+d x) \left (a^2 (4 A+6 C)+18 a b B+3 b^2 (6 A-C)\right )}{6 d}+\frac {\tan (c+d x) \left (a^2 (4 A+6 C)+15 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{6 d}+\frac {1}{2} b^2 x \left (12 a^2 C+8 a b B+2 A b^2+b^2 C\right )+\frac {(3 a B+4 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^4}{3 d} \]

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]^4,x]

[Out]

(b^2*(2*A*b^2 + 8*a*b*B + 12*a^2*C + b^2*C)*x)/2 + (a*(8*A*b^3 + a^3*B + 12*a*b^2*B + 4*a^2*b*(A + 2*C))*ArcTa
nh[Sin[c + d*x]])/(2*d) - (b*(39*a^2*b*B - 6*b^3*B + 4*a*b^2*(11*A - 6*C) + 4*a^3*(2*A + 3*C))*Sin[c + d*x])/(
6*d) - (b^2*(18*a*b*B + 3*b^2*(6*A - C) + a^2*(4*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(6*d) + ((12*A*b^2 + 15*
a*b*B + a^2*(4*A + 6*C))*(a + b*Cos[c + d*x])^2*Tan[c + d*x])/(6*d) + ((4*A*b + 3*a*B)*(a + b*Cos[c + d*x])^3*
Sec[c + d*x]*Tan[c + d*x])/(6*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^2*Tan[c + d*x])/(3*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 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 ^4(c+d x) \, dx &=\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{3} \int (a+b \cos (c+d x))^3 \left (4 A b+3 a B+(2 a A+3 b B+3 a C) \cos (c+d x)-b (2 A-3 C) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac {(4 A b+3 a B) (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{6} \int (a+b \cos (c+d x))^2 \left (12 A b^2+15 a b B+a^2 (4 A+6 C)+\left (3 a^2 B+6 b^2 B+4 a b (A+3 C)\right ) \cos (c+d x)-6 b (2 A b+a B-b C) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {\left (12 A b^2+15 a b B+a^2 (4 A+6 C)\right ) (a+b \cos (c+d x))^2 \tan (c+d x)}{6 d}+\frac {(4 A b+3 a B) (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{6} \int (a+b \cos (c+d x)) \left (3 \left (8 A b^3+a^3 B+12 a b^2 B+4 a^2 b (A+2 C)\right )-b \left (8 a A b+3 a^2 B-6 b^2 B-18 a b C\right ) \cos (c+d x)-2 b \left (18 a b B+3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {b^2 \left (18 a b B+3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {\left (12 A b^2+15 a b B+a^2 (4 A+6 C)\right ) (a+b \cos (c+d x))^2 \tan (c+d x)}{6 d}+\frac {(4 A b+3 a B) (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{12} \int \left (6 a \left (8 A b^3+a^3 B+12 a b^2 B+4 a^2 b (A+2 C)\right )+6 b^2 \left (2 A b^2+8 a b B+12 a^2 C+b^2 C\right ) \cos (c+d x)-2 b \left (39 a^2 b B-6 b^3 B+4 a b^2 (11 A-6 C)+4 a^3 (2 A+3 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {b \left (39 a^2 b B-6 b^3 B+4 a b^2 (11 A-6 C)+4 a^3 (2 A+3 C)\right ) \sin (c+d x)}{6 d}-\frac {b^2 \left (18 a b B+3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {\left (12 A b^2+15 a b B+a^2 (4 A+6 C)\right ) (a+b \cos (c+d x))^2 \tan (c+d x)}{6 d}+\frac {(4 A b+3 a B) (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{12} \int \left (6 a \left (8 A b^3+a^3 B+12 a b^2 B+4 a^2 b (A+2 C)\right )+6 b^2 \left (2 A b^2+8 a b B+12 a^2 C+b^2 C\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac {1}{2} b^2 \left (2 A b^2+8 a b B+12 a^2 C+b^2 C\right ) x-\frac {b \left (39 a^2 b B-6 b^3 B+4 a b^2 (11 A-6 C)+4 a^3 (2 A+3 C)\right ) \sin (c+d x)}{6 d}-\frac {b^2 \left (18 a b B+3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {\left (12 A b^2+15 a b B+a^2 (4 A+6 C)\right ) (a+b \cos (c+d x))^2 \tan (c+d x)}{6 d}+\frac {(4 A b+3 a B) (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{2} \left (a \left (8 A b^3+a^3 B+12 a b^2 B+4 a^2 b (A+2 C)\right )\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{2} b^2 \left (2 A b^2+8 a b B+12 a^2 C+b^2 C\right ) x+\frac {a \left (8 A b^3+a^3 B+12 a b^2 B+4 a^2 b (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {b \left (39 a^2 b B-6 b^3 B+4 a b^2 (11 A-6 C)+4 a^3 (2 A+3 C)\right ) \sin (c+d x)}{6 d}-\frac {b^2 \left (18 a b B+3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {\left (12 A b^2+15 a b B+a^2 (4 A+6 C)\right ) (a+b \cos (c+d x))^2 \tan (c+d x)}{6 d}+\frac {(4 A b+3 a B) (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{6 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end {align*}

