3.284 \(\int \cos ^4(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx\)

Optimal. Leaf size=105 \[ -\frac{(a B+A b) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x)}{d}+\frac{(3 a A+4 b B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (3 a A+4 b B)+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.138741, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {3996, 3787, 2633, 2635, 8} \[ -\frac{(a B+A b) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x)}{d}+\frac{(3 a A+4 b B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (3 a A+4 b B)+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.235245, size = 91, normalized size = 0.87 \[ \frac{-32 (a B+A b) \sin ^3(c+d x)+96 (a B+A b) \sin (c+d x)+24 (a A+b B) \sin (2 (c+d x))+3 a A \sin (4 (c+d x))+36 a A c+36 a A d x+48 b B c+48 b B d x}{96 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(36*a*A*c + 48*b*B*c + 36*a*A*d*x + 48*b*B*d*x + 96*(A*b + a*B)*Sin[c + d*x] - 32*(A*b + a*B)*Sin[c + d*x]^3 +
24*(a*A + b*B)*Sin[2*(c + d*x)] + 3*a*A*Sin[4*(c + d*x)])/(96*d)

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Maple [A]  time = 0.063, size = 107, 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}}+{\frac{Ba \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+Bb \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)^4*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)),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)+1/3*
B*a*(2+cos(d*x+c)^2)*sin(d*x+c)+B*b*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c))

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Maxima [A]  time = 0.960661, size = 136, normalized size = 1.3 \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 - 32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a - 32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b + 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b}{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+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 - 32*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a
- 32*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*b + 24*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*b)/d

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Fricas [A]  time = 0.476038, size = 205, normalized size = 1.95 \begin{align*} \frac{3 \,{\left (3 \, A a + 4 \, B b\right )} d x +{\left (6 \, A a \cos \left (d x + c\right )^{3} + 8 \,{\left (B a + A b\right )} \cos \left (d x + c\right )^{2} + 16 \, B a + 16 \, A b + 3 \,{\left (3 \, A a + 4 \, B b\right )} \cos \left (d x + c\right )\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+B*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/24*(3*(3*A*a + 4*B*b)*d*x + (6*A*a*cos(d*x + c)^3 + 8*(B*a + A*b)*cos(d*x + c)^2 + 16*B*a + 16*A*b + 3*(3*A*
a + 4*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)**4*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 1.23204, size = 367, normalized size = 3.5 \begin{align*} \frac{3 \,{\left (3 \, A a + 4 \, B b\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} - 24 \, B 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} + 12 \, B 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} - 40 \, B 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} + 12 \, B 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} - 40 \, B 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} - 12 \, B 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 ) - 24 \, B 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 ) - 12 \, B 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+B*sec(d*x+c)),x, algorithm="giac")

[Out]

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