3.11 \(\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx\)

Optimal. Leaf size=160 \[ -\frac{a^2 (10 A+9 B) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 A+9 B) \sin (c+d x)}{5 d}+\frac{a^2 (5 A+6 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 (7 A+6 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (7 A+6 B)+\frac{B \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{5 d} \]

[Out]

(a^2*(7*A + 6*B)*x)/8 + (a^2*(10*A + 9*B)*Sin[c + d*x])/(5*d) + (a^2*(7*A + 6*B)*Cos[c + d*x]*Sin[c + d*x])/(8
*d) + (a^2*(5*A + 6*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (B*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])*Sin[c
+ d*x])/(5*d) - (a^2*(10*A + 9*B)*Sin[c + d*x]^3)/(15*d)

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Rubi [A]  time = 0.277828, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {2976, 2968, 3023, 2748, 2635, 8, 2633} \[ -\frac{a^2 (10 A+9 B) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 A+9 B) \sin (c+d x)}{5 d}+\frac{a^2 (5 A+6 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 (7 A+6 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (7 A+6 B)+\frac{B \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(7*A + 6*B)*x)/8 + (a^2*(10*A + 9*B)*Sin[c + d*x])/(5*d) + (a^2*(7*A + 6*B)*Cos[c + d*x]*Sin[c + d*x])/(8
*d) + (a^2*(5*A + 6*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (B*Cos[c + d*x]^3*(a^2 + a^2*Cos[c + d*x])*Sin[c
+ d*x])/(5*d) - (a^2*(10*A + 9*B)*Sin[c + d*x]^3)/(15*d)

Rule 2976

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*S
in[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&
NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, 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 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 ^2(c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx &=\frac{B \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{5} \int \cos ^2(c+d x) (a+a \cos (c+d x)) (a (5 A+3 B)+a (5 A+6 B) \cos (c+d x)) \, dx\\ &=\frac{B \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{5} \int \cos ^2(c+d x) \left (a^2 (5 A+3 B)+\left (a^2 (5 A+3 B)+a^2 (5 A+6 B)\right ) \cos (c+d x)+a^2 (5 A+6 B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 (5 A+6 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{B \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{20} \int \cos ^2(c+d x) \left (5 a^2 (7 A+6 B)+4 a^2 (10 A+9 B) \cos (c+d x)\right ) \, dx\\ &=\frac{a^2 (5 A+6 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{B \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{4} \left (a^2 (7 A+6 B)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{5} \left (a^2 (10 A+9 B)\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac{a^2 (7 A+6 B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a^2 (5 A+6 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{B \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{8} \left (a^2 (7 A+6 B)\right ) \int 1 \, dx-\frac{\left (a^2 (10 A+9 B)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac{1}{8} a^2 (7 A+6 B) x+\frac{a^2 (10 A+9 B) \sin (c+d x)}{5 d}+\frac{a^2 (7 A+6 B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a^2 (5 A+6 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{B \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}-\frac{a^2 (10 A+9 B) \sin ^3(c+d x)}{15 d}\\ \end{align*}

Mathematica [A]  time = 0.39821, size = 108, normalized size = 0.68 \[ \frac{a^2 (60 (12 A+11 B) \sin (c+d x)+240 (A+B) \sin (2 (c+d x))+80 A \sin (3 (c+d x))+15 A \sin (4 (c+d x))+420 A d x+90 B \sin (3 (c+d x))+30 B \sin (4 (c+d x))+6 B \sin (5 (c+d x))+360 B c+360 B d x)}{480 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(360*B*c + 420*A*d*x + 360*B*d*x + 60*(12*A + 11*B)*Sin[c + d*x] + 240*(A + B)*Sin[2*(c + d*x)] + 80*A*Si
n[3*(c + d*x)] + 90*B*Sin[3*(c + d*x)] + 15*A*Sin[4*(c + d*x)] + 30*B*Sin[4*(c + d*x)] + 6*B*Sin[5*(c + d*x)])
)/(480*d)

