\(\int \cos ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx\) [995]

Optimal result

Integrand size = 31, antiderivative size = 231 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {11}{256} a^3 (10 A+3 B) x-\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}+\frac {11 a^3 (10 A+3 B) \cos (c+d x) \sin (c+d x)}{256 d}+\frac {11 a^3 (10 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {11 a^3 (10 A+3 B) \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d} \]

[Out]

11/256*a^3*(10*A+3*B)*x-11/560*a^3*(10*A+3*B)*cos(d*x+c)^7/d+11/256*a^3*(10*A+3*B)*cos(d*x+c)*sin(d*x+c)/d+11/
384*a^3*(10*A+3*B)*cos(d*x+c)^3*sin(d*x+c)/d+11/480*a^3*(10*A+3*B)*cos(d*x+c)^5*sin(d*x+c)/d-1/90*a*(10*A+3*B)
*cos(d*x+c)^7*(a+a*sin(d*x+c))^2/d-1/10*B*cos(d*x+c)^7*(a+a*sin(d*x+c))^3/d-11/720*(10*A+3*B)*cos(d*x+c)^7*(a^
3+a^3*sin(d*x+c))/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2939, 2757, 2748, 2715, 8} \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{720 d}+\frac {11 a^3 (10 A+3 B) \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac {11 a^3 (10 A+3 B) \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac {11 a^3 (10 A+3 B) \sin (c+d x) \cos (c+d x)}{256 d}+\frac {11}{256} a^3 x (10 A+3 B)-\frac {a (10 A+3 B) \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{90 d}-\frac {B \cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d} \]

[In]

Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]

[Out]

(11*a^3*(10*A + 3*B)*x)/256 - (11*a^3*(10*A + 3*B)*Cos[c + d*x]^7)/(560*d) + (11*a^3*(10*A + 3*B)*Cos[c + d*x]
*Sin[c + d*x])/(256*d) + (11*a^3*(10*A + 3*B)*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (11*a^3*(10*A + 3*B)*Cos[
c + d*x]^5*Sin[c + d*x])/(480*d) - (a*(10*A + 3*B)*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(90*d) - (B*Cos[c +
d*x]^7*(a + a*Sin[c + d*x])^3)/(10*d) - (11*(10*A + 3*B)*Cos[c + d*x]^7*(a^3 + a^3*Sin[c + d*x]))/(720*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 2757

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

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))^3}{10 d}+\frac {1}{10} (10 A+3 B) \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 \, dx \\ & = -\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}+\frac {1}{90} (11 a (10 A+3 B)) \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d}+\frac {1}{80} \left (11 a^2 (10 A+3 B)\right ) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d}+\frac {1}{80} \left (11 a^3 (10 A+3 B)\right ) \int \cos ^6(c+d x) \, dx \\ & = -\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}+\frac {11 a^3 (10 A+3 B) \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d}+\frac {1}{96} \left (11 a^3 (10 A+3 B)\right ) \int \cos ^4(c+d x) \, dx \\ & = -\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}+\frac {11 a^3 (10 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {11 a^3 (10 A+3 B) \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d}+\frac {1}{128} \left (11 a^3 (10 A+3 B)\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}+\frac {11 a^3 (10 A+3 B) \cos (c+d x) \sin (c+d x)}{256 d}+\frac {11 a^3 (10 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {11 a^3 (10 A+3 B) \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d}+\frac {1}{256} \left (11 a^3 (10 A+3 B)\right ) \int 1 \, dx \\ & = \frac {11}{256} a^3 (10 A+3 B) x-\frac {11 a^3 (10 A+3 B) \cos ^7(c+d x)}{560 d}+\frac {11 a^3 (10 A+3 B) \cos (c+d x) \sin (c+d x)}{256 d}+\frac {11 a^3 (10 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {11 a^3 (10 A+3 B) \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {a (10 A+3 B) \cos ^7(c+d x) (a+a \sin (c+d x))^2}{90 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 (10 A+3 B) \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{720 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.39 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.03 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {a^3 \cos (c+d x) \left (47800 A+28200 B+\frac {27720 (10 A+3 B) \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(c+d x)}}+320 (221 A+123 B) \cos (2 (c+d x))+160 (167 A+69 B) \cos (4 (c+d x))+3520 A \cos (6 (c+d x))-960 B \cos (6 (c+d x))-280 A \cos (8 (c+d x))-840 B \cos (8 (c+d x))-161490 A \sin (c+d x)-40131 B \sin (c+d x)-19950 A \sin (3 (c+d x))+8631 B \sin (3 (c+d x))+4830 A \sin (5 (c+d x))+9009 B \sin (5 (c+d x))+1890 A \sin (7 (c+d x))+1701 B \sin (7 (c+d x))-126 B \sin (9 (c+d x))\right )}{322560 d} \]

