\(\int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx\) [606]

Optimal result

Integrand size = 29, antiderivative size = 209 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {41 a^3 x}{1024}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {7 a^3 \cos ^9(c+d x)}{9 d}-\frac {3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {41 a^3 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {41 a^3 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {41 a^3 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {41 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d} \]

[Out]

41/1024*a^3*x-4/7*a^3*cos(d*x+c)^7/d+7/9*a^3*cos(d*x+c)^9/d-3/11*a^3*cos(d*x+c)^11/d+41/1024*a^3*cos(d*x+c)*si
n(d*x+c)/d+41/1536*a^3*cos(d*x+c)^3*sin(d*x+c)/d+41/1920*a^3*cos(d*x+c)^5*sin(d*x+c)/d-41/320*a^3*cos(d*x+c)^7
*sin(d*x+c)/d-41/120*a^3*cos(d*x+c)^7*sin(d*x+c)^3/d-1/12*a^3*cos(d*x+c)^7*sin(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2645, 14, 2648, 2715, 8, 276} \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {7 a^3 \cos ^9(c+d x)}{9 d}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac {41 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac {41 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac {41 a^3 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac {41 a^3 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac {41 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {41 a^3 x}{1024} \]

[In]

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

[Out]

(41*a^3*x)/1024 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (7*a^3*Cos[c + d*x]^9)/(9*d) - (3*a^3*Cos[c + d*x]^11)/(11*d)
+ (41*a^3*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (41*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + (41*a^3*Cos[c
+ d*x]^5*Sin[c + d*x])/(1920*d) - (41*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (41*a^3*Cos[c + d*x]^7*Sin[c
+ d*x]^3)/(120*d) - (a^3*Cos[c + d*x]^7*Sin[c + d*x]^5)/(12*d)

Rule 8

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 2648

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

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 2952

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \cos ^6(c+d x) \sin ^3(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^4(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^5(c+d x)+a^3 \cos ^6(c+d x) \sin ^6(c+d x)\right ) \, dx \\ & = a^3 \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+a^3 \int \cos ^6(c+d x) \sin ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx \\ & = -\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{12} \left (5 a^3\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+\frac {1}{10} \left (9 a^3\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {9 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{80} \left (9 a^3\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{8} a^3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {7 a^3 \cos ^9(c+d x)}{9 d}-\frac {3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {3 a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {41 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{64} a^3 \int \cos ^6(c+d x) \, dx+\frac {1}{32} \left (3 a^3\right ) \int \cos ^4(c+d x) \, dx \\ & = -\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {7 a^3 \cos ^9(c+d x)}{9 d}-\frac {3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {3 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {41 a^3 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {41 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{384} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{128} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {7 a^3 \cos ^9(c+d x)}{9 d}-\frac {3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {9 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {41 a^3 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {41 a^3 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {41 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{512} \left (5 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{256} \left (9 a^3\right ) \int 1 \, dx \\ & = \frac {9 a^3 x}{256}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {7 a^3 \cos ^9(c+d x)}{9 d}-\frac {3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {41 a^3 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {41 a^3 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {41 a^3 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {41 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {\left (5 a^3\right ) \int 1 \, dx}{1024} \\ & = \frac {41 a^3 x}{1024}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {7 a^3 \cos ^9(c+d x)}{9 d}-\frac {3 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {41 a^3 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {41 a^3 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {41 a^3 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {41 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {41 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.65 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1247400 c+1136520 d x-1496880 \cos (c+d x)-572880 \cos (3 (c+d x))+83160 \cos (5 (c+d x))+106920 \cos (7 (c+d x))+3080 \cos (9 (c+d x))-7560 \cos (11 (c+d x))+166320 \sin (2 (c+d x))-384615 \sin (4 (c+d x))-83160 \sin (6 (c+d x))+51975 \sin (8 (c+d x))+16632 \sin (10 (c+d x))-1155 \sin (12 (c+d x)))}{28385280 d} \]

[In]

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

[Out]

(a^3*(1247400*c + 1136520*d*x - 1496880*Cos[c + d*x] - 572880*Cos[3*(c + d*x)] + 83160*Cos[5*(c + d*x)] + 1069
20*Cos[7*(c + d*x)] + 3080*Cos[9*(c + d*x)] - 7560*Cos[11*(c + d*x)] + 166320*Sin[2*(c + d*x)] - 384615*Sin[4*
(c + d*x)] - 83160*Sin[6*(c + d*x)] + 51975*Sin[8*(c + d*x)] + 16632*Sin[10*(c + d*x)] - 1155*Sin[12*(c + d*x)
]))/(28385280*d)

