3.293 \(\int \cos ^5(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx\)

Optimal. Leaf size=180 \[ -\frac{\left (4 a^2 A+10 a b B+5 A b^2\right ) \sin ^3(c+d x)}{15 d}+\frac{\left (4 a^2 A+10 a b B+5 A b^2\right ) \sin (c+d x)}{5 d}+\frac{\left (3 a^2 B+6 a A b+4 b^2 B\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left (3 a^2 B+6 a A b+4 b^2 B\right )+\frac{a^2 A \sin (c+d x) \cos ^4(c+d x)}{5 d}+\frac{a (a B+2 A b) \sin (c+d x) \cos ^3(c+d x)}{4 d} \]

[Out]

((6*a*A*b + 3*a^2*B + 4*b^2*B)*x)/8 + ((4*a^2*A + 5*A*b^2 + 10*a*b*B)*Sin[c + d*x])/(5*d) + ((6*a*A*b + 3*a^2*
B + 4*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(2*A*b + a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a^2*A*C
os[c + d*x]^4*Sin[c + d*x])/(5*d) - ((4*a^2*A + 5*A*b^2 + 10*a*b*B)*Sin[c + d*x]^3)/(15*d)

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Rubi [A]  time = 0.268378, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {4024, 4047, 2633, 4045, 2635, 8} \[ -\frac{\left (4 a^2 A+10 a b B+5 A b^2\right ) \sin ^3(c+d x)}{15 d}+\frac{\left (4 a^2 A+10 a b B+5 A b^2\right ) \sin (c+d x)}{5 d}+\frac{\left (3 a^2 B+6 a A b+4 b^2 B\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left (3 a^2 B+6 a A b+4 b^2 B\right )+\frac{a^2 A \sin (c+d x) \cos ^4(c+d x)}{5 d}+\frac{a (a B+2 A b) \sin (c+d x) \cos ^3(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]

[Out]

((6*a*A*b + 3*a^2*B + 4*b^2*B)*x)/8 + ((4*a^2*A + 5*A*b^2 + 10*a*b*B)*Sin[c + d*x])/(5*d) + ((6*a*A*b + 3*a^2*
B + 4*b^2*B)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(2*A*b + a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a^2*A*C
os[c + d*x]^4*Sin[c + d*x])/(5*d) - ((4*a^2*A + 5*A*b^2 + 10*a*b*B)*Sin[c + d*x]^3)/(15*d)

Rule 4024

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

Rule 4047

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 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 4045

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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

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

Rubi steps

\begin{align*} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx &=\frac{a^2 A \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac{1}{5} \int \cos ^4(c+d x) \left (-5 a (2 A b+a B)+\left (A \left (-4 a^2-5 b^2\right )-10 a b B\right ) \sec (c+d x)-5 b^2 B \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 A \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac{1}{5} \int \cos ^4(c+d x) \left (-5 a (2 A b+a B)-5 b^2 B \sec ^2(c+d x)\right ) \, dx-\frac{1}{5} \left (-4 a^2 A-5 A b^2-10 a b B\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac{a (2 A b+a B) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{a^2 A \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac{1}{4} \left (-6 a A b-3 a^2 B-4 b^2 B\right ) \int \cos ^2(c+d x) \, dx-\frac{\left (4 a^2 A+5 A b^2+10 a b B\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac{\left (4 a^2 A+5 A b^2+10 a b B\right ) \sin (c+d x)}{5 d}+\frac{\left (6 a A b+3 a^2 B+4 b^2 B\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a (2 A b+a B) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{a^2 A \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac{\left (4 a^2 A+5 A b^2+10 a b B\right ) \sin ^3(c+d x)}{15 d}-\frac{1}{8} \left (-6 a A b-3 a^2 B-4 b^2 B\right ) \int 1 \, dx\\ &=\frac{1}{8} \left (6 a A b+3 a^2 B+4 b^2 B\right ) x+\frac{\left (4 a^2 A+5 A b^2+10 a b B\right ) \sin (c+d x)}{5 d}+\frac{\left (6 a A b+3 a^2 B+4 b^2 B\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a (2 A b+a B) \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{a^2 A \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac{\left (4 a^2 A+5 A b^2+10 a b B\right ) \sin ^3(c+d x)}{15 d}\\ \end{align*}

Mathematica [A]  time = 0.545049, size = 146, normalized size = 0.81 \[ \frac{60 (c+d x) \left (3 a^2 B+6 a A b+4 b^2 B\right )+60 \left (5 a^2 A+12 a b B+6 A b^2\right ) \sin (c+d x)+120 \left (a^2 B+2 a A b+b^2 B\right ) \sin (2 (c+d x))+10 \left (5 a^2 A+8 a b B+4 A b^2\right ) \sin (3 (c+d x))+6 a^2 A \sin (5 (c+d x))+15 a (a B+2 A b) \sin (4 (c+d x))}{480 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]

[Out]

