\(\int \cos ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx\) [987]

Optimal result

Integrand size = 31, antiderivative size = 105 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {2 (A-B) (a+a \sin (c+d x))^6}{3 a^3 d}-\frac {4 (A-2 B) (a+a \sin (c+d x))^7}{7 a^4 d}+\frac {(A-5 B) (a+a \sin (c+d x))^8}{8 a^5 d}+\frac {B (a+a \sin (c+d x))^9}{9 a^6 d} \]

[Out]

2/3*(A-B)*(a+a*sin(d*x+c))^6/a^3/d-4/7*(A-2*B)*(a+a*sin(d*x+c))^7/a^4/d+1/8*(A-5*B)*(a+a*sin(d*x+c))^8/a^5/d+1
/9*B*(a+a*sin(d*x+c))^9/a^6/d

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2915, 78} \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {B (a \sin (c+d x)+a)^9}{9 a^6 d}+\frac {(A-5 B) (a \sin (c+d x)+a)^8}{8 a^5 d}-\frac {4 (A-2 B) (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac {2 (A-B) (a \sin (c+d x)+a)^6}{3 a^3 d} \]

[In]

Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]

[Out]

(2*(A - B)*(a + a*Sin[c + d*x])^6)/(3*a^3*d) - (4*(A - 2*B)*(a + a*Sin[c + d*x])^7)/(7*a^4*d) + ((A - 5*B)*(a
+ a*Sin[c + d*x])^8)/(8*a^5*d) + (B*(a + a*Sin[c + d*x])^9)/(9*a^6*d)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2915

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a-x)^2 (a+x)^5 \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \left (4 a^2 (A-B) (a+x)^5-4 a (A-2 B) (a+x)^6+(A-5 B) (a+x)^7+\frac {B (a+x)^8}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {2 (A-B) (a+a \sin (c+d x))^6}{3 a^3 d}-\frac {4 (A-2 B) (a+a \sin (c+d x))^7}{7 a^4 d}+\frac {(A-5 B) (a+a \sin (c+d x))^8}{8 a^5 d}+\frac {B (a+a \sin (c+d x))^9}{9 a^6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.67 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^3 (1+\sin (c+d x))^6 \left (111 A-19 B-6 (27 A-19 B) \sin (c+d x)+21 (3 A-7 B) \sin ^2(c+d x)+56 B \sin ^3(c+d x)\right )}{504 d} \]

[In]

Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]

[Out]

(a^3*(1 + Sin[c + d*x])^6*(111*A - 19*B - 6*(27*A - 19*B)*Sin[c + d*x] + 21*(3*A - 7*B)*Sin[c + d*x]^2 + 56*B*
Sin[c + d*x]^3))/(504*d)

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {\left (\sin ^{9}\left (d x +c \right )\right ) B}{9}+\frac {\left (A +3 B \right ) \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (3 A +B \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (A -5 B \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (-5 A -5 B \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (B -5 A \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (A +3 B \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (3 A +B \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}+A \sin \left (d x +c \right )\right )}{d}\)	\(135\)
default	\(\frac {a^{3} \left (\frac {\left (\sin ^{9}\left (d x +c \right )\right ) B}{9}+\frac {\left (A +3 B \right ) \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (3 A +B \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (A -5 B \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (-5 A -5 B \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (B -5 A \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (A +3 B \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (3 A +B \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}+A \sin \left (d x +c \right )\right )}{d}\)	\(135\)
parallelrisch	\(\frac {a^{3} \left (8 \left (-33 A -19 B \right ) \cos \left (2 d x +2 c \right )+4 \left (-25 A -11 B \right ) \cos \left (4 d x +4 c \right )+\frac {8 \left (B -5 A \right ) \cos \left (6 d x +6 c \right )}{3}+\left (A +3 B \right ) \cos \left (8 d x +8 c \right )+\frac {16 \left (17 A -4 B \right ) \sin \left (3 d x +3 c \right )}{3}+16 \left (-A -2 B \right ) \sin \left (5 d x +5 c \right )+\frac {4 \left (-12 A -11 B \right ) \sin \left (7 d x +7 c \right )}{7}+\frac {4 B \sin \left (9 d x +9 c \right )}{9}+88 \left (10 A +3 B \right ) \sin \left (d x +c \right )+\frac {1129 A}{3}+\frac {571 B}{3}\right )}{1024 d}\)	\(164\)
risch	\(\frac {55 \sin \left (d x +c \right ) A \,a^{3}}{64 d}+\frac {33 a^{3} B \sin \left (d x +c \right )}{128 d}+\frac {\sin \left (9 d x +9 c \right ) B \,a^{3}}{2304 d}+\frac {a^{3} \cos \left (8 d x +8 c \right ) A}{1024 d}+\frac {3 a^{3} \cos \left (8 d x +8 c \right ) B}{1024 d}-\frac {3 \sin \left (7 d x +7 c \right ) A \,a^{3}}{448 d}-\frac {11 \sin \left (7 d x +7 c \right ) B \,a^{3}}{1792 d}-\frac {5 a^{3} \cos \left (6 d x +6 c \right ) A}{384 d}+\frac {a^{3} \cos \left (6 d x +6 c \right ) B}{384 d}-\frac {\sin \left (5 d x +5 c \right ) A \,a^{3}}{64 d}-\frac {\sin \left (5 d x +5 c \right ) B \,a^{3}}{32 d}-\frac {25 a^{3} \cos \left (4 d x +4 c \right ) A}{256 d}-\frac {11 a^{3} \cos \left (4 d x +4 c \right ) B}{256 d}+\frac {17 \sin \left (3 d x +3 c \right ) A \,a^{3}}{192 d}-\frac {\sin \left (3 d x +3 c \right ) B \,a^{3}}{48 d}-\frac {33 a^{3} \cos \left (2 d x +2 c \right ) A}{128 d}-\frac {19 a^{3} \cos \left (2 d x +2 c \right ) B}{128 d}\)	\(302\)
norman	\(\frac {\frac {\left (6 A \,a^{3}+2 B \,a^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (6 A \,a^{3}+2 B \,a^{3}\right ) \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (11 A \,a^{3}+9 B \,a^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (11 A \,a^{3}+9 B \,a^{3}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (37 A \,a^{3}+23 B \,a^{3}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (37 A \,a^{3}+23 B \,a^{3}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (55 A \,a^{3}+13 B \,a^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 \left (55 A \,a^{3}+13 B \,a^{3}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 A \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 A \,a^{3} \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{3} \left (5 A +2 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{3} \left (5 A +2 B \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a^{3} \left (7 A +3 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 a^{3} \left (7 A +3 B \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {24 a^{3} \left (23 A +3 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d}+\frac {24 a^{3} \left (23 A +3 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d}+\frac {44 a^{3} \left (159 A +88 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\)	\(468\)