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.68

method	result	size
parallelrisch	\(\frac {3 \left (\frac {41 d x}{6}+\sin \left (2 d x +2 c \right )-\frac {37 \sin \left (4 d x +4 c \right )}{16}-\frac {\sin \left (6 d x +6 c \right )}{2}+\frac {5 \sin \left (8 d x +8 c \right )}{16}-\frac {\cos \left (11 d x +11 c \right )}{22}+\frac {\sin \left (10 d x +10 c \right )}{10}-\frac {\sin \left (12 d x +12 c \right )}{144}-9 \cos \left (d x +c \right )-\frac {31 \cos \left (3 d x +3 c \right )}{9}+\frac {\cos \left (5 d x +5 c \right )}{2}+\frac {9 \cos \left (7 d x +7 c \right )}{14}+\frac {\cos \left (9 d x +9 c \right )}{54}-\frac {23552}{2079}\right ) a^{3}}{512 d}\)	\(142\)
risch	\(\frac {3 a^{3} \sin \left (10 d x +10 c \right )}{5120 d}-\frac {3 a^{3} \cos \left (11 d x +11 c \right )}{11264 d}+\frac {41 a^{3} x}{1024}-\frac {27 a^{3} \cos \left (d x +c \right )}{512 d}-\frac {a^{3} \sin \left (12 d x +12 c \right )}{24576 d}+\frac {a^{3} \cos \left (9 d x +9 c \right )}{9216 d}+\frac {15 a^{3} \sin \left (8 d x +8 c \right )}{8192 d}+\frac {27 a^{3} \cos \left (7 d x +7 c \right )}{7168 d}-\frac {3 a^{3} \sin \left (6 d x +6 c \right )}{1024 d}+\frac {3 a^{3} \cos \left (5 d x +5 c \right )}{1024 d}-\frac {111 a^{3} \sin \left (4 d x +4 c \right )}{8192 d}-\frac {31 a^{3} \cos \left (3 d x +3 c \right )}{1536 d}+\frac {3 a^{3} \sin \left (2 d x +2 c \right )}{512 d}\)	\(209\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )}{d}\)	\(272\)
default	\(\frac {a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )}{d}\)	\(272\)

[In]

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

[Out]

3/512*(41/6*d*x+sin(2*d*x+2*c)-37/16*sin(4*d*x+4*c)-1/2*sin(6*d*x+6*c)+5/16*sin(8*d*x+8*c)-1/22*cos(11*d*x+11*
c)+1/10*sin(10*d*x+10*c)-1/144*sin(12*d*x+12*c)-9*cos(d*x+c)-31/9*cos(3*d*x+3*c)+1/2*cos(5*d*x+5*c)+9/14*cos(7
*d*x+7*c)+1/54*cos(9*d*x+9*c)-23552/2079)*a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.66 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {967680 \, a^{3} \cos \left (d x + c\right )^{11} - 2759680 \, a^{3} \cos \left (d x + c\right )^{9} + 2027520 \, a^{3} \cos \left (d x + c\right )^{7} - 142065 \, a^{3} d x + 231 \, {\left (1280 \, a^{3} \cos \left (d x + c\right )^{11} - 7808 \, a^{3} \cos \left (d x + c\right )^{9} + 8496 \, a^{3} \cos \left (d x + c\right )^{7} - 328 \, a^{3} \cos \left (d x + c\right )^{5} - 410 \, a^{3} \cos \left (d x + c\right )^{3} - 615 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \]

[In]

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

[Out]

-1/3548160*(967680*a^3*cos(d*x + c)^11 - 2759680*a^3*cos(d*x + c)^9 + 2027520*a^3*cos(d*x + c)^7 - 142065*a^3*
d*x + 231*(1280*a^3*cos(d*x + c)^11 - 7808*a^3*cos(d*x + c)^9 + 8496*a^3*cos(d*x + c)^7 - 328*a^3*cos(d*x + c)
^5 - 410*a^3*cos(d*x + c)^3 - 615*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. 699 vs. \(2 (201) = 402\).