(60*(6*a*A*b + 3*a^2*B + 4*b^2*B)*(c + d*x) + 60*(5*a^2*A + 6*A*b^2 + 12*a*b*B)*Sin[c + d*x] + 120*(2*a*A*b +
a^2*B + b^2*B)*Sin[2*(c + d*x)] + 10*(5*a^2*A + 4*A*b^2 + 8*a*b*B)*Sin[3*(c + d*x)] + 15*a*(2*A*b + a*B)*Sin[4
*(c + d*x)] + 6*a^2*A*Sin[5*(c + d*x)])/(480*d)

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Maple [A]  time = 0.068, size = 184, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{2}A\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+B{a}^{2} \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +2\,Aab \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{\frac{2\,Bab \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{\frac{A{b}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+B{b}^{2} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)),x)

[Out]

1/d*(1/5*a^2*A*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+B*a^2*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x
+c)+3/8*d*x+3/8*c)+2*A*a*b*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+2/3*B*a*b*(2+cos(d*x+c
)^2)*sin(d*x+c)+1/3*A*b^2*(2+cos(d*x+c)^2)*sin(d*x+c)+B*b^2*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c))

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Maxima [A]  time = 0.963242, size = 238, normalized size = 1.32 \begin{align*} \frac{32 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} + 15 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 30 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b - 320 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b - 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{2} + 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{2}}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)),x, algorithm="maxima")

[Out]

1/480*(32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^2 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c
) + 8*sin(2*d*x + 2*c))*B*a^2 + 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a*b - 320*(sin(d*
x + c)^3 - 3*sin(d*x + c))*B*a*b - 160*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*b^2 + 120*(2*d*x + 2*c + sin(2*d*x
+ 2*c))*B*b^2)/d

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Fricas [A]  time = 0.50198, size = 350, normalized size = 1.94 \begin{align*} \frac{15 \,{\left (3 \, B a^{2} + 6 \, A a b + 4 \, B b^{2}\right )} d x +{\left (24 \, A a^{2} \cos \left (d x + c\right )^{4} + 30 \,{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{3} + 64 \, A a^{2} + 160 \, B a b + 80 \, A b^{2} + 8 \,{\left (4 \, A a^{2} + 10 \, B a b + 5 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left (3 \, B a^{2} + 6 \, A a b + 4 \, B b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/120*(15*(3*B*a^2 + 6*A*a*b + 4*B*b^2)*d*x + (24*A*a^2*cos(d*x + c)^4 + 30*(B*a^2 + 2*A*a*b)*cos(d*x + c)^3 +
64*A*a^2 + 160*B*a*b + 80*A*b^2 + 8*(4*A*a^2 + 10*B*a*b + 5*A*b^2)*cos(d*x + c)^2 + 15*(3*B*a^2 + 6*A*a*b + 4
*B*b^2)*cos(d*x + c))*sin(d*x + c))/d

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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(cos(d*x+c)**5*(a+b*sec(d*x+c))**2*(A+B*sec(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 1.19317, size = 657, normalized size = 3.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^2*(A+B*sec(d*x+c)),x, algorithm="giac")

[Out]

1/120*(15*(3*B*a^2 + 6*A*a*b + 4*B*b^2)*(d*x + c) + 2*(120*A*a^2*tan(1/2*d*x + 1/2*c)^9 - 75*B*a^2*tan(1/2*d*x
+ 1/2*c)^9 - 150*A*a*b*tan(1/2*d*x + 1/2*c)^9 + 240*B*a*b*tan(1/2*d*x + 1/2*c)^9 + 120*A*b^2*tan(1/2*d*x + 1/
2*c)^9 - 60*B*b^2*tan(1/2*d*x + 1/2*c)^9 + 160*A*a^2*tan(1/2*d*x + 1/2*c)^7 - 30*B*a^2*tan(1/2*d*x + 1/2*c)^7
- 60*A*a*b*tan(1/2*d*x + 1/2*c)^7 + 640*B*a*b*tan(1/2*d*x + 1/2*c)^7 + 320*A*b^2*tan(1/2*d*x + 1/2*c)^7 - 120*
B*b^2*tan(1/2*d*x + 1/2*c)^7 + 464*A*a^2*tan(1/2*d*x + 1/2*c)^5 + 800*B*a*b*tan(1/2*d*x + 1/2*c)^5 + 400*A*b^2
*tan(1/2*d*x + 1/2*c)^5 + 160*A*a^2*tan(1/2*d*x + 1/2*c)^3 + 30*B*a^2*tan(1/2*d*x + 1/2*c)^3 + 60*A*a*b*tan(1/
2*d*x + 1/2*c)^3 + 640*B*a*b*tan(1/2*d*x + 1/2*c)^3 + 320*A*b^2*tan(1/2*d*x + 1/2*c)^3 + 120*B*b^2*tan(1/2*d*x
+ 1/2*c)^3 + 120*A*a^2*tan(1/2*d*x + 1/2*c) + 75*B*a^2*tan(1/2*d*x + 1/2*c) + 150*A*a*b*tan(1/2*d*x + 1/2*c)
+ 240*B*a*b*tan(1/2*d*x + 1/2*c) + 120*A*b^2*tan(1/2*d*x + 1/2*c) + 60*B*b^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*
x + 1/2*c)^2 + 1)^5)/d