3.961 \(\int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx\)

Optimal. Leaf size=138 \[ -\frac{a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac{a (8 A+B) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a (8 A+B) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a (8 A+B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} a x (8 A+B)-\frac{B \cos ^7(c+d x) (a \sin (c+d x)+a)}{8 d} \]

[Out]

(5*a*(8*A + B)*x)/128 - (a*(8*A + B)*Cos[c + d*x]^7)/(56*d) + (5*a*(8*A + B)*Cos[c + d*x]*Sin[c + d*x])/(128*d
) + (5*a*(8*A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a*(8*A + B)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (
B*Cos[c + d*x]^7*(a + a*Sin[c + d*x]))/(8*d)

________________________________________________________________________________________

Rubi [A]  time = 0.140802, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2860, 2669, 2635, 8} \[ -\frac{a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac{a (8 A+B) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a (8 A+B) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a (8 A+B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} a x (8 A+B)-\frac{B \cos ^7(c+d x) (a \sin (c+d x)+a)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]

[Out]

(5*a*(8*A + B)*x)/128 - (a*(8*A + B)*Cos[c + d*x]^7)/(56*d) + (5*a*(8*A + B)*Cos[c + d*x]*Sin[c + d*x])/(128*d
) + (5*a*(8*A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a*(8*A + B)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (
B*Cos[c + d*x]^7*(a + a*Sin[c + d*x]))/(8*d)

Rule 2860

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[(d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(f*g*(m + p + 1)), x]
+ Dist[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; Fre
eQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 1, 0]

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^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 ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d}+\frac{1}{8} (8 A+B) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac{a (8 A+B) \cos ^7(c+d x)}{56 d}-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d}+\frac{1}{8} (a (8 A+B)) \int \cos ^6(c+d x) \, dx\\ &=-\frac{a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac{a (8 A+B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d}+\frac{1}{48} (5 a (8 A+B)) \int \cos ^4(c+d x) \, dx\\ &=-\frac{a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac{5 a (8 A+B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{a (8 A+B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d}+\frac{1}{64} (5 a (8 A+B)) \int \cos ^2(c+d x) \, dx\\ &=-\frac{a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac{5 a (8 A+B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{5 a (8 A+B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{a (8 A+B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d}+\frac{1}{128} (5 a (8 A+B)) \int 1 \, dx\\ &=\frac{5}{128} a (8 A+B) x-\frac{a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac{5 a (8 A+B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{5 a (8 A+B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{a (8 A+B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.880102, size = 164, normalized size = 1.19 \[ -\frac{a (1680 (A+B) \cos (c+d x)+1008 (A+B) \cos (3 (c+d x))-5040 A \sin (2 (c+d x))-1008 A \sin (4 (c+d x))-112 A \sin (6 (c+d x))+336 A \cos (5 (c+d x))+48 A \cos (7 (c+d x))-6720 A d x-336 B \sin (2 (c+d x))+168 B \sin (4 (c+d x))+112 B \sin (6 (c+d x))+21 B \sin (8 (c+d x))+336 B \cos (5 (c+d x))+48 B \cos (7 (c+d x))-840 B d x)}{21504 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]

[Out]

-(a*(-6720*A*d*x - 840*B*d*x + 1680*(A + B)*Cos[c + d*x] + 1008*(A + B)*Cos[3*(c + d*x)] + 336*A*Cos[5*(c + d*
x)] + 336*B*Cos[5*(c + d*x)] + 48*A*Cos[7*(c + d*x)] + 48*B*Cos[7*(c + d*x)] - 5040*A*Sin[2*(c + d*x)] - 336*B
*Sin[2*(c + d*x)] - 1008*A*Sin[4*(c + d*x)] + 168*B*Sin[4*(c + d*x)] - 112*A*Sin[6*(c + d*x)] + 112*B*Sin[6*(c
+ d*x)] + 21*B*Sin[8*(c + d*x)]))/(21504*d)

________________________________________________________________________________________

Maple [A]  time = 0.056, size = 138, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( aB \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8}}+{\frac{\sin \left ( dx+c \right ) }{48} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{128}}+{\frac{5\,c}{128}} \right ) -{\frac{aA \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7}}-{\frac{aB \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7}}+aA \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x)

[Out]

1/d*(a*B*(-1/8*sin(d*x+c)*cos(d*x+c)^7+1/48*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/128*d
*x+5/128*c)-1/7*a*A*cos(d*x+c)^7-1/7*a*B*cos(d*x+c)^7+a*A*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))
*sin(d*x+c)+5/16*d*x+5/16*c))