[In]

Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]

[Out]

-1/322560*(a^3*Cos[c + d*x]*(47800*A + 28200*B + (27720*(10*A + 3*B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]])/S
qrt[Cos[c + d*x]^2] + 320*(221*A + 123*B)*Cos[2*(c + d*x)] + 160*(167*A + 69*B)*Cos[4*(c + d*x)] + 3520*A*Cos[
6*(c + d*x)] - 960*B*Cos[6*(c + d*x)] - 280*A*Cos[8*(c + d*x)] - 840*B*Cos[8*(c + d*x)] - 161490*A*Sin[c + d*x
] - 40131*B*Sin[c + d*x] - 19950*A*Sin[3*(c + d*x)] + 8631*B*Sin[3*(c + d*x)] + 4830*A*Sin[5*(c + d*x)] + 9009
*B*Sin[5*(c + d*x)] + 1890*A*Sin[7*(c + d*x)] + 1701*B*Sin[7*(c + d*x)] - 126*B*Sin[9*(c + d*x)]))/d

Maple [A] (verified)

Time = 1.73 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.81

method	result	size
parallelrisch	\(-\frac {9 a^{3} \left (\left (\frac {812 A}{27}+\frac {140 B}{9}\right ) \cos \left (3 d x +3 c \right )+\left (\frac {28 A}{3}+\frac {28 B}{9}\right ) \cos \left (5 d x +5 c \right )+\left (A -\frac {5 B}{9}\right ) \cos \left (7 d x +7 c \right )+\left (-\frac {7 A}{81}-\frac {7 B}{27}\right ) \cos \left (9 d x +9 c \right )+\left (-56 A -\frac {175 B}{18}\right ) \sin \left (2 d x +2 c \right )+\left (-\frac {14 A}{3}+\frac {49 B}{9}\right ) \sin \left (4 d x +4 c \right )+\left (\frac {56 A}{27}+\frac {119 B}{36}\right ) \sin \left (6 d x +6 c \right )+\left (\frac {7 A}{12}+\frac {35 B}{72}\right ) \sin \left (8 d x +8 c \right )-\frac {7 B \sin \left (10 d x +10 c \right )}{180}+\left (\frac {154 A}{3}+\frac {266 B}{9}\right ) \cos \left (d x +c \right )-\frac {770 d x A}{9}-\frac {77 d x B}{3}+\frac {7424 A}{81}+\frac {1280 B}{27}\right )}{1792 d}\)	\(186\)
risch	\(\frac {55 a^{3} x A}{128}+\frac {33 a^{3} B x}{256}-\frac {33 A \,a^{3} \cos \left (d x +c \right )}{128 d}-\frac {19 a^{3} \cos \left (d x +c \right ) B}{128 d}+\frac {B \,a^{3} \sin \left (10 d x +10 c \right )}{5120 d}+\frac {a^{3} \cos \left (9 d x +9 c \right ) A}{2304 d}+\frac {a^{3} \cos \left (9 d x +9 c \right ) B}{768 d}-\frac {3 \sin \left (8 d x +8 c \right ) A \,a^{3}}{1024 d}-\frac {5 \sin \left (8 d x +8 c \right ) B \,a^{3}}{2048 d}-\frac {9 a^{3} \cos \left (7 d x +7 c \right ) A}{1792 d}+\frac {5 a^{3} \cos \left (7 d x +7 c \right ) B}{1792 d}-\frac {\sin \left (6 d x +6 c \right ) A \,a^{3}}{96 d}-\frac {17 \sin \left (6 d x +6 c \right ) B \,a^{3}}{1024 d}-\frac {3 a^{3} \cos \left (5 d x +5 c \right ) A}{64 d}-\frac {a^{3} \cos \left (5 d x +5 c \right ) B}{64 d}+\frac {3 \sin \left (4 d x +4 c \right ) A \,a^{3}}{128 d}-\frac {7 \sin \left (4 d x +4 c \right ) B \,a^{3}}{256 d}-\frac {29 a^{3} \cos \left (3 d x +3 c \right ) A}{192 d}-\frac {5 a^{3} \cos \left (3 d x +3 c \right ) B}{64 d}+\frac {9 \sin \left (2 d x +2 c \right ) A \,a^{3}}{32 d}+\frac {25 \sin \left (2 d x +2 c \right ) B \,a^{3}}{512 d}\)	\(352\)
derivativedivides	\(\frac {A \,a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 A \,a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+3 B \,a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-\frac {3 A \,a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+3 B \,a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+A \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {B \,a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d}\)	\(363\)
default	\(\frac {A \,a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+B \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 A \,a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+3 B \,a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-\frac {3 A \,a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+3 B \,a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+A \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {B \,a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d}\)	\(363\)