[In]

int(cos(d*x+c)^5*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

a^3/d*(1/9*sin(d*x+c)^9*B+1/8*(A+3*B)*sin(d*x+c)^8+1/7*(3*A+B)*sin(d*x+c)^7+1/6*(A-5*B)*sin(d*x+c)^6+1/5*(-5*A
-5*B)*sin(d*x+c)^5+1/4*(B-5*A)*sin(d*x+c)^4+1/3*(A+3*B)*sin(d*x+c)^3+1/2*(3*A+B)*sin(d*x+c)^2+A*sin(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.23 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {63 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{8} - 336 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{6} + 8 \, {\left (7 \, B a^{3} \cos \left (d x + c\right )^{8} - {\left (27 \, A + 37 \, B\right )} a^{3} \cos \left (d x + c\right )^{6} + 6 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{4} + 8 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + 16 \, {\left (3 \, A + B\right )} a^{3}\right )} \sin \left (d x + c\right )}{504 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/504*(63*(A + 3*B)*a^3*cos(d*x + c)^8 - 336*(A + B)*a^3*cos(d*x + c)^6 + 8*(7*B*a^3*cos(d*x + c)^8 - (27*A +
37*B)*a^3*cos(d*x + c)^6 + 6*(3*A + B)*a^3*cos(d*x + c)^4 + 8*(3*A + B)*a^3*cos(d*x + c)^2 + 16*(3*A + B)*a^3)
*sin(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (99) = 198\).

Time = 0.95 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.98 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {8 A a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {4 A a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {8 A a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {A a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {4 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} - \frac {A a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} + \frac {A a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {A a^{3} \cos ^{8}{\left (c + d x \right )}}{24 d} - \frac {A a^{3} \cos ^{6}{\left (c + d x \right )}}{2 d} + \frac {8 B a^{3} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 B a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {8 B a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {B a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {4 B a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {B a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {B a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{2 d} - \frac {B a^{3} \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {B a^{3} \cos ^{6}{\left (c + d x \right )}}{6 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\left (c \right )}\right ) \left (a \sin {\left (c \right )} + a\right )^{3} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**5*(a+a*sin(d*x+c))**3*(A+B*sin(d*x+c)),x)

[Out]