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Maple [A]  time = 0.076, size = 186, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{2}A \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{B{a}^{2}\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 ) }+{\frac{2\,{a}^{2}A \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,B{a}^{2} \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 ) +{a}^{2}A \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{\frac{B{a}^{2} \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)^2*(a+cos(d*x+c)*a)^2*(A+B*cos(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 0.975302, size = 240, normalized size = 1.5 \begin{align*} -\frac{320 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \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 )} A a^{2} - 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 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^{2} + 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + 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 )} B a^{2}}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.39993, size = 271, normalized size = 1.69 \begin{align*} \frac{15 \,{\left (7 \, A + 6 \, B\right )} a^{2} d x +{\left (24 \, B a^{2} \cos \left (d x + c\right )^{4} + 30 \,{\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \,{\left (10 \, A + 9 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \,{\left (7 \, A + 6 \, B\right )} a^{2} \cos \left (d x + c\right ) + 16 \,{\left (10 \, A + 9 \, B\right )} a^{2}\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)^2*(a+a*cos(d*x+c))^2*(A+B*cos(d*x+c)),x, algorithm="fricas")

[Out]

1/120*(15*(7*A + 6*B)*a^2*d*x + (24*B*a^2*cos(d*x + c)^4 + 30*(A + 2*B)*a^2*cos(d*x + c)^3 + 8*(10*A + 9*B)*a^
2*cos(d*x + c)^2 + 15*(7*A + 6*B)*a^2*cos(d*x + c) + 16*(10*A + 9*B)*a^2)*sin(d*x + c))/d

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Sympy [A]  time = 3.41671, size = 459, normalized size = 2.87 \begin{align*} \begin{cases} \frac{3 A a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 A a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{A a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{3 A a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{A a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{3 A a^{2} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{4 A a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{5 A a^{2} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{2 A a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{A a^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{3 B a^{2} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac{3 B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac{3 B a^{2} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac{8 B a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{3 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{4 d} + \frac{2 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{B a^{2} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{5 B a^{2} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac{B a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (A + B \cos{\left (c \right )}\right ) \left (a \cos{\left (c \right )} + a\right )^{2} \cos ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2*(a+a*cos(d*x+c))**2*(A+B*cos(d*x+c)),x)

[Out]

Piecewise((3*A*a**2*x*sin(c + d*x)**4/8 + 3*A*a**2*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + A*a**2*x*sin(c + d*x)
**2/2 + 3*A*a**2*x*cos(c + d*x)**4/8 + A*a**2*x*cos(c + d*x)**2/2 + 3*A*a**2*sin(c + d*x)**3*cos(c + d*x)/(8*d
) + 4*A*a**2*sin(c + d*x)**3/(3*d) + 5*A*a**2*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 2*A*a**2*sin(c + d*x)*cos(c
+ d*x)**2/d + A*a**2*sin(c + d*x)*cos(c + d*x)/(2*d) + 3*B*a**2*x*sin(c + d*x)**4/4 + 3*B*a**2*x*sin(c + d*x)
**2*cos(c + d*x)**2/2 + 3*B*a**2*x*cos(c + d*x)**4/4 + 8*B*a**2*sin(c + d*x)**5/(15*d) + 4*B*a**2*sin(c + d*x)
**3*cos(c + d*x)**2/(3*d) + 3*B*a**2*sin(c + d*x)**3*cos(c + d*x)/(4*d) + 2*B*a**2*sin(c + d*x)**3/(3*d) + B*a
**2*sin(c + d*x)*cos(c + d*x)**4/d + 5*B*a**2*sin(c + d*x)*cos(c + d*x)**3/(4*d) + B*a**2*sin(c + d*x)*cos(c +
d*x)**2/d, Ne(d, 0)), (x*(A + B*cos(c))*(a*cos(c) + a)**2*cos(c)**2, True))

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Giac [A]  time = 1.22221, size = 185, normalized size = 1.16 \begin{align*} \frac{B a^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{1}{8} \,{\left (7 \, A a^{2} + 6 \, B a^{2}\right )} x + \frac{{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{{\left (8 \, A a^{2} + 9 \, B a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (A a^{2} + B a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac{{\left (12 \, A a^{2} + 11 \, B a^{2}\right )} \sin \left (d x + c\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80*B*a^2*sin(5*d*x + 5*c)/d + 1/8*(7*A*a^2 + 6*B*a^2)*x + 1/32*(A*a^2 + 2*B*a^2)*sin(4*d*x + 4*c)/d + 1/48*(
8*A*a^2 + 9*B*a^2)*sin(3*d*x + 3*c)/d + 1/2*(A*a^2 + B*a^2)*sin(2*d*x + 2*c)/d + 1/8*(12*A*a^2 + 11*B*a^2)*sin
(d*x + c)/d