\(\int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx\) [953]

Optimal result

Integrand size = 29, antiderivative size = 134 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {8 (A-B) (a+a \sin (c+d x))^5}{5 a^4 d}-\frac {2 (3 A-5 B) (a+a \sin (c+d x))^6}{3 a^5 d}+\frac {6 (A-3 B) (a+a \sin (c+d x))^7}{7 a^6 d}-\frac {(A-7 B) (a+a \sin (c+d x))^8}{8 a^7 d}-\frac {B (a+a \sin (c+d x))^9}{9 a^8 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2915, 78} \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {B (a \sin (c+d x)+a)^9}{9 a^8 d}-\frac {(A-7 B) (a \sin (c+d x)+a)^8}{8 a^7 d}+\frac {6 (A-3 B) (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac {2 (3 A-5 B) (a \sin (c+d x)+a)^6}{3 a^5 d}+\frac {8 (A-B) (a \sin (c+d x)+a)^5}{5 a^4 d} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.74 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a \left (\frac {8}{5} (A-B) (1+\sin (c+d x))^5-\frac {2}{3} (3 A-5 B) (1+\sin (c+d x))^6+\frac {6}{7} (A-3 B) (1+\sin (c+d x))^7-\frac {1}{8} (A-7 B) (1+\sin (c+d x))^8-\frac {1}{9} B (1+\sin (c+d x))^9\right )}{d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.05

method	result	size
derivativedivides	\(-\frac {a \left (\frac {B \left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (A +B \right ) \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (A -3 B \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (-3 B -3 A \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (3 B -3 A \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (3 B +3 A \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (3 A -B \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (-A -B \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}-A \sin \left (d x +c \right )\right )}{d}\)	\(141\)
default	\(-\frac {a \left (\frac {B \left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (A +B \right ) \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (A -3 B \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (-3 B -3 A \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (3 B -3 A \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (3 B +3 A \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (3 A -B \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (-A -B \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}-A \sin \left (d x +c \right )\right )}{d}\)	\(141\)
parallelrisch	\(-\frac {\left (56 \left (A +B \right ) \cos \left (2 d x +2 c \right )+28 \left (A +B \right ) \cos \left (4 d x +4 c \right )+8 \left (A +B \right ) \cos \left (6 d x +6 c \right )+\left (A +B \right ) \cos \left (8 d x +8 c \right )+\frac {16 \left (-7 A +2 B \right ) \sin \left (5 d x +5 c \right )}{5}+\frac {4 \left (-4 A +5 B \right ) \sin \left (7 d x +7 c \right )}{7}-112 A \sin \left (3 d x +3 c \right )+\frac {4 B \sin \left (9 d x +9 c \right )}{9}+56 \left (-10 A -B \right ) \sin \left (d x +c \right )-93 A -93 B \right ) a}{1024 d}\)	\(144\)
risch	\(\frac {35 a A \sin \left (d x +c \right )}{64 d}+\frac {7 a B \sin \left (d x +c \right )}{128 d}-\frac {B a \sin \left (9 d x +9 c \right )}{2304 d}-\frac {a \cos \left (8 d x +8 c \right ) A}{1024 d}-\frac {a \cos \left (8 d x +8 c \right ) B}{1024 d}+\frac {\sin \left (7 d x +7 c \right ) a A}{448 d}-\frac {5 \sin \left (7 d x +7 c \right ) B a}{1792 d}-\frac {a \cos \left (6 d x +6 c \right ) A}{128 d}-\frac {a \cos \left (6 d x +6 c \right ) B}{128 d}+\frac {7 \sin \left (5 d x +5 c \right ) a A}{320 d}-\frac {\sin \left (5 d x +5 c \right ) B a}{160 d}-\frac {7 a \cos \left (4 d x +4 c \right ) A}{256 d}-\frac {7 a \cos \left (4 d x +4 c \right ) B}{256 d}+\frac {7 a A \sin \left (3 d x +3 c \right )}{64 d}-\frac {7 a \cos \left (2 d x +2 c \right ) A}{128 d}-\frac {7 a \cos \left (2 d x +2 c \right ) B}{128 d}\)	\(252\)
norman	\(\frac {\frac {\left (2 a A +2 B a \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (2 a A +2 B a \right ) \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 \left (a A +B a \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 \left (a A +B a \right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 \left (a A +B a \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 \left (a A +B a \right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (a A +B a \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (a A +B a \right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a A \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a \left (3 A +B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 a \left (3 A +B \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 a \left (17 A -2 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 a \left (17 A -2 B \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 a \left (221 A +79 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {8 a \left (221 A +79 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {4 a \left (4617 A -712 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\)	\(402\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.72 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {315 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{8} + 8 \, {\left (35 \, B a \cos \left (d x + c\right )^{8} - 5 \, {\left (9 \, A + B\right )} a \cos \left (d x + c\right )^{6} - 6 \, {\left (9 \, A + B\right )} a \cos \left (d x + c\right )^{4} - 8 \, {\left (9 \, A + B\right )} a \cos \left (d x + c\right )^{2} - 16 \, {\left (9 \, A + B\right )} a\right )} \sin \left (d x + c\right )}{2520 \, d} \]

