3.971 \(\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^6(c+d x) \, dx\)

Optimal. Leaf size=314 \[ \frac{\tan (c+d x) \left (2 a^2 b^2 (56 A+85 C)+4 a^4 (4 A+5 C)+80 a^3 b B+95 a b^3 B+12 A b^4\right )}{30 d}+\frac{\left (4 a^3 b (3 A+4 C)+24 a^2 b^2 B+3 a^4 B+16 a b^3 (A+2 C)+8 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec (c+d x) \left (4 a^2 b (29 A+40 C)+45 a^3 B+130 a b^2 B+24 A b^3\right )}{120 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{60 d}+\frac{(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}+b^4 C x \]

[Out]

b^4*C*x + ((3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*ArcTanh[Sin[c + d*x]]
)/(8*d) + ((12*A*b^4 + 80*a^3*b*B + 95*a*b^3*B + 4*a^4*(4*A + 5*C) + 2*a^2*b^2*(56*A + 85*C))*Tan[c + d*x])/(3
0*d) + (a*(24*A*b^3 + 45*a^3*B + 130*a*b^2*B + 4*a^2*b*(29*A + 40*C))*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((1
2*A*b^2 + 35*a*b*B + 4*a^2*(4*A + 5*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((4*A*b +
5*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^4*
Tan[c + d*x])/(5*d)

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Rubi [A]  time = 1.04766, antiderivative size = 314, normalized size of antiderivative = 1., 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, 3031, 3021, 2735, 3770} \[ \frac{\tan (c+d x) \left (2 a^2 b^2 (56 A+85 C)+4 a^4 (4 A+5 C)+80 a^3 b B+95 a b^3 B+12 A b^4\right )}{30 d}+\frac{\left (4 a^3 b (3 A+4 C)+24 a^2 b^2 B+3 a^4 B+16 a b^3 (A+2 C)+8 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec (c+d x) \left (4 a^2 b (29 A+40 C)+45 a^3 B+130 a b^2 B+24 A b^3\right )}{120 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{60 d}+\frac{(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}+b^4 C x \]

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]^6,x]

[Out]

b^4*C*x + ((3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*ArcTanh[Sin[c + d*x]]
)/(8*d) + ((12*A*b^4 + 80*a^3*b*B + 95*a*b^3*B + 4*a^4*(4*A + 5*C) + 2*a^2*b^2*(56*A + 85*C))*Tan[c + d*x])/(3
0*d) + (a*(24*A*b^3 + 45*a^3*B + 130*a*b^2*B + 4*a^2*b*(29*A + 40*C))*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((1
2*A*b^2 + 35*a*b*B + 4*a^2*(4*A + 5*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(60*d) + ((4*A*b +
5*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^4*
Tan[c + d*x])/(5*d)

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 3031

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[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*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] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3021

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

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 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 ^6(c+d x) \, dx &=\frac{A (a+b \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{5} \int (a+b \cos (c+d x))^3 \left (4 A b+5 a B+(4 a A+5 b B+5 a C) \cos (c+d x)+5 b C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac{(4 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{20} \int (a+b \cos (c+d x))^2 \left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)+\left (28 a A b+15 a^2 B+20 b^2 B+40 a b C\right ) \cos (c+d x)+20 b^2 C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac{(4 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{60} \int (a+b \cos (c+d x)) \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)+\left (115 a^2 b B+60 b^3 B+36 a b^2 (3 A+5 C)+8 a^3 (4 A+5 C)\right ) \cos (c+d x)+60 b^3 C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac{\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac{(4 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac{1}{120} \int \left (-4 \left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right )-15 \left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right ) \cos (c+d x)-120 b^4 C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{\left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right ) \tan (c+d x)}{30 d}+\frac{a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac{\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac{(4 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac{1}{120} \int \left (-15 \left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right )-120 b^4 C \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=b^4 C x+\frac{\left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right ) \tan (c+d x)}{30 d}+\frac{a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac{\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac{(4 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac{1}{8} \left (-3 a^4 B-24 a^2 b^2 B-8 b^4 B-16 a b^3 (A+2 C)-4 a^3 b (3 A+4 C)\right ) \int \sec (c+d x) \, dx\\ &=b^4 C x+\frac{\left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right ) \tan (c+d x)}{30 d}+\frac{a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac{\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac{(4 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 1.80237, size = 230, normalized size = 0.73 \[ \frac{40 a^2 \tan ^3(c+d x) \left (a^2 (2 A+C)+4 a b B+6 A b^2\right )+15 \left (4 a^3 b (3 A+4 C)+24 a^2 b^2 B+3 a^4 B+16 a b^3 (A+2 C)+8 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))+15 \tan (c+d x) \left (a \sec (c+d x) \left (4 a^2 b (3 A+4 C)+3 a^3 B+24 a b^2 B+16 A b^3\right )+8 \left (6 a^2 b^2 (A+C)+a^4 (A+C)+4 a^3 b B+4 a b^3 B+A b^4\right )+2 a^3 (a B+4 A b) \sec ^3(c+d x)\right )+24 a^4 A \tan ^5(c+d x)+120 b^4 C d x}{120 d} \]

