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

Optimal. Leaf size=95 \[ \frac{a (3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} a x (3 A+4 C)+\frac{b (A+C) \sin (c+d x)}{d}-\frac{A b \sin ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.172779, antiderivative size = 95, 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 = {4075, 4047, 2635, 8, 4044, 3013} \[ \frac{a (3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} a x (3 A+4 C)+\frac{b (A+C) \sin (c+d x)}{d}-\frac{A b \sin ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]

[Out]

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

Rule 4075

Int[((A_.) + 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[A*b*n + a*(C*n + 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, C}, x] && LtQ[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 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 4044

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Int[(C + A*Sin[e + f*
x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[{e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]

Rule 3013

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Dist[f^(-1), Subst[I
nt[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2
, 0]

Rubi steps

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

Mathematica [A]  time = 0.209407, size = 84, normalized size = 0.88 \[ \frac{24 a (A+C) \sin (2 (c+d x))+3 a A \sin (4 (c+d x))+36 a A c+36 a A d x+48 a c C+48 a C d x+24 b (3 A+4 C) \sin (c+d x)+8 A b \sin (3 (c+d x))}{96 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]

[Out]

(36*a*A*c + 48*a*c*C + 36*a*A*d*x + 48*a*C*d*x + 24*b*(3*A + 4*C)*Sin[c + d*x] + 24*a*(A + C)*Sin[2*(c + d*x)]
 + 8*A*b*Sin[3*(c + d*x)] + 3*a*A*Sin[4*(c + d*x)])/(96*d)

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Maple [A]  time = 0.066, size = 96, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( Aa \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{Ab \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+aC \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +C\sin \left ( dx+c \right ) b \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*(a+b*sec(d*x+c))*(A+C*sec(d*x+c)^2),x)

[Out]

1/d*(A*a*(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*A*b*(2+cos(d*x+c)^2)*sin(d*x+c)+a*C*
(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+C*sin(d*x+c)*b)

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Maxima [A]  time = 0.95724, size = 122, normalized size = 1.28 \begin{align*} \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 + 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b + 96 \, C b \sin \left (d x + c\right )}{96 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))*(A+C*sec(d*x+c)^2),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 + 24*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a
- 32*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*b + 96*C*b*sin(d*x + c))/d

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Fricas [A]  time = 0.49443, size = 189, normalized size = 1.99 \begin{align*} \frac{3 \,{\left (3 \, A + 4 \, C\right )} a d x +{\left (6 \, A a \cos \left (d x + c\right )^{3} + 8 \, A b \cos \left (d x + c\right )^{2} + 3 \,{\left (3 \, A + 4 \, C\right )} a \cos \left (d x + c\right ) + 8 \,{\left (2 \, A + 3 \, C\right )} b\right )} \sin \left (d x + c\right )}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))*(A+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/24*(3*(3*A + 4*C)*a*d*x + (6*A*a*cos(d*x + c)^3 + 8*A*b*cos(d*x + c)^2 + 3*(3*A + 4*C)*a*cos(d*x + c) + 8*(2
*A + 3*C)*b)*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)**4*(a+b*sec(d*x+c))*(A+C*sec(d*x+c)**2),x)

[Out]

Timed out

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Giac [B]  time = 1.20293, size = 367, normalized size = 3.86 \begin{align*} \frac{3 \,{\left (3 \, A a + 4 \, C a\right )}{\left (d x + c\right )} - \frac{2 \,{\left (15 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 12 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 24 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 24 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 9 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 12 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 40 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 72 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 40 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 72 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 15 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 12 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 24 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 24 \, 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 )}^{4}}}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))*(A+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

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