________________________________________________________________________________________

Maxima [A]  time = 1.03704, size = 167, normalized size = 1.21 \begin{align*} -\frac{3072 \, A a \cos \left (d x + c\right )^{7} + 3072 \, B a \cos \left (d x + c\right )^{7} + 112 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 7 \,{\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a}{21504 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/21504*(3072*A*a*cos(d*x + c)^7 + 3072*B*a*cos(d*x + c)^7 + 112*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*si
n(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*A*a - 7*(64*sin(2*d*x + 2*c)^3 + 120*d*x + 120*c - 3*sin(8*d*x + 8*c) -
24*sin(4*d*x + 4*c))*B*a)/d

________________________________________________________________________________________

Fricas [A]  time = 1.91591, size = 267, normalized size = 1.93 \begin{align*} -\frac{384 \,{\left (A + B\right )} a \cos \left (d x + c\right )^{7} - 105 \,{\left (8 \, A + B\right )} a d x + 7 \,{\left (48 \, B a \cos \left (d x + c\right )^{7} - 8 \,{\left (8 \, A + B\right )} a \cos \left (d x + c\right )^{5} - 10 \,{\left (8 \, A + B\right )} a \cos \left (d x + c\right )^{3} - 15 \,{\left (8 \, A + B\right )} a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/2688*(384*(A + B)*a*cos(d*x + c)^7 - 105*(8*A + B)*a*d*x + 7*(48*B*a*cos(d*x + c)^7 - 8*(8*A + B)*a*cos(d*x
+ c)^5 - 10*(8*A + B)*a*cos(d*x + c)^3 - 15*(8*A + B)*a*cos(d*x + c))*sin(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 13.8277, size = 416, normalized size = 3.01 \begin{align*} \begin{cases} \frac{5 A a x \sin ^{6}{\left (c + d x \right )}}{16} + \frac{15 A a x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{15 A a x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac{5 A a x \cos ^{6}{\left (c + d x \right )}}{16} + \frac{5 A a \sin ^{5}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{16 d} + \frac{5 A a \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac{11 A a \sin{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac{A a \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac{5 B a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac{5 B a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac{15 B a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{5 B a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac{5 B a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac{5 B a \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{55 B a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac{73 B a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac{5 B a \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac{B a \cos ^{7}{\left (c + d x \right )}}{7 d} & \text{for}\: d \neq 0 \\x \left (A + B \sin{\left (c \right )}\right ) \left (a \sin{\left (c \right )} + a\right ) \cos ^{6}{\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)**6*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x)

[Out]

Piecewise((5*A*a*x*sin(c + d*x)**6/16 + 15*A*a*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*A*a*x*sin(c + d*x)**2
*cos(c + d*x)**4/16 + 5*A*a*x*cos(c + d*x)**6/16 + 5*A*a*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*A*a*sin(c + d
*x)**3*cos(c + d*x)**3/(6*d) + 11*A*a*sin(c + d*x)*cos(c + d*x)**5/(16*d) - A*a*cos(c + d*x)**7/(7*d) + 5*B*a*
x*sin(c + d*x)**8/128 + 5*B*a*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 15*B*a*x*sin(c + d*x)**4*cos(c + d*x)**4/
64 + 5*B*a*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 5*B*a*x*cos(c + d*x)**8/128 + 5*B*a*sin(c + d*x)**7*cos(c +
d*x)/(128*d) + 55*B*a*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 73*B*a*sin(c + d*x)**3*cos(c + d*x)**5/(384*d)
- 5*B*a*sin(c + d*x)*cos(c + d*x)**7/(128*d) - B*a*cos(c + d*x)**7/(7*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*sin
(c) + a)*cos(c)**6, True))

________________________________________________________________________________________

Giac [A]  time = 1.327, size = 238, normalized size = 1.72 \begin{align*} \frac{5}{128} \,{\left (8 \, A a + B a\right )} x - \frac{B a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac{{\left (A a + B a\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac{{\left (A a + B a\right )} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac{3 \,{\left (A a + B a\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac{5 \,{\left (A a + B a\right )} \cos \left (d x + c\right )}{64 \, d} + \frac{{\left (A a - B a\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{{\left (6 \, A a - B a\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{{\left (15 \, A a + B a\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

5/128*(8*A*a + B*a)*x - 1/1024*B*a*sin(8*d*x + 8*c)/d - 1/448*(A*a + B*a)*cos(7*d*x + 7*c)/d - 1/64*(A*a + B*a
)*cos(5*d*x + 5*c)/d - 3/64*(A*a + B*a)*cos(3*d*x + 3*c)/d - 5/64*(A*a + B*a)*cos(d*x + c)/d + 1/192*(A*a - B*
a)*sin(6*d*x + 6*c)/d + 1/128*(6*A*a - B*a)*sin(4*d*x + 4*c)/d + 1/64*(15*A*a + B*a)*sin(2*d*x + 2*c)/d