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Mathematica [A]  time = 2.27, size = 351, normalized size = 1.16 \[ \frac {\sec ^3(c+d x) \left (36 b^2 (c+d x) \cos (c+d x) \left (12 a^2 C+8 a b B+2 A b^2+b^2 C\right )+12 b^2 (c+d x) \cos (3 (c+d x)) \left (12 a^2 C+8 a b B+2 A b^2+b^2 C\right )-48 a \cos ^3(c+d x) \left (a^3 B+4 a^2 b (A+2 C)+12 a b^2 B+8 A b^3\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+2 \sin (c+d x) \left (32 a^4 A+24 a^4 C+96 a^3 b B+144 a^2 A b^2+12 \cos (c+d x) \left (2 a^4 B+8 a^3 A b+12 a b^3 C+3 b^4 B\right )+4 \cos (2 (c+d x)) \left (a^4 (4 A+6 C)+24 a^3 b B+36 a^2 A b^2+3 b^4 C\right )+48 a b^3 C \cos (3 (c+d x))+12 b^4 B \cos (3 (c+d x))+3 b^4 C \cos (4 (c+d x))+9 b^4 C\right )\right )}{96 d} \]

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]^4,x]

[Out]

(Sec[c + d*x]^3*(36*b^2*(2*A*b^2 + 8*a*b*B + 12*a^2*C + b^2*C)*(c + d*x)*Cos[c + d*x] + 12*b^2*(2*A*b^2 + 8*a*
b*B + 12*a^2*C + b^2*C)*(c + d*x)*Cos[3*(c + d*x)] - 48*a*(8*A*b^3 + a^3*B + 12*a*b^2*B + 4*a^2*b*(A + 2*C))*C
os[c + d*x]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*(32*a^
4*A + 144*a^2*A*b^2 + 96*a^3*b*B + 24*a^4*C + 9*b^4*C + 12*(8*a^3*A*b + 2*a^4*B + 3*b^4*B + 12*a*b^3*C)*Cos[c
+ d*x] + 4*(36*a^2*A*b^2 + 24*a^3*b*B + 3*b^4*C + a^4*(4*A + 6*C))*Cos[2*(c + d*x)] + 12*b^4*B*Cos[3*(c + d*x)
] + 48*a*b^3*C*Cos[3*(c + d*x)] + 3*b^4*C*Cos[4*(c + d*x)])*Sin[c + d*x]))/(96*d)

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fricas [A]  time = 0.49, size = 269, normalized size = 0.89 \[ \frac {6 \, {\left (12 \, C a^{2} b^{2} + 8 \, B a b^{3} + {\left (2 \, A + C\right )} b^{4}\right )} d x \cos \left (d x + c\right )^{3} + 3 \, {\left (B a^{4} + 4 \, {\left (A + 2 \, C\right )} a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (B a^{4} + 4 \, {\left (A + 2 \, C\right )} a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, C b^{4} \cos \left (d x + c\right )^{4} + 2 \, A a^{4} + 6 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{4} + 12 \, B a^{3} b + 18 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]