[In]

int(cos(d*x+c)^6*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-9/1792*a^3*((812/27*A+140/9*B)*cos(3*d*x+3*c)+(28/3*A+28/9*B)*cos(5*d*x+5*c)+(A-5/9*B)*cos(7*d*x+7*c)+(-7/81*
A-7/27*B)*cos(9*d*x+9*c)+(-56*A-175/18*B)*sin(2*d*x+2*c)+(-14/3*A+49/9*B)*sin(4*d*x+4*c)+(56/27*A+119/36*B)*si
n(6*d*x+6*c)+(7/12*A+35/72*B)*sin(8*d*x+8*c)-7/180*B*sin(10*d*x+10*c)+(154/3*A+266/9*B)*cos(d*x+c)-770/9*d*x*A
-77/3*d*x*B+7424/81*A+1280/27*B)/d

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.67 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {8960 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{9} - 46080 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{7} + 3465 \, {\left (10 \, A + 3 \, B\right )} a^{3} d x + 21 \, {\left (384 \, B a^{3} \cos \left (d x + c\right )^{9} - 48 \, {\left (30 \, A + 41 \, B\right )} a^{3} \cos \left (d x + c\right )^{7} + 88 \, {\left (10 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} + 110 \, {\left (10 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 165 \, {\left (10 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/80640*(8960*(A + 3*B)*a^3*cos(d*x + c)^9 - 46080*(A + B)*a^3*cos(d*x + c)^7 + 3465*(10*A + 3*B)*a^3*d*x + 21
*(384*B*a^3*cos(d*x + c)^9 - 48*(30*A + 41*B)*a^3*cos(d*x + c)^7 + 88*(10*A + 3*B)*a^3*cos(d*x + c)^5 + 110*(1
0*A + 3*B)*a^3*cos(d*x + c)^3 + 165*(10*A + 3*B)*a^3*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. 1042 vs. \(2 (219) = 438\).

Time = 1.35 (sec) , antiderivative size = 1042, normalized size of antiderivative = 4.51 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)**6*(a+a*sin(d*x+c))**3*(A+B*sin(d*x+c)),x)

[Out]

