\(\int \cos ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx\) [986]

Optimal result

Integrand size = 31, antiderivative size = 134 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {8 (A-B) (a+a \sin (c+d x))^7}{7 a^4 d}-\frac {(3 A-5 B) (a+a \sin (c+d x))^8}{2 a^5 d}+\frac {2 (A-3 B) (a+a \sin (c+d x))^9}{3 a^6 d}-\frac {(A-7 B) (a+a \sin (c+d x))^{10}}{10 a^7 d}-\frac {B (a+a \sin (c+d x))^{11}}{11 a^8 d} \]

[Out]

8/7*(A-B)*(a+a*sin(d*x+c))^7/a^4/d-1/2*(3*A-5*B)*(a+a*sin(d*x+c))^8/a^5/d+2/3*(A-3*B)*(a+a*sin(d*x+c))^9/a^6/d
-1/10*(A-7*B)*(a+a*sin(d*x+c))^10/a^7/d-1/11*B*(a+a*sin(d*x+c))^11/a^8/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 134, 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} \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {B (a \sin (c+d x)+a)^{11}}{11 a^8 d}-\frac {(A-7 B) (a \sin (c+d x)+a)^{10}}{10 a^7 d}+\frac {2 (A-3 B) (a \sin (c+d x)+a)^9}{3 a^6 d}-\frac {(3 A-5 B) (a \sin (c+d x)+a)^8}{2 a^5 d}+\frac {8 (A-B) (a \sin (c+d x)+a)^7}{7 a^4 d} \]

[In]

Int[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]

[Out]

(8*(A - B)*(a + a*Sin[c + d*x])^7)/(7*a^4*d) - ((3*A - 5*B)*(a + a*Sin[c + d*x])^8)/(2*a^5*d) + (2*(A - 3*B)*(
a + a*Sin[c + d*x])^9)/(3*a^6*d) - ((A - 7*B)*(a + a*Sin[c + d*x])^10)/(10*a^7*d) - (B*(a + a*Sin[c + d*x])^11
)/(11*a^8*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)^3 (a+x)^6 \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \left (8 a^3 (A-B) (a+x)^6-4 a^2 (3 A-5 B) (a+x)^7+6 a (A-3 B) (a+x)^8+(-A+7 B) (a+x)^9-\frac {B (a+x)^{10}}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {8 (A-B) (a+a \sin (c+d x))^7}{7 a^4 d}-\frac {(3 A-5 B) (a+a \sin (c+d x))^8}{2 a^5 d}+\frac {2 (A-3 B) (a+a \sin (c+d x))^9}{3 a^6 d}-\frac {(A-7 B) (a+a \sin (c+d x))^{10}}{10 a^7 d}-\frac {B (a+a \sin (c+d x))^{11}}{11 a^8 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.64 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3 (1+\sin (c+d x))^7 \left (-484 A+78 B+14 (77 A-39 B) \sin (c+d x)+(-847 A+1029 B) \sin ^2(c+d x)+21 (11 A-37 B) \sin ^3(c+d x)+210 B \sin ^4(c+d x)\right )}{2310 d} \]

[In]

Integrate[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]

[Out]

-1/2310*(a^3*(1 + Sin[c + d*x])^7*(-484*A + 78*B + 14*(77*A - 39*B)*Sin[c + d*x] + (-847*A + 1029*B)*Sin[c + d
*x]^2 + 21*(11*A - 37*B)*Sin[c + d*x]^3 + 210*B*Sin[c + d*x]^4))/d