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)^4,x, algorithm="fricas")

[Out]

1/12*(6*(12*C*a^2*b^2 + 8*B*a*b^3 + (2*A + C)*b^4)*d*x*cos(d*x + c)^3 + 3*(B*a^4 + 4*(A + 2*C)*a^3*b + 12*B*a^
2*b^2 + 8*A*a*b^3)*cos(d*x + c)^3*log(sin(d*x + c) + 1) - 3*(B*a^4 + 4*(A + 2*C)*a^3*b + 12*B*a^2*b^2 + 8*A*a*
b^3)*cos(d*x + c)^3*log(-sin(d*x + c) + 1) + 2*(3*C*b^4*cos(d*x + c)^4 + 2*A*a^4 + 6*(4*C*a*b^3 + B*b^4)*cos(d
*x + c)^3 + 2*((2*A + 3*C)*a^4 + 12*B*a^3*b + 18*A*a^2*b^2)*cos(d*x + c)^2 + 3*(B*a^4 + 4*A*a^3*b)*cos(d*x + c
))*sin(d*x + c))/(d*cos(d*x + c)^3)

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giac [A]  time = 0.35, size = 550, normalized size = 1.82 \[ \frac {3 \, {\left (12 \, C a^{2} b^{2} + 8 \, B a b^{3} + 2 \, A b^{4} + C b^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b + 8 \, C a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (B a^{4} + 4 \, A a^{3} b + 8 \, C a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {6 \, {\left (8 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac {2 \, {\left (6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, 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)^4,x, algorithm="giac")

[Out]

1/6*(3*(12*C*a^2*b^2 + 8*B*a*b^3 + 2*A*b^4 + C*b^4)*(d*x + c) + 3*(B*a^4 + 4*A*a^3*b + 8*C*a^3*b + 12*B*a^2*b^
2 + 8*A*a*b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(B*a^4 + 4*A*a^3*b + 8*C*a^3*b + 12*B*a^2*b^2 + 8*A*a*b^
3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 6*(8*C*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 2*B*b^4*tan(1/2*d*x + 1/2*c)^3 -
C*b^4*tan(1/2*d*x + 1/2*c)^3 + 8*C*a*b^3*tan(1/2*d*x + 1/2*c) + 2*B*b^4*tan(1/2*d*x + 1/2*c) + C*b^4*tan(1/2*
d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^2 - 2*(6*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 3*B*a^4*tan(1/2*d*x + 1/2*c
)^5 + 6*C*a^4*tan(1/2*d*x + 1/2*c)^5 - 12*A*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 24*B*a^3*b*tan(1/2*d*x + 1/2*c)^5 +
36*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 - 4*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 12*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 48*B
*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 72*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 6*A*a^4*tan(1/2*d*x + 1/2*c) + 3*B*a^4*t
an(1/2*d*x + 1/2*c) + 6*C*a^4*tan(1/2*d*x + 1/2*c) + 12*A*a^3*b*tan(1/2*d*x + 1/2*c) + 24*B*a^3*b*tan(1/2*d*x
+ 1/2*c) + 36*A*a^2*b^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^3)/d

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maple [A]  time = 0.38, size = 377, normalized size = 1.24 \[ \frac {2 A \,a^{4} \tan \left (d x +c \right )}{3 d}+\frac {A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {a^{4} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {a^{4} C \tan \left (d x +c \right )}{d}+\frac {2 A \,a^{3} b \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {2 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 B \,a^{3} b \tan \left (d x +c \right )}{d}+\frac {4 a^{3} b C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {6 A \,a^{2} b^{2} \tan \left (d x +c \right )}{d}+\frac {6 a^{2} b^{2} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+6 C \,a^{2} b^{2} x +\frac {6 C \,a^{2} b^{2} c}{d}+\frac {4 a A \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+4 B x a \,b^{3}+\frac {4 B a \,b^{3} c}{d}+\frac {4 C a \,b^{3} \sin \left (d x +c \right )}{d}+A x \,b^{4}+\frac {A \,b^{4} c}{d}+\frac {B \,b^{4} \sin \left (d x +c \right )}{d}+\frac {C \,b^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {b^{4} C x}{2}+\frac {C \,b^{4} c}{2 d} \]