Piecewise((15*A*a**3*x*sin(c + d*x)**8/128 + 15*A*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 5*A*a**3*x*sin(c
+ d*x)**6/16 + 45*A*a**3*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*A*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/1
6 + 15*A*a**3*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 15*A*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 15*A*a**
3*x*cos(c + d*x)**8/128 + 5*A*a**3*x*cos(c + d*x)**6/16 + 15*A*a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 55*
A*a**3*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) + 5*A*a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 73*A*a**3*sin(
c + d*x)**3*cos(c + d*x)**5/(128*d) + 5*A*a**3*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) - A*a**3*sin(c + d*x)**2*
cos(c + d*x)**7/(7*d) - 15*A*a**3*sin(c + d*x)*cos(c + d*x)**7/(128*d) + 11*A*a**3*sin(c + d*x)*cos(c + d*x)**
5/(16*d) - 2*A*a**3*cos(c + d*x)**9/(63*d) - 3*A*a**3*cos(c + d*x)**7/(7*d) + 3*B*a**3*x*sin(c + d*x)**10/256
+ 15*B*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 15*B*a**3*x*sin(c + d*x)**8/128 + 15*B*a**3*x*sin(c + d*x)
**6*cos(c + d*x)**4/128 + 15*B*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 15*B*a**3*x*sin(c + d*x)**4*cos(c +
d*x)**6/128 + 45*B*a**3*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**8/25
6 + 15*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 3*B*a**3*x*cos(c + d*x)**10/256 + 15*B*a**3*x*cos(c + d*x
)**8/128 + 3*B*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*B*a**3*sin(c + d*x)**7*cos(c + d*x)**3/(128*d) +
15*B*a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) + B*a**3*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 55*B*a**3*sin
(c + d*x)**5*cos(c + d*x)**3/(128*d) - 7*B*a**3*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) + 73*B*a**3*sin(c + d*
x)**3*cos(c + d*x)**5/(128*d) - 3*B*a**3*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 3*B*a**3*sin(c + d*x)*cos(c +
d*x)**9/(256*d) - 15*B*a**3*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 2*B*a**3*cos(c + d*x)**9/(21*d) - B*a**3*c
os(c + d*x)**7/(7*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*sin(c) + a)**3*cos(c)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.23 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=-\frac {276480 \, A a^{3} \cos \left (d x + c\right )^{7} + 92160 \, B a^{3} \cos \left (d x + c\right )^{7} - 10240 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} A a^{3} - 630 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} A a^{3} + 3360 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 30720 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} B a^{3} - 63 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{3} - 630 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{3}}{645120 \, d} \]

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/645120*(276480*A*a^3*cos(d*x + c)^7 + 92160*B*a^3*cos(d*x + c)^7 - 10240*(7*cos(d*x + c)^9 - 9*cos(d*x + c)
^7)*A*a^3 - 630*(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))*A*a^3 + 3
360*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*A*a^3 - 30720*(7*cos(d*x
+ c)^9 - 9*cos(d*x + c)^7)*B*a^3 - 63*(32*sin(2*d*x + 2*c)^5 + 120*d*x + 120*c + 5*sin(8*d*x + 8*c) - 40*sin(
4*d*x + 4*c))*B*a^3 - 630*(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^3)/d

Giac [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.18 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {B a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {11}{256} \, {\left (10 \, A a^{3} + 3 \, B a^{3}\right )} x + \frac {{\left (A a^{3} + 3 \, B a^{3}\right )} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {{\left (9 \, A a^{3} - 5 \, B a^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {{\left (3 \, A a^{3} + B a^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {{\left (29 \, A a^{3} + 15 \, B a^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {{\left (33 \, A a^{3} + 19 \, B a^{3}\right )} \cos \left (d x + c\right )}{128 \, d} - \frac {{\left (6 \, A a^{3} + 5 \, B a^{3}\right )} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {{\left (32 \, A a^{3} + 51 \, B a^{3}\right )} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} + \frac {{\left (6 \, A a^{3} - 7 \, B a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {{\left (144 \, A a^{3} + 25 \, B a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