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.17

method	result	size
derivativedivides	\(-\frac {a^{3} \left (\frac {B \left (\sin ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (A +3 B \right ) \left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\sin ^{9}\left (d x +c \right )\right ) A}{3}-B \left (\sin ^{8}\left (d x +c \right )\right )+\frac {\left (-6 B -8 A \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (6 B -6 A \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (8 B +6 A \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+2 \left (\sin ^{4}\left (d x +c \right )\right ) A -B \left (\sin ^{3}\left (d x +c \right )\right )+\frac {\left (-3 A -B \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}-A \sin \left (d x +c \right )\right )}{d}\)	\(157\)
default	\(-\frac {a^{3} \left (\frac {B \left (\sin ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (A +3 B \right ) \left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (\sin ^{9}\left (d x +c \right )\right ) A}{3}-B \left (\sin ^{8}\left (d x +c \right )\right )+\frac {\left (-6 B -8 A \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (6 B -6 A \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (8 B +6 A \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+2 \left (\sin ^{4}\left (d x +c \right )\right ) A -B \left (\sin ^{3}\left (d x +c \right )\right )+\frac {\left (-3 A -B \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}-A \sin \left (d x +c \right )\right )}{d}\)	\(157\)
parallelrisch	\(-\frac {a^{3} \left (\left (-\frac {A}{10}-\frac {3 B}{10}\right ) \cos \left (10 d x +10 c \right )+\left (91 A +49 B \right ) \cos \left (2 d x +2 c \right )+\left (44 A +20 B \right ) \cos \left (4 d x +4 c \right )+\left (\frac {23 A}{2}+\frac {5 B}{2}\right ) \cos \left (6 d x +6 c \right )+\left (A -B \right ) \cos \left (8 d x +8 c \right )+\left (-56 A +B \right ) \sin \left (3 d x +3 c \right )+\left (-\frac {8 A}{5}+\frac {107 B}{10}\right ) \sin \left (5 d x +5 c \right )+\left (\frac {22 A}{7}+\frac {61 B}{14}\right ) \sin \left (7 d x +7 c \right )+\left (\frac {2 A}{3}+\frac {B}{2}\right ) \sin \left (9 d x +9 c \right )-\frac {B \sin \left (11 d x +11 c \right )}{22}+\left (-364 A -91 B \right ) \sin \left (d x +c \right )-\frac {737 A}{5}-\frac {351 B}{5}\right )}{512 d}\)	\(191\)
risch	\(\frac {91 \sin \left (d x +c \right ) A \,a^{3}}{128 d}+\frac {91 a^{3} B \sin \left (d x +c \right )}{512 d}+\frac {B \,a^{3} \sin \left (11 d x +11 c \right )}{11264 d}+\frac {a^{3} \cos \left (10 d x +10 c \right ) A}{5120 d}+\frac {3 a^{3} \cos \left (10 d x +10 c \right ) B}{5120 d}-\frac {\sin \left (9 d x +9 c \right ) A \,a^{3}}{768 d}-\frac {\sin \left (9 d x +9 c \right ) B \,a^{3}}{1024 d}-\frac {a^{3} \cos \left (8 d x +8 c \right ) A}{512 d}+\frac {a^{3} \cos \left (8 d x +8 c \right ) B}{512 d}-\frac {11 \sin \left (7 d x +7 c \right ) A \,a^{3}}{1792 d}-\frac {61 \sin \left (7 d x +7 c \right ) B \,a^{3}}{7168 d}-\frac {23 a^{3} \cos \left (6 d x +6 c \right ) A}{1024 d}-\frac {5 a^{3} \cos \left (6 d x +6 c \right ) B}{1024 d}+\frac {\sin \left (5 d x +5 c \right ) A \,a^{3}}{320 d}-\frac {107 \sin \left (5 d x +5 c \right ) B \,a^{3}}{5120 d}-\frac {11 a^{3} \cos \left (4 d x +4 c \right ) A}{128 d}-\frac {5 a^{3} \cos \left (4 d x +4 c \right ) B}{128 d}+\frac {7 \sin \left (3 d x +3 c \right ) A \,a^{3}}{64 d}-\frac {\sin \left (3 d x +3 c \right ) B \,a^{3}}{512 d}-\frac {91 a^{3} \cos \left (2 d x +2 c \right ) A}{512 d}-\frac {49 a^{3} \cos \left (2 d x +2 c \right ) B}{512 d}\)	\(374\)

[In]

int(cos(d*x+c)^7*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-a^3/d*(1/11*B*sin(d*x+c)^11+1/10*(A+3*B)*sin(d*x+c)^10+1/3*sin(d*x+c)^9*A-B*sin(d*x+c)^8+1/7*(-6*B-8*A)*sin(d
*x+c)^7+1/6*(6*B-6*A)*sin(d*x+c)^6+1/5*(8*B+6*A)*sin(d*x+c)^5+2*sin(d*x+c)^4*A-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 = 0.32 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.16 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {231 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{10} - 1155 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{8} + 2 \, {\left (105 \, B a^{3} \cos \left (d x + c\right )^{10} - 35 \, {\left (11 \, A + 15 \, B\right )} a^{3} \cos \left (d x + c\right )^{8} + 20 \, {\left (11 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{6} + 24 \, {\left (11 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} + 32 \, {\left (11 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 64 \, {\left (11 \, A + 3 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{2310 \, d} \]

[In]

integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/2310*(231*(A + 3*B)*a^3*cos(d*x + c)^10 - 1155*(A + B)*a^3*cos(d*x + c)^8 + 2*(105*B*a^3*cos(d*x + c)^10 - 3
5*(11*A + 15*B)*a^3*cos(d*x + c)^8 + 20*(11*A + 3*B)*a^3*cos(d*x + c)^6 + 24*(11*A + 3*B)*a^3*cos(d*x + c)^4 +
32*(11*A + 3*B)*a^3*cos(d*x + c)^2 + 64*(11*A + 3*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. 530 vs. \(2 (124) = 248\).