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)^4,x)

[Out]

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

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maxima [A]  time = 0.36, size = 335, normalized size = 1.11 \[ \frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 72 \, {\left (d x + c\right )} C a^{2} b^{2} + 48 \, {\left (d x + c\right )} B a b^{3} + 12 \, {\left (d x + c\right )} A b^{4} + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{4} - 3 \, B a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a^{3} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, B a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A a b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, C a b^{3} \sin \left (d x + c\right ) + 12 \, B b^{4} \sin \left (d x + c\right ) + 12 \, C a^{4} \tan \left (d x + c\right ) + 48 \, B a^{3} b \tan \left (d x + c\right ) + 72 \, A a^{2} b^{2} \tan \left (d x + c\right )}{12 \, d} \]

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)^4,x, algorithm="maxima")

[Out]

1/12*(4*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 72*(d*x + c)*C*a^2*b^2 + 48*(d*x + c)*B*a*b^3 + 12*(d*x + c)
*A*b^4 + 3*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*b^4 - 3*B*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x
+ c) + 1) + log(sin(d*x + c) - 1)) - 12*A*a^3*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1)
+ log(sin(d*x + c) - 1)) + 24*C*a^3*b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 36*B*a^2*b^2*(log(sin(
d*x + c) + 1) - log(sin(d*x + c) - 1)) + 24*A*a*b^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 48*C*a*b
^3*sin(d*x + c) + 12*B*b^4*sin(d*x + c) + 12*C*a^4*tan(d*x + c) + 48*B*a^3*b*tan(d*x + c) + 72*A*a^2*b^2*tan(d
*x + c))/d

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mupad [B]  time = 5.73, size = 4849, normalized size = 16.00 \[ \text {result too large to display} \]

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)^4,x)

[Out]

