\(\int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx\) [961]

Optimal result

Integrand size = 29, antiderivative size = 138 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {5}{128} a (8 A+B) x-\frac {a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac {5 a (8 A+B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a (8 A+B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a (8 A+B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))}{8 d} \]

[Out]

5/128*a*(8*A+B)*x-1/56*a*(8*A+B)*cos(d*x+c)^7/d+5/128*a*(8*A+B)*cos(d*x+c)*sin(d*x+c)/d+5/192*a*(8*A+B)*cos(d*
x+c)^3*sin(d*x+c)/d+1/48*a*(8*A+B)*cos(d*x+c)^5*sin(d*x+c)/d-1/8*B*cos(d*x+c)^7*(a+a*sin(d*x+c))/d

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2939, 2748, 2715, 8} \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {a (8 A+B) \cos ^7(c+d x)}{56 d}+\frac {a (8 A+B) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 a (8 A+B) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 a (8 A+B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5}{128} a x (8 A+B)-\frac {B \cos ^7(c+d x) (a \sin (c+d x)+a)}{8 d} \]

[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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

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] && Integ
erQ[2*n]

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[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 2939

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] /; F
reeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 3.29 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.32 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a \cos (c+d x) \left (-480 A-480 B-\frac {840 (8 A+B) \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(c+d x)}}-720 (A+B) \cos (2 (c+d x))-288 (A+B) \cos (4 (c+d x))-48 A \cos (6 (c+d x))-48 B \cos (6 (c+d x))+4144 A \sin (c+d x)+413 B \sin (c+d x)+896 A \sin (3 (c+d x))-77 B \sin (3 (c+d x))+112 A \sin (5 (c+d x))-91 B \sin (5 (c+d x))-21 B \sin (7 (c+d x))\right )}{10752 d} \]

[In]

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

[Out]

(a*Cos[c + d*x]*(-480*A - 480*B - (840*(8*A + B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]])/Sqrt[Cos[c + d*x]^2]
- 720*(A + B)*Cos[2*(c + d*x)] - 288*(A + B)*Cos[4*(c + d*x)] - 48*A*Cos[6*(c + d*x)] - 48*B*Cos[6*(c + d*x)]
+ 4144*A*Sin[c + d*x] + 413*B*Sin[c + d*x] + 896*A*Sin[3*(c + d*x)] - 77*B*Sin[3*(c + d*x)] + 112*A*Sin[5*(c +
d*x)] - 91*B*Sin[5*(c + d*x)] - 21*B*Sin[7*(c + d*x)]))/(10752*d)

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {B a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {a A \left (\cos ^{7}\left (d x +c \right )\right )}{7}-\frac {B a \left (\cos ^{7}\left (d x +c \right )\right )}{7}+a A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(138\)
default	\(\frac {B a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {a A \left (\cos ^{7}\left (d x +c \right )\right )}{7}-\frac {B a \left (\cos ^{7}\left (d x +c \right )\right )}{7}+a A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(138\)
parallelrisch	\(-\frac {a \left (21 \left (A +B \right ) \cos \left (3 d x +3 c \right )+7 \left (A +B \right ) \cos \left (5 d x +5 c \right )+\left (A +B \right ) \cos \left (7 d x +7 c \right )+7 \left (-15 A -B \right ) \sin \left (2 d x +2 c \right )+7 \left (-3 A +\frac {B}{2}\right ) \sin \left (4 d x +4 c \right )+\frac {7 \left (-A +B \right ) \sin \left (6 d x +6 c \right )}{3}+\frac {7 B \sin \left (8 d x +8 c \right )}{16}+35 \left (A +B \right ) \cos \left (d x +c \right )-140 d x A -\frac {35 d x B}{2}+64 A +64 B \right )}{448 d}\)	\(140\)
risch	\(\frac {5 a x A}{16}+\frac {5 a B x}{128}-\frac {5 a A \cos \left (d x +c \right )}{64 d}-\frac {5 a \cos \left (d x +c \right ) B}{64 d}-\frac {B a \sin \left (8 d x +8 c \right )}{1024 d}-\frac {a \cos \left (7 d x +7 c \right ) A}{448 d}-\frac {a \cos \left (7 d x +7 c \right ) B}{448 d}+\frac {\sin \left (6 d x +6 c \right ) a A}{192 d}-\frac {\sin \left (6 d x +6 c \right ) B a}{192 d}-\frac {a \cos \left (5 d x +5 c \right ) A}{64 d}-\frac {a \cos \left (5 d x +5 c \right ) B}{64 d}+\frac {3 \sin \left (4 d x +4 c \right ) a A}{64 d}-\frac {\sin \left (4 d x +4 c \right ) B a}{128 d}-\frac {3 a \cos \left (3 d x +3 c \right ) A}{64 d}-\frac {3 a \cos \left (3 d x +3 c \right ) B}{64 d}+\frac {15 \sin \left (2 d x +2 c \right ) a A}{64 d}+\frac {\sin \left (2 d x +2 c \right ) B a}{64 d}\)	\(246\)
norman	\(\frac {\frac {a \left (88 A -5 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {5 a \left (136 A +353 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {5 a \left (136 A +353 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\left (\frac {5}{16} a A +\frac {5}{128} B a \right ) x +\left (\frac {35}{4} a A +\frac {35}{32} B a \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35}{2} a A +\frac {35}{16} B a \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 a A +2 B a}{7 d}+\frac {a \left (488 A +397 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {a \left (904 A -895 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {2 \left (a A +B a \right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {10 \left (a A +B a \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (a A +B a \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d}+\left (\frac {5}{2} a A +\frac {5}{16} B a \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a A +\frac {5}{16} B a \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{16} a A +\frac {5}{128} B a \right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 \left (3 a A +3 B a \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (5 a A +5 B a \right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (3 a A +3 B a \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \left (904 A -895 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {a \left (488 A +397 B \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {a \left (88 A -5 B \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\left (\frac {175}{8} a A +\frac {175}{64} B a \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35}{2} a A +\frac {35}{16} B a \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35}{4} a A +\frac {35}{32} B a \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 \left (a A +B a \right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\)	\(576\)