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]^6,x]

[Out]

(120*b^4*C*d*x + 15*(3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*ArcTanh[Sin[
c + d*x]] + 15*(8*(A*b^4 + 4*a^3*b*B + 4*a*b^3*B + a^4*(A + C) + 6*a^2*b^2*(A + C)) + a*(16*A*b^3 + 3*a^3*B +
24*a*b^2*B + 4*a^2*b*(3*A + 4*C))*Sec[c + d*x] + 2*a^3*(4*A*b + a*B)*Sec[c + d*x]^3)*Tan[c + d*x] + 40*a^2*(6*
A*b^2 + 4*a*b*B + a^2*(2*A + C))*Tan[c + d*x]^3 + 24*a^4*A*Tan[c + d*x]^5)/(120*d)

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Maple [A]  time = 0.086, size = 572, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

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)^6,x)

[Out]

1/d*b^4*B*ln(sec(d*x+c)+tan(d*x+c))+2/d*a*A*b^3*sec(d*x+c)*tan(d*x+c)+2/d*a^2*A*b^2*tan(d*x+c)*sec(d*x+c)^2+1/
d*A*a^3*b*tan(d*x+c)*sec(d*x+c)^3+2/d*a^3*b*C*sec(d*x+c)*tan(d*x+c)+3/2/d*A*a^3*b*sec(d*x+c)*tan(d*x+c)+1/d*A*
b^4*tan(d*x+c)+1/d*C*b^4*c+2/3/d*a^4*C*tan(d*x+c)+3/8/d*a^4*B*ln(sec(d*x+c)+tan(d*x+c))+8/15/d*A*a^4*tan(d*x+c
)+b^4*C*x+3/d*a^2*b^2*B*ln(sec(d*x+c)+tan(d*x+c))+8/3/d*a^3*b*B*tan(d*x+c)+4/d*C*a*b^3*ln(sec(d*x+c)+tan(d*x+c
))+6/d*a^2*b^2*C*tan(d*x+c)+2/d*a*A*b^3*ln(sec(d*x+c)+tan(d*x+c))+4/d*a^2*A*b^2*tan(d*x+c)+2/d*a^3*b*C*ln(sec(
d*x+c)+tan(d*x+c))+3/2/d*A*a^3*b*ln(sec(d*x+c)+tan(d*x+c))+1/4/d*a^4*B*tan(d*x+c)*sec(d*x+c)^3+3/8/d*a^4*B*sec
(d*x+c)*tan(d*x+c)+1/5/d*A*a^4*tan(d*x+c)*sec(d*x+c)^4+1/3/d*a^4*C*tan(d*x+c)*sec(d*x+c)^2+4/15/d*A*a^4*tan(d*
x+c)*sec(d*x+c)^2+3/d*a^2*b^2*B*sec(d*x+c)*tan(d*x+c)+4/3/d*a^3*b*B*tan(d*x+c)*sec(d*x+c)^2+4/d*a*b^3*B*tan(d*
x+c)

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Maxima [A]  time = 1.03842, size = 690, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

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)^6,x, algorithm="maxima")

[Out]

1/240*(16*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^4 + 80*(tan(d*x + c)^3 + 3*tan(d*x + c)
)*C*a^4 + 320*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^3*b + 480*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^2*b^2 + 24
0*(d*x + c)*C*b^4 - 15*B*a^4*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) -
3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 60*A*a^3*b*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*
x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 240*C*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)) - 360*B*a^2*b^2*(2*sin(d*x + c)/(s
in(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 240*A*a*b^3*(2*sin(d*x + c)/(sin(d*x + c
)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 480*C*a*b^3*(log(sin(d*x + c) + 1) - log(sin(d*x +
c) - 1)) + 120*B*b^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 1440*C*a^2*b^2*tan(d*x + c) + 960*B*a*
b^3*tan(d*x + c) + 240*A*b^4*tan(d*x + c))/d