(atan(((((B*a^4)/2 + 6*B*a^2*b^2 + 4*A*a*b^3 + 2*A*a^3*b + 4*C*a^3*b)*(32*A*b^4 + 16*B*a^4 + 16*C*b^4 + 192*B*
a^2*b^2 + 192*C*a^2*b^2 + 128*A*a*b^3 + 64*A*a^3*b + 128*B*a*b^3 + 128*C*a^3*b) + tan(c/2 + (d*x)/2)*(32*A^2*b
^8 + 8*B^2*a^8 + 8*C^2*b^8 + 512*A^2*a^2*b^6 + 512*A^2*a^4*b^4 + 128*A^2*a^6*b^2 + 512*B^2*a^2*b^6 + 1152*B^2*
a^4*b^4 + 192*B^2*a^6*b^2 + 192*C^2*a^2*b^6 + 1152*C^2*a^4*b^4 + 512*C^2*a^6*b^2 + 32*A*C*b^8 + 256*A*B*a*b^7
+ 64*A*B*a^7*b + 128*B*C*a*b^7 + 128*B*C*a^7*b + 1536*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 384*A*C*a^2*b^6 + 1024*A
*C*a^4*b^4 + 512*A*C*a^6*b^2 + 1536*B*C*a^3*b^5 + 1536*B*C*a^5*b^3))*((B*a^4)/2 + 6*B*a^2*b^2 + 4*A*a*b^3 + 2*
A*a^3*b + 4*C*a^3*b)*1i - (((B*a^4)/2 + 6*B*a^2*b^2 + 4*A*a*b^3 + 2*A*a^3*b + 4*C*a^3*b)*(32*A*b^4 + 16*B*a^4
+ 16*C*b^4 + 192*B*a^2*b^2 + 192*C*a^2*b^2 + 128*A*a*b^3 + 64*A*a^3*b + 128*B*a*b^3 + 128*C*a^3*b) - tan(c/2 +
(d*x)/2)*(32*A^2*b^8 + 8*B^2*a^8 + 8*C^2*b^8 + 512*A^2*a^2*b^6 + 512*A^2*a^4*b^4 + 128*A^2*a^6*b^2 + 512*B^2*
a^2*b^6 + 1152*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 192*C^2*a^2*b^6 + 1152*C^2*a^4*b^4 + 512*C^2*a^6*b^2 + 32*A*C*b
^8 + 256*A*B*a*b^7 + 64*A*B*a^7*b + 128*B*C*a*b^7 + 128*B*C*a^7*b + 1536*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 384*A
*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 1536*B*C*a^3*b^5 + 1536*B*C*a^5*b^3))*((B*a^4)/2 + 6*B*a^2*b
^2 + 4*A*a*b^3 + 2*A*a^3*b + 4*C*a^3*b)*1i)/(1024*A^3*a^2*b^10 - (((B*a^4)/2 + 6*B*a^2*b^2 + 4*A*a*b^3 + 2*A*a
^3*b + 4*C*a^3*b)*(32*A*b^4 + 16*B*a^4 + 16*C*b^4 + 192*B*a^2*b^2 + 192*C*a^2*b^2 + 128*A*a*b^3 + 64*A*a^3*b +
128*B*a*b^3 + 128*C*a^3*b) - tan(c/2 + (d*x)/2)*(32*A^2*b^8 + 8*B^2*a^8 + 8*C^2*b^8 + 512*A^2*a^2*b^6 + 512*A
^2*a^4*b^4 + 128*A^2*a^6*b^2 + 512*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 192*C^2*a^2*b^6 + 1152*C
^2*a^4*b^4 + 512*C^2*a^6*b^2 + 32*A*C*b^8 + 256*A*B*a*b^7 + 64*A*B*a^7*b + 128*B*C*a*b^7 + 128*B*C*a^7*b + 153
6*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 384*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 1536*B*C*a^3*b^5 + 15
36*B*C*a^5*b^3))*((B*a^4)/2 + 6*B*a^2*b^2 + 4*A*a*b^3 + 2*A*a^3*b + 4*C*a^3*b) - 256*A^3*a*b^11 - (((B*a^4)/2
