\(\int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx\) [300]

Optimal result

Integrand size = 31, antiderivative size = 179 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {1}{8} \left (3 a^3 A+12 a A b^2+12 a^2 b B+8 b^3 B\right ) x+\frac {\left (6 a^2 A b+3 A b^3+2 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{3 d}+\frac {a \left (3 a^2 A+10 A b^2+12 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (3 A b+2 a B) \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d} \]

[Out]

1/8*(3*A*a^3+12*A*a*b^2+12*B*a^2*b+8*B*b^3)*x+1/3*(6*A*a^2*b+3*A*b^3+2*B*a^3+9*B*a*b^2)*sin(d*x+c)/d+1/8*a*(3*
A*a^2+10*A*b^2+12*B*a*b)*cos(d*x+c)*sin(d*x+c)/d+1/6*a^2*(3*A*b+2*B*a)*cos(d*x+c)^2*sin(d*x+c)/d+1/4*a*A*cos(d
*x+c)^3*(a+b*sec(d*x+c))^2*sin(d*x+c)/d

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4110, 4159, 4132, 2717, 4130, 8} \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a \left (3 a^2 A+12 a b B+10 A b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a^2 (2 a B+3 A b) \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac {\left (2 a^3 B+6 a^2 A b+9 a b^2 B+3 A b^3\right ) \sin (c+d x)}{3 d}+\frac {1}{8} x \left (3 a^3 A+12 a^2 b B+12 a A b^2+8 b^3 B\right )+\frac {a A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d} \]

[In]

Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]

[Out]

((3*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 8*b^3*B)*x)/8 + ((6*a^2*A*b + 3*A*b^3 + 2*a^3*B + 9*a*b^2*B)*Sin[c + d*x
])/(3*d) + (a*(3*a^2*A + 10*A*b^2 + 12*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(3*A*b + 2*a*B)*Cos[c +
d*x]^2*Sin[c + d*x])/(6*d) + (a*A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4110

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x]
+ Dist[1/(d*n), Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*Simp[a*(a*B*n - A*b*(m - n - 1)) + (
2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x], x] /; Free
Q[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LeQ[n, -1]

Rule 4130

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e
+ f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

Rule 4132

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rule 4159

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x]
+ Dist[1/(d*n), Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B*b) + A*a*(n + 1))*Csc[e + f*x]
+ b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}-\frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (-2 a (3 A b+2 a B)-\left (3 a^2 A+4 A b^2+8 a b B\right ) \sec (c+d x)-b (a A+4 b B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 (3 A b+2 a B) \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {1}{12} \int \cos ^2(c+d x) \left (3 a \left (3 a^2 A+10 A b^2+12 a b B\right )+4 \left (6 a^2 A b+3 A b^3+2 a^3 B+9 a b^2 B\right ) \sec (c+d x)+3 b^2 (a A+4 b B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 (3 A b+2 a B) \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {1}{12} \int \cos ^2(c+d x) \left (3 a \left (3 a^2 A+10 A b^2+12 a b B\right )+3 b^2 (a A+4 b B) \sec ^2(c+d x)\right ) \, dx+\frac {1}{3} \left (6 a^2 A b+3 A b^3+2 a^3 B+9 a b^2 B\right ) \int \cos (c+d x) \, dx \\ & = \frac {\left (6 a^2 A b+3 A b^3+2 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{3 d}+\frac {a \left (3 a^2 A+10 A b^2+12 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (3 A b+2 a B) \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d}+\frac {1}{8} \left (3 a^3 A+12 a A b^2+12 a^2 b B+8 b^3 B\right ) \int 1 \, dx \\ & = \frac {1}{8} \left (3 a^3 A+12 a A b^2+12 a^2 b B+8 b^3 B\right ) x+\frac {\left (6 a^2 A b+3 A b^3+2 a^3 B+9 a b^2 B\right ) \sin (c+d x)}{3 d}+\frac {a \left (3 a^2 A+10 A b^2+12 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (3 A b+2 a B) \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {a A \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.41 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.78 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {12 \left (3 a^3 A+12 a A b^2+12 a^2 b B+8 b^3 B\right ) (c+d x)+24 \left (9 a^2 A b+4 A b^3+3 a^3 B+12 a b^2 B\right ) \sin (c+d x)+24 a \left (a^2 A+3 A b^2+3 a b B\right ) \sin (2 (c+d x))+8 a^2 (3 A b+a B) \sin (3 (c+d x))+3 a^3 A \sin (4 (c+d x))}{96 d} \]

[In]

Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]

[Out]

(12*(3*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 8*b^3*B)*(c + d*x) + 24*(9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*
Sin[c + d*x] + 24*a*(a^2*A + 3*A*b^2 + 3*a*b*B)*Sin[2*(c + d*x)] + 8*a^2*(3*A*b + a*B)*Sin[3*(c + d*x)] + 3*a^
3*A*Sin[4*(c + d*x)])/(96*d)

Maple [A] (verified)