Time = 1.82 (sec) , antiderivative size = 530, normalized size of antiderivative = 3.96 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {16 A a^{3} \sin ^{9}{\left (c + d x \right )}}{105 d} + \frac {24 A a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {16 A a^{3} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {6 A a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {8 A a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {A a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} + \frac {2 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {A a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac {A a^{3} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {A a^{3} \cos ^{10}{\left (c + d x \right )}}{40 d} - \frac {3 A a^{3} \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac {16 B a^{3} \sin ^{11}{\left (c + d x \right )}}{1155 d} + \frac {8 B a^{3} \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{105 d} + \frac {16 B a^{3} \sin ^{9}{\left (c + d x \right )}}{105 d} + \frac {6 B a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{35 d} + \frac {24 B a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {B a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{5 d} + \frac {6 B a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {B a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {3 B a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {3 B a^{3} \cos ^{10}{\left (c + d x \right )}}{40 d} - \frac {B a^{3} \cos ^{8}{\left (c + d x \right )}}{8 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 ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**7*(a+a*sin(d*x+c))**3*(A+B*sin(d*x+c)),x)

[Out]

Piecewise((16*A*a**3*sin(c + d*x)**9/(105*d) + 24*A*a**3*sin(c + d*x)**7*cos(c + d*x)**2/(35*d) + 16*A*a**3*si
n(c + d*x)**7/(35*d) + 6*A*a**3*sin(c + d*x)**5*cos(c + d*x)**4/(5*d) + 8*A*a**3*sin(c + d*x)**5*cos(c + d*x)*
*2/(5*d) + A*a**3*sin(c + d*x)**3*cos(c + d*x)**6/d + 2*A*a**3*sin(c + d*x)**3*cos(c + d*x)**4/d - A*a**3*sin(
c + d*x)**2*cos(c + d*x)**8/(8*d) + A*a**3*sin(c + d*x)*cos(c + d*x)**6/d - A*a**3*cos(c + d*x)**10/(40*d) - 3
*A*a**3*cos(c + d*x)**8/(8*d) + 16*B*a**3*sin(c + d*x)**11/(1155*d) + 8*B*a**3*sin(c + d*x)**9*cos(c + d*x)**2
/(105*d) + 16*B*a**3*sin(c + d*x)**9/(105*d) + 6*B*a**3*sin(c + d*x)**7*cos(c + d*x)**4/(35*d) + 24*B*a**3*sin
(c + d*x)**7*cos(c + d*x)**2/(35*d) + B*a**3*sin(c + d*x)**5*cos(c + d*x)**6/(5*d) + 6*B*a**3*sin(c + d*x)**5*
cos(c + d*x)**4/(5*d) + B*a**3*sin(c + d*x)**3*cos(c + d*x)**6/d - 3*B*a**3*sin(c + d*x)**2*cos(c + d*x)**8/(8
*d) - 3*B*a**3*cos(c + d*x)**10/(40*d) - B*a**3*cos(c + d*x)**8/(8*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*sin(c)
+ a)**3*cos(c)**7, True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.36 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {210 \, B a^{3} \sin \left (d x + c\right )^{11} + 231 \, {\left (A + 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{10} + 770 \, A a^{3} \sin \left (d x + c\right )^{9} - 2310 \, B a^{3} \sin \left (d x + c\right )^{8} - 660 \, {\left (4 \, A + 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{7} - 2310 \, {\left (A - B\right )} a^{3} \sin \left (d x + c\right )^{6} + 924 \, {\left (3 \, A + 4 \, B\right )} a^{3} \sin \left (d x + c\right )^{5} + 4620 \, A a^{3} \sin \left (d x + c\right )^{4} - 2310 \, B a^{3} \sin \left (d x + c\right )^{3} - 1155 \, {\left (3 \, A + B\right )} a^{3} \sin \left (d x + c\right )^{2} - 2310 \, A a^{3} \sin \left (d x + c\right )}{2310 \, d} \]

[In]

integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/2310*(210*B*a^3*sin(d*x + c)^11 + 231*(A + 3*B)*a^3*sin(d*x + c)^10 + 770*A*a^3*sin(d*x + c)^9 - 2310*B*a^3
*sin(d*x + c)^8 - 660*(4*A + 3*B)*a^3*sin(d*x + c)^7 - 2310*(A - B)*a^3*sin(d*x + c)^6 + 924*(3*A + 4*B)*a^3*s
in(d*x + c)^5 + 4620*A*a^3*sin(d*x + c)^4 - 2310*B*a^3*sin(d*x + c)^3 - 1155*(3*A + B)*a^3*sin(d*x + c)^2 - 23
10*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. 283 vs. \(2 (124) = 248\).