+ 6*B*a^2*b^2 + 4*A*a*b^3 + 2*A*a^3*b + 4*C*a^3*b)*(32*A*b^4 + 16*B*a^4 + 16*C*b^4 + 192*B*a^2*b^2 + 192*C*a^2
*b^2 + 128*A*a*b^3 + 64*A*a^3*b + 128*B*a*b^3 + 128*C*a^3*b) + tan(c/2 + (d*x)/2)*(32*A^2*b^8 + 8*B^2*a^8 + 8*
C^2*b^8 + 512*A^2*a^2*b^6 + 512*A^2*a^4*b^4 + 128*A^2*a^6*b^2 + 512*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 192*B^2*a
^6*b^2 + 192*C^2*a^2*b^6 + 1152*C^2*a^4*b^4 + 512*C^2*a^6*b^2 + 32*A*C*b^8 + 256*A*B*a*b^7 + 64*A*B*a^7*b + 12
8*B*C*a*b^7 + 128*B*C*a^7*b + 1536*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 384*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 512*A*
C*a^6*b^2 + 1536*B*C*a^3*b^5 + 1536*B*C*a^5*b^3))*((B*a^4)/2 + 6*B*a^2*b^2 + 4*A*a*b^3 + 2*A*a^3*b + 4*C*a^3*b
) - 128*A^3*a^3*b^9 + 1024*A^3*a^4*b^8 + 256*A^3*a^6*b^6 - 6144*B^3*a^4*b^8 + 9216*B^3*a^5*b^7 - 512*B^3*a^6*b
^6 + 1536*B^3*a^7*b^5 + 64*B^3*a^9*b^3 - 64*C^3*a^3*b^9 - 1536*C^3*a^5*b^7 + 512*C^3*a^6*b^6 - 9216*C^3*a^7*b^
5 + 6144*C^3*a^8*b^4 - 64*A*C^2*a*b^11 - 256*A^2*C*a*b^11 - 7168*A*B^2*a^3*b^9 + 14592*A*B^2*a^4*b^8 - 2304*A*
B^2*a^5*b^7 + 7552*A*B^2*a^6*b^6 + 528*A*B^2*a^8*b^4 - 2432*A^2*B*a^2*b^10 + 7168*A^2*B*a^3*b^9 - 1056*A^2*B*a
^4*b^8 + 5888*A^2*B*a^5*b^7 + 1152*A^2*B*a^7*b^5 - 1824*A*C^2*a^3*b^9 + 1024*A*C^2*a^4*b^8 - 13056*A*C^2*a^5*b
^7 + 13824*A*C^2*a^6*b^6 - 4608*A*C^2*a^7*b^5 + 6144*A*C^2*a^8*b^4 + 512*A^2*C*a^2*b^10 - 3456*A^2*C*a^3*b^9 +
8704*A^2*C*a^4*b^8 - 1536*A^2*C*a^5*b^7 + 7296*A^2*C*a^6*b^6 + 1536*A^2*C*a^8*b^4 - 96*B*C^2*a^2*b^10 - 3336*
B*C^2*a^4*b^8 + 1536*B*C^2*a^5*b^7 - 26304*B*C^2*a^6*b^6 + 22656*B*C^2*a^7*b^5 - 1152*B*C^2*a^8*b^4 + 1536*B*C
^2*a^9*b^3 - 1536*B^2*C*a^3*b^9 + 1152*B^2*C*a^4*b^8 - 22656*B^2*C*a^5*b^7 + 26304*B^2*C*a^6*b^6 - 1536*B^2*C*
a^7*b^5 + 3336*B^2*C*a^8*b^4 + 96*B^2*C*a^10*b^2 - 1408*A*B*C*a^2*b^10 + 1536*A*B*C*a^3*b^9 - 19488*A*B*C*a^4*
b^8 + 30592*A*B*C*a^5*b^7 - 6528*A*B*C*a^6*b^6 + 15168*A*B*C*a^7*b^5 + 768*A*B*C*a^9*b^3))*(B*a^4*1i + B*a^2*b
^2*12i + A*a*b^3*8i + A*a^3*b*4i + C*a^3*b*8i))/d - (tan(c/2 + (d*x)/2)^3*((8*A*a^4)/3 + 2*B*a^4 - 4*B*b^4 - 4
*C*b^4 + 8*A*a^3*b - 16*C*a*b^3) + tan(c/2 + (d*x)/2)^7*((8*A*a^4)/3 - 2*B*a^4 + 4*B*b^4 - 4*C*b^4 - 8*A*a^3*b
+ 16*C*a*b^3) + tan(c/2 + (d*x)/2)^9*(2*A*a^4 - B*a^4 - 2*B*b^4 + 2*C*a^4 + C*b^4 + 12*A*a^2*b^2 - 4*A*a^3*b