Time = 2.45 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.78

method	result	size
parallelrisch	\(\frac {24 a \left (A \,a^{2}+3 A \,b^{2}+3 B a b \right ) \sin \left (2 d x +2 c \right )+\left (24 A \,a^{2} b +8 B \,a^{3}\right ) \sin \left (3 d x +3 c \right )+3 a^{3} A \sin \left (4 d x +4 c \right )+\left (216 A \,a^{2} b +96 A \,b^{3}+72 B \,a^{3}+288 B a \,b^{2}\right ) \sin \left (d x +c \right )+36 \left (a^{3} A +4 A a \,b^{2}+4 B \,a^{2} b +\frac {8}{3} B \,b^{3}\right ) d x}{96 d}\)	\(139\)
derivativedivides	\(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \,a^{2} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+3 A a \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,b^{3} \sin \left (d x +c \right )+3 B a \,b^{2} \sin \left (d x +c \right )+B \,b^{3} \left (d x +c \right )}{d}\)	\(180\)
default	\(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \,a^{2} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+3 A a \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,b^{3} \sin \left (d x +c \right )+3 B a \,b^{2} \sin \left (d x +c \right )+B \,b^{3} \left (d x +c \right )}{d}\)	\(180\)
risch	\(\frac {3 a^{3} A x}{8}+\frac {3 A a \,b^{2} x}{2}+\frac {3 B \,a^{2} b x}{2}+x B \,b^{3}+\frac {9 \sin \left (d x +c \right ) A \,a^{2} b}{4 d}+\frac {\sin \left (d x +c \right ) A \,b^{3}}{d}+\frac {3 a^{3} B \sin \left (d x +c \right )}{4 d}+\frac {3 \sin \left (d x +c \right ) B a \,b^{2}}{d}+\frac {a^{3} A \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{2} b}{4 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3}}{12 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} A}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) A a \,b^{2}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{2} b}{4 d}\)	\(203\)
norman	\(\frac {\left (-\frac {3}{8} a^{3} A -\frac {3}{2} A a \,b^{2}-\frac {3}{2} B \,a^{2} b -B \,b^{3}\right ) x +\left (-\frac {9}{8} a^{3} A -\frac {9}{2} A a \,b^{2}-\frac {9}{2} B \,a^{2} b -3 B \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-\frac {9}{8} a^{3} A -\frac {9}{2} A a \,b^{2}-\frac {9}{2} B \,a^{2} b -3 B \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-\frac {3}{8} a^{3} A -\frac {3}{2} A a \,b^{2}-\frac {3}{2} B \,a^{2} b -B \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {3}{8} a^{3} A +\frac {3}{2} A a \,b^{2}+\frac {3}{2} B \,a^{2} b +B \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {3}{8} a^{3} A +\frac {3}{2} A a \,b^{2}+\frac {3}{2} B \,a^{2} b +B \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {9}{8} a^{3} A +\frac {9}{2} A a \,b^{2}+\frac {9}{2} B \,a^{2} b +3 B \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {9}{8} a^{3} A +\frac {9}{2} A a \,b^{2}+\frac {9}{2} B \,a^{2} b +3 B \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {a \left (7 A \,a^{2}-12 A \,b^{2}-12 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {\left (5 a^{3} A -24 A \,a^{2} b +12 A a \,b^{2}-8 A \,b^{3}-8 B \,a^{3}+12 B \,a^{2} b -24 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{4 d}-\frac {\left (5 a^{3} A +24 A \,a^{2} b +12 A a \,b^{2}+8 A \,b^{3}+8 B \,a^{3}+12 B \,a^{2} b +24 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (81 a^{3} A -24 A \,a^{2} b -36 A a \,b^{2}-72 A \,b^{3}-8 B \,a^{3}-36 B \,a^{2} b -216 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}-\frac {\left (81 a^{3} A +24 A \,a^{2} b -36 A a \,b^{2}+72 A \,b^{3}+8 B \,a^{3}-36 B \,a^{2} b +216 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}+\frac {a \left (27 A \,a^{2}-48 A a b +36 A \,b^{2}-16 B \,a^{2}+36 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{6 d}+\frac {a \left (27 A \,a^{2}+48 A a b +36 A \,b^{2}+16 B \,a^{2}+36 B a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\)	\(716\)

[In]

int(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/96*(24*a*(A*a^2+3*A*b^2+3*B*a*b)*sin(2*d*x+2*c)+(24*A*a^2*b+8*B*a^3)*sin(3*d*x+3*c)+3*a^3*A*sin(4*d*x+4*c)+(
216*A*a^2*b+96*A*b^3+72*B*a^3+288*B*a*b^2)*sin(d*x+c)+36*(a^3*A+4*A*a*b^2+4*B*a^2*b+8/3*B*b^3)*d*x)/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.76 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {3 \, {\left (3 \, A a^{3} + 12 \, B a^{2} b + 12 \, A a b^{2} + 8 \, B b^{3}\right )} d x + {\left (6 \, A a^{3} \cos \left (d x + c\right )^{3} + 16 \, B a^{3} + 48 \, A a^{2} b + 72 \, B a b^{2} + 24 \, A b^{3} + 8 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{2} + 9 \, {\left (A a^{3} + 4 \, B a^{2} b + 4 \, A a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/24*(3*(3*A*a^3 + 12*B*a^2*b + 12*A*a*b^2 + 8*B*b^3)*d*x + (6*A*a^3*cos(d*x + c)^3 + 16*B*a^3 + 48*A*a^2*b +
72*B*a*b^2 + 24*A*b^3 + 8*(B*a^3 + 3*A*a^2*b)*cos(d*x + c)^2 + 9*(A*a^3 + 4*B*a^2*b + 4*A*a*b^2)*cos(d*x + c))
*sin(d*x + c))/d