1/5120*B*a^3*sin(10*d*x + 10*c)/d + 11/256*(10*A*a^3 + 3*B*a^3)*x + 1/2304*(A*a^3 + 3*B*a^3)*cos(9*d*x + 9*c)/
d - 1/1792*(9*A*a^3 - 5*B*a^3)*cos(7*d*x + 7*c)/d - 1/64*(3*A*a^3 + B*a^3)*cos(5*d*x + 5*c)/d - 1/192*(29*A*a^
3 + 15*B*a^3)*cos(3*d*x + 3*c)/d - 1/128*(33*A*a^3 + 19*B*a^3)*cos(d*x + c)/d - 1/2048*(6*A*a^3 + 5*B*a^3)*sin
(8*d*x + 8*c)/d - 1/3072*(32*A*a^3 + 51*B*a^3)*sin(6*d*x + 6*c)/d + 1/256*(6*A*a^3 - 7*B*a^3)*sin(4*d*x + 4*c)
/d + 1/512*(144*A*a^3 + 25*B*a^3)*sin(2*d*x + 2*c)/d

Mupad [B] (verification not implemented)

Time = 11.39 (sec) , antiderivative size = 711, normalized size of antiderivative = 3.08 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^6*(A + B*sin(c + d*x))*(a + a*sin(c + d*x))^3,x)

[Out]

(11*a^3*atan((11*a^3*tan(c/2 + (d*x)/2)*(10*A + 3*B))/(128*((55*A*a^3)/64 + (33*B*a^3)/128)))*(10*A + 3*B))/(1
28*d) - (11*a^3*(10*A + 3*B)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(128*d) - ((58*A*a^3)/63 - tan(c/2 + (d*x)/
2)*((73*A*a^3)/64 - (33*B*a^3)/128) + (10*B*a^3)/21 + tan(c/2 + (d*x)/2)^18*(6*A*a^3 + 2*B*a^3) + tan(c/2 + (d
*x)/2)^16*(22*A*a^3 + 18*B*a^3) + tan(c/2 + (d*x)/2)^8*(84*A*a^3 + 28*B*a^3) + tan(c/2 + (d*x)/2)^14*((136*A*a
^3)/3 + 8*B*a^3) + tan(c/2 + (d*x)/2)^4*((136*A*a^3)/7 + (24*B*a^3)/7) + tan(c/2 + (d*x)/2)^10*(116*A*a^3 + 60
*B*a^3) + tan(c/2 + (d*x)/2)^19*((73*A*a^3)/64 - (33*B*a^3)/128) + tan(c/2 + (d*x)/2)^2*((202*A*a^3)/63 + (58*
B*a^3)/21) + tan(c/2 + (d*x)/2)^12*((328*A*a^3)/3 + 72*B*a^3) - tan(c/2 + (d*x)/2)^7*((341*A*a^3)/16 - (333*B*
a^3)/32) + tan(c/2 + (d*x)/2)^13*((341*A*a^3)/16 - (333*B*a^3)/32) + tan(c/2 + (d*x)/2)^6*((456*A*a^3)/7 + (34
4*B*a^3)/7) - tan(c/2 + (d*x)/2)^5*((449*A*a^3)/48 + (577*B*a^3)/160) + tan(c/2 + (d*x)/2)^15*((449*A*a^3)/48
+ (577*B*a^3)/160) - tan(c/2 + (d*x)/2)^3*((2117*A*a^3)/192 + (705*B*a^3)/128) + tan(c/2 + (d*x)/2)^17*((2117*
A*a^3)/192 + (705*B*a^3)/128) - tan(c/2 + (d*x)/2)^9*((699*A*a^3)/32 + (2749*B*a^3)/64) + tan(c/2 + (d*x)/2)^1
1*((699*A*a^3)/32 + (2749*B*a^3)/64))/(d*(10*tan(c/2 + (d*x)/2)^2 + 45*tan(c/2 + (d*x)/2)^4 + 120*tan(c/2 + (d
*x)/2)^6 + 210*tan(c/2 + (d*x)/2)^8 + 252*tan(c/2 + (d*x)/2)^10 + 210*tan(c/2 + (d*x)/2)^12 + 120*tan(c/2 + (d
*x)/2)^14 + 45*tan(c/2 + (d*x)/2)^16 + 10*tan(c/2 + (d*x)/2)^18 + tan(c/2 + (d*x)/2)^20 + 1))