+ 8*B*a^3*b - 8*C*a*b^3) + tan(c/2 + (d*x)/2)*(2*A*a^4 + B*a^4 + 2*B*b^4 + 2*C*a^4 + C*b^4 + 12*A*a^2*b^2 + 4*
A*a^3*b + 8*B*a^3*b + 8*C*a*b^3) - tan(c/2 + (d*x)/2)^5*(4*C*a^4 - (4*A*a^4)/3 - 6*C*b^4 + 24*A*a^2*b^2 + 16*B
*a^3*b))/(d*(tan(c/2 + (d*x)/2)^2 + 2*tan(c/2 + (d*x)/2)^4 - 2*tan(c/2 + (d*x)/2)^6 - tan(c/2 + (d*x)/2)^8 + t
an(c/2 + (d*x)/2)^10 - 1)) + (b^2*atan(((b^2*(tan(c/2 + (d*x)/2)*(32*A^2*b^8 + 8*B^2*a^8 + 8*C^2*b^8 + 512*A^2
*a^2*b^6 + 512*A^2*a^4*b^4 + 128*A^2*a^6*b^2 + 512*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 192*C^2*
a^2*b^6 + 1152*C^2*a^4*b^4 + 512*C^2*a^6*b^2 + 32*A*C*b^8 + 256*A*B*a*b^7 + 64*A*B*a^7*b + 128*B*C*a*b^7 + 128
*B*C*a^7*b + 1536*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 384*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 1536*
B*C*a^3*b^5 + 1536*B*C*a^5*b^3) - (b^2*(2*A*b^2 + 12*C*a^2 + C*b^2 + 8*B*a*b)*(32*A*b^4 + 16*B*a^4 + 16*C*b^4
+ 192*B*a^2*b^2 + 192*C*a^2*b^2 + 128*A*a*b^3 + 64*A*a^3*b + 128*B*a*b^3 + 128*C*a^3*b)*1i)/2)*(2*A*b^2 + 12*C
*a^2 + C*b^2 + 8*B*a*b))/2 + (b^2*(tan(c/2 + (d*x)/2)*(32*A^2*b^8 + 8*B^2*a^8 + 8*C^2*b^8 + 512*A^2*a^2*b^6 +
512*A^2*a^4*b^4 + 128*A^2*a^6*b^2 + 512*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 192*C^2*a^2*b^6 + 1
152*C^2*a^4*b^4 + 512*C^2*a^6*b^2 + 32*A*C*b^8 + 256*A*B*a*b^7 + 64*A*B*a^7*b + 128*B*C*a*b^7 + 128*B*C*a^7*b
+ 1536*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 384*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 1536*B*C*a^3*b^5
+ 1536*B*C*a^5*b^3) + (b^2*(2*A*b^2 + 12*C*a^2 + C*b^2 + 8*B*a*b)*(32*A*b^4 + 16*B*a^4 + 16*C*b^4 + 192*B*a^2
*b^2 + 192*C*a^2*b^2 + 128*A*a*b^3 + 64*A*a^3*b + 128*B*a*b^3 + 128*C*a^3*b)*1i)/2)*(2*A*b^2 + 12*C*a^2 + C*b^
2 + 8*B*a*b))/2)/((b^2*(tan(c/2 + (d*x)/2)*(32*A^2*b^8 + 8*B^2*a^8 + 8*C^2*b^8 + 512*A^2*a^2*b^6 + 512*A^2*a^4
*b^4 + 128*A^2*a^6*b^2 + 512*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 192*C^2*a^2*b^6 + 1152*C^2*a^4
*b^4 + 512*C^2*a^6*b^2 + 32*A*C*b^8 + 256*A*B*a*b^7 + 64*A*B*a^7*b + 128*B*C*a*b^7 + 128*B*C*a^7*b + 1536*A*B*
a^3*b^5 + 896*A*B*a^5*b^3 + 384*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 1536*B*C*a^3*b^5 + 1536*B*C