Time = 0.44 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.11 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {B a^{3} \sin \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {{\left (A a^{3} + 3 \, B a^{3}\right )} \cos \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac {{\left (A a^{3} - B a^{3}\right )} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac {{\left (23 \, A a^{3} + 5 \, B a^{3}\right )} \cos \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {{\left (11 \, A a^{3} + 5 \, B a^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{128 \, d} - \frac {7 \, {\left (13 \, A a^{3} + 7 \, B a^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )}{512 \, d} - \frac {{\left (4 \, A a^{3} + 3 \, B a^{3}\right )} \sin \left (9 \, d x + 9 \, c\right )}{3072 \, d} - \frac {{\left (44 \, A a^{3} + 61 \, B a^{3}\right )} \sin \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {{\left (16 \, A a^{3} - 107 \, B a^{3}\right )} \sin \left (5 \, d x + 5 \, c\right )}{5120 \, d} + \frac {{\left (56 \, A a^{3} - B a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{512 \, d} + \frac {91 \, {\left (4 \, A a^{3} + B a^{3}\right )} \sin \left (d x + c\right )}{512 \, d} \]

[In]

integrate(cos(d*x+c)^7*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

1/11264*B*a^3*sin(11*d*x + 11*c)/d + 1/5120*(A*a^3 + 3*B*a^3)*cos(10*d*x + 10*c)/d - 1/512*(A*a^3 - B*a^3)*cos
(8*d*x + 8*c)/d - 1/1024*(23*A*a^3 + 5*B*a^3)*cos(6*d*x + 6*c)/d - 1/128*(11*A*a^3 + 5*B*a^3)*cos(4*d*x + 4*c)
/d - 7/512*(13*A*a^3 + 7*B*a^3)*cos(2*d*x + 2*c)/d - 1/3072*(4*A*a^3 + 3*B*a^3)*sin(9*d*x + 9*c)/d - 1/7168*(4
4*A*a^3 + 61*B*a^3)*sin(7*d*x + 7*c)/d + 1/5120*(16*A*a^3 - 107*B*a^3)*sin(5*d*x + 5*c)/d + 1/512*(56*A*a^3 -
B*a^3)*sin(3*d*x + 3*c)/d + 91/512*(4*A*a^3 + B*a^3)*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.32 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^2\,\left (3\,A+B\right )}{2}-\frac {A\,a^3\,{\sin \left (c+d\,x\right )}^9}{3}-2\,A\,a^3\,{\sin \left (c+d\,x\right )}^4+a^3\,{\sin \left (c+d\,x\right )}^6\,\left (A-B\right )-\frac {a^3\,{\sin \left (c+d\,x\right )}^{10}\,\left (A+3\,B\right )}{10}+B\,a^3\,{\sin \left (c+d\,x\right )}^3+B\,a^3\,{\sin \left (c+d\,x\right )}^8-\frac {B\,a^3\,{\sin \left (c+d\,x\right )}^{11}}{11}-\frac {2\,a^3\,{\sin \left (c+d\,x\right )}^5\,\left (3\,A+4\,B\right )}{5}+\frac {2\,a^3\,{\sin \left (c+d\,x\right )}^7\,\left (4\,A+3\,B\right )}{7}+A\,a^3\,\sin \left (c+d\,x\right )}{d} \]

[In]

int(cos(c + d*x)^7*(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*a^3*sin(c + d*x)^9)/3 - 2*A*a^3*sin(c + d*x)^4 + a^3*sin(c + d*x)^6*(A
- B) - (a^3*sin(c + d*x)^10*(A + 3*B))/10 + B*a^3*sin(c + d*x)^3 + B*a^3*sin(c + d*x)^8 - (B*a^3*sin(c + d*x)^
11)/11 - (2*a^3*sin(c + d*x)^5*(3*A + 4*B))/5 + (2*a^3*sin(c + d*x)^7*(4*A + 3*B))/7 + A*a^3*sin(c + d*x))/d