[In]

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

[Out]

1/d*(B*a*(-1/8*cos(d*x+c)^7*sin(d*x+c)+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*B*a*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))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.70 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {384 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{7} - 105 \, {\left (8 \, A + B\right )} a d x + 7 \, {\left (48 \, B a \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, A + B\right )} a \cos \left (d x + c\right )^{5} - 10 \, {\left (8 \, A + B\right )} a \cos \left (d x + c\right )^{3} - 15 \, {\left (8 \, A + B\right )} a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, d} \]

[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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (131) = 262\).

Time = 0.66 (sec) , antiderivative size = 416, normalized size of antiderivative = 3.01 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {5 A a x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 A a x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 A a x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 A a x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 A a \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 A a \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 A a \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {A a \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac {5 B a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {5 B a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 B a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {5 B a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {5 B a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {5 B a \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 B a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {73 B a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {5 B a \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {B a \cos ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\left (c \right )}\right ) \left (a \sin {\left (c \right )} + a\right ) \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.90 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {3072 \, A a \cos \left (d x + c\right )^{7} + 3072 \, B a \cos \left (d x + c\right )^{7} + 112 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 7 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a}{21504 \, d} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.28 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {5}{128} \, {\left (8 \, A a + B a\right )} x - \frac {B a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (A a + B a\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {{\left (A a + B a\right )} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {3 \, {\left (A a + B a\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {5 \, {\left (A a + B a\right )} \cos \left (d x + c\right )}{64 \, d} + \frac {{\left (A a - B a\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (6 \, A a - B a\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (15 \, A a + B a\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 11.40 (sec) , antiderivative size = 504, normalized size of antiderivative = 3.65 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {5\,a\,\mathrm {atan}\left (\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,A+B\right )}{64\,\left (\frac {5\,A\,a}{8}+\frac {5\,B\,a}{64}\right )}\right )\,\left (8\,A+B\right )}{64\,d}-\frac {5\,a\,\left (8\,A+B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{64\,d}-\frac {\left (\frac {11\,A\,a}{8}-\frac {5\,B\,a}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+\left (2\,A\,a+2\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\left (\frac {61\,A\,a}{24}+\frac {397\,B\,a}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (2\,A\,a+2\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\left (\frac {113\,A\,a}{24}-\frac {895\,B\,a}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (10\,A\,a+10\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\left (\frac {85\,A\,a}{24}+\frac {1765\,B\,a}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (10\,A\,a+10\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\left (-\frac {85\,A\,a}{24}-\frac {1765\,B\,a}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (6\,A\,a+6\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {895\,B\,a}{192}-\frac {113\,A\,a}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (6\,A\,a+6\,B\,a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-\frac {61\,A\,a}{24}-\frac {397\,B\,a}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {2\,A\,a}{7}+\frac {2\,B\,a}{7}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (\frac {5\,B\,a}{64}-\frac {11\,A\,a}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {2\,A\,a}{7}+\frac {2\,B\,a}{7}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

[In]

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

[Out]

(5*a*atan((5*a*tan(c/2 + (d*x)/2)*(8*A + B))/(64*((5*A*a)/8 + (5*B*a)/64)))*(8*A + B))/(64*d) - (5*a*(8*A + B)
*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(64*d) - ((2*A*a)/7 + (2*B*a)/7 - tan(c/2 + (d*x)/2)*((11*A*a)/8 - (5*B
*a)/64) + tan(c/2 + (d*x)/2)^4*(6*A*a + 6*B*a) + tan(c/2 + (d*x)/2)^12*(2*A*a + 2*B*a) + tan(c/2 + (d*x)/2)^6*
(6*A*a + 6*B*a) + tan(c/2 + (d*x)/2)^14*(2*A*a + 2*B*a) + tan(c/2 + (d*x)/2)^2*((2*A*a)/7 + (2*B*a)/7) + tan(c
/2 + (d*x)/2)^8*(10*A*a + 10*B*a) + tan(c/2 + (d*x)/2)^10*(10*A*a + 10*B*a) + tan(c/2 + (d*x)/2)^15*((11*A*a)/
8 - (5*B*a)/64) - tan(c/2 + (d*x)/2)^3*((61*A*a)/24 + (397*B*a)/192) + tan(c/2 + (d*x)/2)^13*((61*A*a)/24 + (3
97*B*a)/192) - tan(c/2 + (d*x)/2)^5*((113*A*a)/24 - (895*B*a)/192) + tan(c/2 + (d*x)/2)^11*((113*A*a)/24 - (89
5*B*a)/192) - tan(c/2 + (d*x)/2)^7*((85*A*a)/24 + (1765*B*a)/192) + tan(c/2 + (d*x)/2)^9*((85*A*a)/24 + (1765*
B*a)/192))/(d*(8*tan(c/2 + (d*x)/2)^2 + 28*tan(c/2 + (d*x)/2)^4 + 56*tan(c/2 + (d*x)/2)^6 + 70*tan(c/2 + (d*x)
/2)^8 + 56*tan(c/2 + (d*x)/2)^10 + 28*tan(c/2 + (d*x)/2)^12 + 8*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^16
+ 1))