*a^5*b^3) - (b^2*(2*A*b^2 + 12*C*a^2 + C*b^2 + 8*B*a*b)*(32*A*b^4 + 16*B*a^4 + 16*C*b^4 + 192*B*a^2*b^2 + 192*
C*a^2*b^2 + 128*A*a*b^3 + 64*A*a^3*b + 128*B*a*b^3 + 128*C*a^3*b)*1i)/2)*(2*A*b^2 + 12*C*a^2 + C*b^2 + 8*B*a*b
)*1i)/2 - 256*A^3*a*b^11 - (b^2*(tan(c/2 + (d*x)/2)*(32*A^2*b^8 + 8*B^2*a^8 + 8*C^2*b^8 + 512*A^2*a^2*b^6 + 51
2*A^2*a^4*b^4 + 128*A^2*a^6*b^2 + 512*B^2*a^2*b^6 + 1152*B^2*a^4*b^4 + 192*B^2*a^6*b^2 + 192*C^2*a^2*b^6 + 115
2*C^2*a^4*b^4 + 512*C^2*a^6*b^2 + 32*A*C*b^8 + 256*A*B*a*b^7 + 64*A*B*a^7*b + 128*B*C*a*b^7 + 128*B*C*a^7*b +
1536*A*B*a^3*b^5 + 896*A*B*a^5*b^3 + 384*A*C*a^2*b^6 + 1024*A*C*a^4*b^4 + 512*A*C*a^6*b^2 + 1536*B*C*a^3*b^5 +
1536*B*C*a^5*b^3) + (b^2*(2*A*b^2 + 12*C*a^2 + C*b^2 + 8*B*a*b)*(32*A*b^4 + 16*B*a^4 + 16*C*b^4 + 192*B*a^2*b
^2 + 192*C*a^2*b^2 + 128*A*a*b^3 + 64*A*a^3*b + 128*B*a*b^3 + 128*C*a^3*b)*1i)/2)*(2*A*b^2 + 12*C*a^2 + C*b^2
+ 8*B*a*b)*1i)/2 + 1024*A^3*a^2*b^10 - 128*A^3*a^3*b^9 + 1024*A^3*a^4*b^8 + 256*A^3*a^6*b^6 - 6144*B^3*a^4*b^8
+ 9216*B^3*a^5*b^7 - 512*B^3*a^6*b^6 + 1536*B^3*a^7*b^5 + 64*B^3*a^9*b^3 - 64*C^3*a^3*b^9 - 1536*C^3*a^5*b^7
+ 512*C^3*a^6*b^6 - 9216*C^3*a^7*b^5 + 6144*C^3*a^8*b^4 - 64*A*C^2*a*b^11 - 256*A^2*C*a*b^11 - 7168*A*B^2*a^3*
b^9 + 14592*A*B^2*a^4*b^8 - 2304*A*B^2*a^5*b^7 + 7552*A*B^2*a^6*b^6 + 528*A*B^2*a^8*b^4 - 2432*A^2*B*a^2*b^10
+ 7168*A^2*B*a^3*b^9 - 1056*A^2*B*a^4*b^8 + 5888*A^2*B*a^5*b^7 + 1152*A^2*B*a^7*b^5 - 1824*A*C^2*a^3*b^9 + 102
4*A*C^2*a^4*b^8 - 13056*A*C^2*a^5*b^7 + 13824*A*C^2*a^6*b^6 - 4608*A*C^2*a^7*b^5 + 6144*A*C^2*a^8*b^4 + 512*A^
2*C*a^2*b^10 - 3456*A^2*C*a^3*b^9 + 8704*A^2*C*a^4*b^8 - 1536*A^2*C*a^5*b^7 + 7296*A^2*C*a^6*b^6 + 1536*A^2*C*
a^8*b^4 - 96*B*C^2*a^2*b^10 - 3336*B*C^2*a^4*b^8 + 1536*B*C^2*a^5*b^7 - 26304*B*C^2*a^6*b^6 + 22656*B*C^2*a^7*
b^5 - 1152*B*C^2*a^8*b^4 + 1536*B*C^2*a^9*b^3 - 1536*B^2*C*a^3*b^9 + 1152*B^2*C*a^4*b^8 - 22656*B^2*C*a^5*b^7
+ 26304*B^2*C*a^6*b^6 - 1536*B^2*C*a^7*b^5 + 3336*B^2*C*a^8*b^4 + 96*B^2*C*a^10*b^2 - 1408*A*B*C*a^2*b^10 + 15
36*A*B*C*a^3*b^9 - 19488*A*B*C*a^4*b^8 + 30592*A*B*C*a^5*b^7 - 6528*A*B*C*a^6*b^6 + 15168*A*B*C*a^7*b^5 + 768*
A*B*C*a^9*b^3))*(2*A*b^2 + 12*C*a^2 + C*b^2 + 8*B*a*b))/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)**4,x)

[Out]

Timed out

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