Sympy [F(-1)]

Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**4*(a+b*sec(d*x+c))**3*(A+B*sec(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.96 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 96 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{2} + 96 \, {\left (d x + c\right )} B b^{3} + 288 \, B a b^{2} \sin \left (d x + c\right ) + 96 \, A b^{3} \sin \left (d x + c\right )}{96 \, d} \]

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)),x, algorithm="maxima")

[Out]

1/96*(3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 - 32*(sin(d*x + c)^3 - 3*sin(d*x + c))*B
*a^3 - 96*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^2*b + 72*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^2*b + 72*(2*d*x
+ 2*c + sin(2*d*x + 2*c))*A*a*b^2 + 96*(d*x + c)*B*b^3 + 288*B*a*b^2*sin(d*x + c) + 96*A*b^3*sin(d*x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (169) = 338\).

Time = 0.34 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.99 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {3 \, {\left (3 \, A a^{3} + 12 \, B a^{2} b + 12 \, A a b^{2} + 8 \, B b^{3}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 120 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 216 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 216 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, A b^{3} \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 )}^{4}}}{24 \, d} \]

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)),x, algorithm="giac")

[Out]

1/24*(3*(3*A*a^3 + 12*B*a^2*b + 12*A*a*b^2 + 8*B*b^3)*(d*x + c) - 2*(15*A*a^3*tan(1/2*d*x + 1/2*c)^7 - 24*B*a^
3*tan(1/2*d*x + 1/2*c)^7 - 72*A*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 36*B*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 36*A*a*b^2*
tan(1/2*d*x + 1/2*c)^7 - 72*B*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 24*A*b^3*tan(1/2*d*x + 1/2*c)^7 - 9*A*a^3*tan(1/2
*d*x + 1/2*c)^5 - 40*B*a^3*tan(1/2*d*x + 1/2*c)^5 - 120*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 36*B*a^2*b*tan(1/2*d*
x + 1/2*c)^5 + 36*A*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 216*B*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 72*A*b^3*tan(1/2*d*x +
1/2*c)^5 + 9*A*a^3*tan(1/2*d*x + 1/2*c)^3 - 40*B*a^3*tan(1/2*d*x + 1/2*c)^3 - 120*A*a^2*b*tan(1/2*d*x + 1/2*c
)^3 - 36*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 36*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 216*B*a*b^2*tan(1/2*d*x + 1/2*c)
^3 - 72*A*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*A*a^3*tan(1/2*d*x + 1/2*c) - 24*B*a^3*tan(1/2*d*x + 1/2*c) - 72*A*a^
2*b*tan(1/2*d*x + 1/2*c) - 36*B*a^2*b*tan(1/2*d*x + 1/2*c) - 36*A*a*b^2*tan(1/2*d*x + 1/2*c) - 72*B*a*b^2*tan(
1/2*d*x + 1/2*c) - 24*A*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d

Mupad [B] (verification not implemented)

Time = 14.93 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.13 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {3\,A\,a^3\,x}{8}+B\,b^3\,x+\frac {3\,A\,a\,b^2\,x}{2}+\frac {3\,B\,a^2\,b\,x}{2}+\frac {A\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {3\,B\,a^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {A\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {A\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {B\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {3\,A\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {A\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {3\,B\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {9\,A\,a^2\,b\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,B\,a\,b^2\,\sin \left (c+d\,x\right )}{d} \]

[In]

int(cos(c + d*x)^4*(A + B/cos(c + d*x))*(a + b/cos(c + d*x))^3,x)

[Out]

(3*A*a^3*x)/8 + B*b^3*x + (3*A*a*b^2*x)/2 + (3*B*a^2*b*x)/2 + (A*b^3*sin(c + d*x))/d + (3*B*a^3*sin(c + d*x))/
(4*d) + (A*a^3*sin(2*c + 2*d*x))/(4*d) + (A*a^3*sin(4*c + 4*d*x))/(32*d) + (B*a^3*sin(3*c + 3*d*x))/(12*d) + (
3*A*a*b^2*sin(2*c + 2*d*x))/(4*d) + (A*a^2*b*sin(3*c + 3*d*x))/(4*d) + (3*B*a^2*b*sin(2*c + 2*d*x))/(4*d) + (9
*A*a^2*b*sin(c + d*x))/(4*d) + (3*B*a*b^2*sin(c + d*x))/d