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

Optimal. Leaf size=136 \[ -\frac{(4 a B+5 b C) \sin ^3(c+d x)}{15 d}+\frac{(4 a B+5 b C) \sin (c+d x)}{5 d}+\frac{(a C+b B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 (a C+b B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} x (a C+b B)+\frac{a B \sin (c+d x) \cos ^4(c+d x)}{5 d} \]

[Out]

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

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Rubi [A]  time = 0.199866, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4072, 3996, 3787, 2635, 8, 2633} \[ -\frac{(4 a B+5 b C) \sin ^3(c+d x)}{15 d}+\frac{(4 a B+5 b C) \sin (c+d x)}{5 d}+\frac{(a C+b B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 (a C+b B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} x (a C+b B)+\frac{a B \sin (c+d x) \cos ^4(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

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

Rule 4072

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

Rule 3996

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

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.239587, size = 88, normalized size = 0.65 \[ \frac{-160 (2 a B+b C) \sin ^3(c+d x)+480 (a B+b C) \sin (c+d x)+15 (a C+b B) (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))+96 a B \sin ^5(c+d x)}{480 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(480*(a*B + b*C)*Sin[c + d*x] - 160*(2*a*B + b*C)*Sin[c + d*x]^3 + 96*a*B*Sin[c + d*x]^5 + 15*(b*B + a*C)*(12*
(c + d*x) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(480*d)

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Maple [A]  time = 0.067, size = 128, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{B\sin \left ( dx+c \right ) a}{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 ) }+Bb \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 ) +aC \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 ) +{\frac{Cb \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*(a+b*sec(d*x+c))*(B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

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

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Maxima [A]  time = 0.97131, size = 167, normalized size = 1.23 \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 )} B a + 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 )} C a + 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 b - 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C b}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.50771, size = 248, normalized size = 1.82 \begin{align*} \frac{45 \,{\left (C a + B b\right )} d x +{\left (24 \, B a \cos \left (d x + c\right )^{4} + 30 \,{\left (C a + B b\right )} \cos \left (d x + c\right )^{3} + 8 \,{\left (4 \, B a + 5 \, C b\right )} \cos \left (d x + c\right )^{2} + 64 \, B a + 80 \, C b + 45 \,{\left (C a + B b\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)^6*(a+b*sec(d*x+c))*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/120*(45*(C*a + B*b)*d*x + (24*B*a*cos(d*x + c)^4 + 30*(C*a + B*b)*cos(d*x + c)^3 + 8*(4*B*a + 5*C*b)*cos(d*x
+ c)^2 + 64*B*a + 80*C*b + 45*(C*a + B*b)*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)**6*(a+b*sec(d*x+c))*(B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Timed out

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Giac [B]  time = 1.17965, size = 405, normalized size = 2.98 \begin{align*} \frac{45 \,{\left (C a + B b\right )}{\left (d x + c\right )} + \frac{2 \,{\left (120 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 75 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 75 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 120 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 160 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 30 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 30 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 320 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 464 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 400 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 160 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 30 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 30 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 320 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 120 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 75 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 75 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 120 \, C b \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 )}^{5}}}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/120*(45*(C*a + B*b)*(d*x + c) + 2*(120*B*a*tan(1/2*d*x + 1/2*c)^9 - 75*C*a*tan(1/2*d*x + 1/2*c)^9 - 75*B*b*t
an(1/2*d*x + 1/2*c)^9 + 120*C*b*tan(1/2*d*x + 1/2*c)^9 + 160*B*a*tan(1/2*d*x + 1/2*c)^7 - 30*C*a*tan(1/2*d*x +
1/2*c)^7 - 30*B*b*tan(1/2*d*x + 1/2*c)^7 + 320*C*b*tan(1/2*d*x + 1/2*c)^7 + 464*B*a*tan(1/2*d*x + 1/2*c)^5 +
400*C*b*tan(1/2*d*x + 1/2*c)^5 + 160*B*a*tan(1/2*d*x + 1/2*c)^3 + 30*C*a*tan(1/2*d*x + 1/2*c)^3 + 30*B*b*tan(1
/2*d*x + 1/2*c)^3 + 320*C*b*tan(1/2*d*x + 1/2*c)^3 + 120*B*a*tan(1/2*d*x + 1/2*c) + 75*C*a*tan(1/2*d*x + 1/2*c
) + 75*B*b*tan(1/2*d*x + 1/2*c) + 120*C*b*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d