Piecewise((8*A*a**3*sin(c + d*x)**7/(35*d) + 4*A*a**3*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 8*A*a**3*sin(c +
d*x)**5/(15*d) + A*a**3*sin(c + d*x)**3*cos(c + d*x)**4/d + 4*A*a**3*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) -
A*a**3*sin(c + d*x)**2*cos(c + d*x)**6/(6*d) + A*a**3*sin(c + d*x)*cos(c + d*x)**4/d - A*a**3*cos(c + d*x)**8/
(24*d) - A*a**3*cos(c + d*x)**6/(2*d) + 8*B*a**3*sin(c + d*x)**9/(315*d) + 4*B*a**3*sin(c + d*x)**7*cos(c + d*
x)**2/(35*d) + 8*B*a**3*sin(c + d*x)**7/(35*d) + B*a**3*sin(c + d*x)**5*cos(c + d*x)**4/(5*d) + 4*B*a**3*sin(c
+ d*x)**5*cos(c + d*x)**2/(5*d) + B*a**3*sin(c + d*x)**3*cos(c + d*x)**4/d - B*a**3*sin(c + d*x)**2*cos(c + d
*x)**6/(2*d) - B*a**3*cos(c + d*x)**8/(8*d) - B*a**3*cos(c + d*x)**6/(6*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*si
n(c) + a)**3*cos(c)**5, True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.50 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {56 \, B a^{3} \sin \left (d x + c\right )^{9} + 63 \, {\left (A + 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{8} + 72 \, {\left (3 \, A + B\right )} a^{3} \sin \left (d x + c\right )^{7} + 84 \, {\left (A - 5 \, B\right )} a^{3} \sin \left (d x + c\right )^{6} - 504 \, {\left (A + B\right )} a^{3} \sin \left (d x + c\right )^{5} - 126 \, {\left (5 \, A - B\right )} a^{3} \sin \left (d x + c\right )^{4} + 168 \, {\left (A + 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{3} + 252 \, {\left (3 \, A + B\right )} a^{3} \sin \left (d x + c\right )^{2} + 504 \, A a^{3} \sin \left (d x + c\right )}{504 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/504*(56*B*a^3*sin(d*x + c)^9 + 63*(A + 3*B)*a^3*sin(d*x + c)^8 + 72*(3*A + B)*a^3*sin(d*x + c)^7 + 84*(A - 5
*B)*a^3*sin(d*x + c)^6 - 504*(A + B)*a^3*sin(d*x + c)^5 - 126*(5*A - B)*a^3*sin(d*x + c)^4 + 168*(A + 3*B)*a^3
*sin(d*x + c)^3 + 252*(3*A + B)*a^3*sin(d*x + c)^2 + 504*A*a^3*sin(d*x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (97) = 194\).

Time = 0.44 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.19 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {B a^{3} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {{\left (A a^{3} + 3 \, B a^{3}\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (5 \, A a^{3} - B a^{3}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {{\left (25 \, A a^{3} + 11 \, B a^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {{\left (33 \, A a^{3} + 19 \, B a^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac {{\left (12 \, A a^{3} + 11 \, B a^{3}\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {{\left (A a^{3} + 2 \, B a^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{64 \, d} + \frac {{\left (17 \, A a^{3} - 4 \, B a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {11 \, {\left (10 \, A a^{3} + 3 \, B a^{3}\right )} \sin \left (d x + c\right )}{128 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

1/2304*B*a^3*sin(9*d*x + 9*c)/d + 1/1024*(A*a^3 + 3*B*a^3)*cos(8*d*x + 8*c)/d - 1/384*(5*A*a^3 - B*a^3)*cos(6*
d*x + 6*c)/d - 1/256*(25*A*a^3 + 11*B*a^3)*cos(4*d*x + 4*c)/d - 1/128*(33*A*a^3 + 19*B*a^3)*cos(2*d*x + 2*c)/d
- 1/1792*(12*A*a^3 + 11*B*a^3)*sin(7*d*x + 7*c)/d - 1/64*(A*a^3 + 2*B*a^3)*sin(5*d*x + 5*c)/d + 1/192*(17*A*a
^3 - 4*B*a^3)*sin(3*d*x + 3*c)/d + 11/128*(10*A*a^3 + 3*B*a^3)*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.49 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^2\,\left (3\,A+B\right )}{2}+\frac {a^3\,{\sin \left (c+d\,x\right )}^3\,\left (A+3\,B\right )}{3}+\frac {a^3\,{\sin \left (c+d\,x\right )}^7\,\left (3\,A+B\right )}{7}+\frac {a^3\,{\sin \left (c+d\,x\right )}^6\,\left (A-5\,B\right )}{6}+\frac {a^3\,{\sin \left (c+d\,x\right )}^8\,\left (A+3\,B\right )}{8}+\frac {B\,a^3\,{\sin \left (c+d\,x\right )}^9}{9}-\frac {a^3\,{\sin \left (c+d\,x\right )}^4\,\left (5\,A-B\right )}{4}+A\,a^3\,\sin \left (c+d\,x\right )-a^3\,{\sin \left (c+d\,x\right )}^5\,\left (A+B\right )}{d} \]

[In]

int(cos(c + d*x)^5*(A + B*sin(c + d*x))*(a + a*sin(c + d*x))^3,x)

[Out]

((a^3*sin(c + d*x)^2*(3*A + B))/2 + (a^3*sin(c + d*x)^3*(A + 3*B))/3 + (a^3*sin(c + d*x)^7*(3*A + B))/7 + (a^3
*sin(c + d*x)^6*(A - 5*B))/6 + (a^3*sin(c + d*x)^8*(A + 3*B))/8 + (B*a^3*sin(c + d*x)^9)/9 - (a^3*sin(c + d*x)
^4*(5*A - B))/4 + A*a^3*sin(c + d*x) - a^3*sin(c + d*x)^5*(A + B))/d