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Fricas [A]  time = 2.01, size = 824, normalized size = 2.62 \begin{align*} \frac{240 \, C b^{4} d x \cos \left (d x + c\right )^{5} + 15 \,{\left (3 \, B a^{4} + 4 \,{\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \,{\left (A + 2 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (3 \, B a^{4} + 4 \,{\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \,{\left (A + 2 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (24 \, A a^{4} + 8 \,{\left (2 \,{\left (4 \, A + 5 \, C\right )} a^{4} + 40 \, B a^{3} b + 30 \,{\left (2 \, A + 3 \, C\right )} a^{2} b^{2} + 60 \, B a b^{3} + 15 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 15 \,{\left (3 \, B a^{4} + 4 \,{\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 8 \,{\left ({\left (4 \, A + 5 \, C\right )} a^{4} + 20 \, B a^{3} b + 30 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 30 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \end{align*}

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)^6,x, algorithm="fricas")

[Out]

1/240*(240*C*b^4*d*x*cos(d*x + c)^5 + 15*(3*B*a^4 + 4*(3*A + 4*C)*a^3*b + 24*B*a^2*b^2 + 16*(A + 2*C)*a*b^3 +
8*B*b^4)*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 15*(3*B*a^4 + 4*(3*A + 4*C)*a^3*b + 24*B*a^2*b^2 + 16*(A + 2*C
)*a*b^3 + 8*B*b^4)*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 2*(24*A*a^4 + 8*(2*(4*A + 5*C)*a^4 + 40*B*a^3*b + 3
0*(2*A + 3*C)*a^2*b^2 + 60*B*a*b^3 + 15*A*b^4)*cos(d*x + c)^4 + 15*(3*B*a^4 + 4*(3*A + 4*C)*a^3*b + 24*B*a^2*b
^2 + 16*A*a*b^3)*cos(d*x + c)^3 + 8*((4*A + 5*C)*a^4 + 20*B*a^3*b + 30*A*a^2*b^2)*cos(d*x + c)^2 + 30*(B*a^4 +
4*A*a^3*b)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.4293, size = 1539, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}

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)^6,x, algorithm="giac")

[Out]

1/120*(120*(d*x + c)*C*b^4 + 15*(3*B*a^4 + 12*A*a^3*b + 16*C*a^3*b + 24*B*a^2*b^2 + 16*A*a*b^3 + 32*C*a*b^3 +
8*B*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(3*B*a^4 + 12*A*a^3*b + 16*C*a^3*b + 24*B*a^2*b^2 + 16*A*a*b^
3 + 32*C*a*b^3 + 8*B*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(120*A*a^4*tan(1/2*d*x + 1/2*c)^9 - 75*B*a^4*
tan(1/2*d*x + 1/2*c)^9 + 120*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 300*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 480*B*a^3*b*t
an(1/2*d*x + 1/2*c)^9 - 240*C*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 720*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 360*B*a^2*
b^2*tan(1/2*d*x + 1/2*c)^9 + 720*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 240*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 480*B
*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 120*A*b^4*tan(1/2*d*x + 1/2*c)^9 - 160*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 30*B*a^4
*tan(1/2*d*x + 1/2*c)^7 - 320*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 120*A*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 1280*B*a^3*b
*tan(1/2*d*x + 1/2*c)^7 + 480*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 1920*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 720*B*a
^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 2880*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 480*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 1
920*B*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 480*A*b^4*tan(1/2*d*x + 1/2*c)^7 + 464*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 400
*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 1600*B*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 2400*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 +
4320*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 2880*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 720*A*b^4*tan(1/2*d*x + 1/2*c)^5
- 160*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 30*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 320*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 120
*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 1280*B*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 480*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 1
920*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 720*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 2880*C*a^2*b^2*tan(1/2*d*x + 1/2
*c)^3 - 480*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 1920*B*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 480*A*b^4*tan(1/2*d*x + 1/2
*c)^3 + 120*A*a^4*tan(1/2*d*x + 1/2*c) + 75*B*a^4*tan(1/2*d*x + 1/2*c) + 120*C*a^4*tan(1/2*d*x + 1/2*c) + 300*
A*a^3*b*tan(1/2*d*x + 1/2*c) + 480*B*a^3*b*tan(1/2*d*x + 1/2*c) + 240*C*a^3*b*tan(1/2*d*x + 1/2*c) + 720*A*a^2
*b^2*tan(1/2*d*x + 1/2*c) + 360*B*a^2*b^2*tan(1/2*d*x + 1/2*c) + 720*C*a^2*b^2*tan(1/2*d*x + 1/2*c) + 240*A*a*
b^3*tan(1/2*d*x + 1/2*c) + 480*B*a*b^3*tan(1/2*d*x + 1/2*c) + 120*A*b^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1
/2*c)^2 - 1)^5)/d