[In]

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

[Out]

-1/2520*(315*(A + B)*a*cos(d*x + c)^8 + 8*(35*B*a*cos(d*x + c)^8 - 5*(9*A + B)*a*cos(d*x + c)^6 - 6*(9*A + B)*
a*cos(d*x + c)^4 - 8*(9*A + B)*a*cos(d*x + c)^2 - 16*(9*A + B)*a)*sin(d*x + c))/d

Sympy [A] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.70 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {16 A a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 A a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 A a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {A a \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {A a \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac {16 B a \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {8 B a \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {2 B a \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {B a \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {B a \cos ^{8}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\left (c \right )}\right ) \left (a \sin {\left (c \right )} + a\right ) \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((16*A*a*sin(c + d*x)**7/(35*d) + 8*A*a*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2*A*a*sin(c + d*x)**3
*cos(c + d*x)**4/d + A*a*sin(c + d*x)*cos(c + d*x)**6/d - A*a*cos(c + d*x)**8/(8*d) + 16*B*a*sin(c + d*x)**9/(
315*d) + 8*B*a*sin(c + d*x)**7*cos(c + d*x)**2/(35*d) + 2*B*a*sin(c + d*x)**5*cos(c + d*x)**4/(5*d) + B*a*sin(
c + d*x)**3*cos(c + d*x)**6/(3*d) - B*a*cos(c + d*x)**8/(8*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*sin(c) + a)*cos
(c)**7, True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {280 \, B a \sin \left (d x + c\right )^{9} + 315 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{8} + 360 \, {\left (A - 3 \, B\right )} a \sin \left (d x + c\right )^{7} - 1260 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{6} - 1512 \, {\left (A - B\right )} a \sin \left (d x + c\right )^{5} + 1890 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{4} + 840 \, {\left (3 \, A - B\right )} a \sin \left (d x + c\right )^{3} - 1260 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{2} - 2520 \, A a \sin \left (d x + c\right )}{2520 \, d} \]

[In]

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

[Out]

-1/2520*(280*B*a*sin(d*x + c)^9 + 315*(A + B)*a*sin(d*x + c)^8 + 360*(A - 3*B)*a*sin(d*x + c)^7 - 1260*(A + B)
*a*sin(d*x + c)^6 - 1512*(A - B)*a*sin(d*x + c)^5 + 1890*(A + B)*a*sin(d*x + c)^4 + 840*(3*A - B)*a*sin(d*x +
c)^3 - 1260*(A + B)*a*sin(d*x + c)^2 - 2520*A*a*sin(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.36 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {B a \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {7 \, A a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {{\left (A a + B a\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (A a + B a\right )} \cos \left (6 \, d x + 6 \, c\right )}{128 \, d} - \frac {7 \, {\left (A a + B a\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {7 \, {\left (A a + B a\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {{\left (4 \, A a - 5 \, B a\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (7 \, A a - 2 \, B a\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, {\left (10 \, A a + B a\right )} \sin \left (d x + c\right )}{128 \, d} \]

[In]

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

[Out]

-1/2304*B*a*sin(9*d*x + 9*c)/d + 7/64*A*a*sin(3*d*x + 3*c)/d - 1/1024*(A*a + B*a)*cos(8*d*x + 8*c)/d - 1/128*(
A*a + B*a)*cos(6*d*x + 6*c)/d - 7/256*(A*a + B*a)*cos(4*d*x + 4*c)/d - 7/128*(A*a + B*a)*cos(2*d*x + 2*c)/d +
1/1792*(4*A*a - 5*B*a)*sin(7*d*x + 7*c)/d + 1/320*(7*A*a - 2*B*a)*sin(5*d*x + 5*c)/d + 7/128*(10*A*a + B*a)*si
n(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00 \[ \int \cos ^7(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {\frac {B\,a\,{\sin \left (c+d\,x\right )}^9}{9}+\frac {a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {a\,\left (A-3\,B\right )\,{\sin \left (c+d\,x\right )}^7}{7}-\frac {a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^6}{2}-\frac {3\,a\,\left (A-B\right )\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {3\,a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^4}{4}+\frac {a\,\left (3\,A-B\right )\,{\sin \left (c+d\,x\right )}^3}{3}-\frac {a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^2}{2}-A\,a\,\sin \left (c+d\,x\right )}{d} \]

[In]

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

[Out]

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