Time = 2.65 (sec) , antiderivative size = 699, normalized size of antiderivative = 3.34 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {5 a^{3} x \sin ^{12}{\left (c + d x \right )}}{1024} + \frac {15 a^{3} x \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{512} + \frac {9 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {75 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{1024} + \frac {45 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {25 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{256} + \frac {45 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {75 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{1024} + \frac {45 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{512} + \frac {45 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {5 a^{3} x \cos ^{12}{\left (c + d x \right )}}{1024} + \frac {9 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {5 a^{3} \sin ^{11}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{1024 d} + \frac {85 a^{3} \sin ^{9}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3072 d} + \frac {9 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {33 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{512 d} + \frac {21 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {33 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{512 d} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {85 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{3072 d} - \frac {21 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {4 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{21 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {5 a^{3} \sin {\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{1024 d} - \frac {9 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {8 a^{3} \cos ^{11}{\left (c + d x \right )}}{231 d} - \frac {2 a^{3} \cos ^{9}{\left (c + d x \right )}}{63 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{3}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((5*a**3*x*sin(c + d*x)**12/1024 + 15*a**3*x*sin(c + d*x)**10*cos(c + d*x)**2/512 + 9*a**3*x*sin(c +
d*x)**10/256 + 75*a**3*x*sin(c + d*x)**8*cos(c + d*x)**4/1024 + 45*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256
+ 25*a**3*x*sin(c + d*x)**6*cos(c + d*x)**6/256 + 45*a**3*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 75*a**3*x*si
n(c + d*x)**4*cos(c + d*x)**8/1024 + 45*a**3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 15*a**3*x*sin(c + d*x)**2
*cos(c + d*x)**10/512 + 45*a**3*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 5*a**3*x*cos(c + d*x)**12/1024 + 9*a**
3*x*cos(c + d*x)**10/256 + 5*a**3*sin(c + d*x)**11*cos(c + d*x)/(1024*d) + 85*a**3*sin(c + d*x)**9*cos(c + d*x
)**3/(3072*d) + 9*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 33*a**3*sin(c + d*x)**7*cos(c + d*x)**5/(512*d)
+ 21*a**3*sin(c + d*x)**7*cos(c + d*x)**3/(128*d) - 33*a**3*sin(c + d*x)**5*cos(c + d*x)**7/(512*d) + 3*a**3*s
in(c + d*x)**5*cos(c + d*x)**5/(10*d) - 3*a**3*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 85*a**3*sin(c + d*x)**3
*cos(c + d*x)**9/(3072*d) - 21*a**3*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) - 4*a**3*sin(c + d*x)**2*cos(c + d
*x)**9/(21*d) - a**3*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 5*a**3*sin(c + d*x)*cos(c + d*x)**11/(1024*d) - 9
*a**3*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 8*a**3*cos(c + d*x)**11/(231*d) - 2*a**3*cos(c + d*x)**9/(63*d),
Ne(d, 0)), (x*(a*sin(c) + a)**3*sin(c)**3*cos(c)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.78 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {122880 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 450560 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 8316 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 1155 \, {\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{28385280 \, d} \]

[In]

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

[Out]

-1/28385280*(122880*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^3 - 450560*(7*cos(d*x + c)
^9 - 9*cos(d*x + c)^7)*a^3 - 8316*(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))*a^3 - 1155*(4*sin(4*d*x + 4*c)^3 + 120*d*x + 120*c + 9*sin(8*d*x + 8*c) - 48*sin(4*d*x + 4*c))*a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {41}{1024} \, a^{3} x - \frac {3 \, a^{3} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {a^{3} \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac {27 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {3 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} - \frac {31 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac {27 \, a^{3} \cos \left (d x + c\right )}{512 \, d} - \frac {a^{3} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} + \frac {3 \, a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {15 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac {3 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {111 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac {3 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

[In]

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

[Out]

41/1024*a^3*x - 3/11264*a^3*cos(11*d*x + 11*c)/d + 1/9216*a^3*cos(9*d*x + 9*c)/d + 27/7168*a^3*cos(7*d*x + 7*c
)/d + 3/1024*a^3*cos(5*d*x + 5*c)/d - 31/1536*a^3*cos(3*d*x + 3*c)/d - 27/512*a^3*cos(d*x + c)/d - 1/24576*a^3
*sin(12*d*x + 12*c)/d + 3/5120*a^3*sin(10*d*x + 10*c)/d + 15/8192*a^3*sin(8*d*x + 8*c)/d - 3/1024*a^3*sin(6*d*
x + 6*c)/d - 111/8192*a^3*sin(4*d*x + 4*c)/d + 3/512*a^3*sin(2*d*x + 2*c)/d

Mupad [B] (verification not implemented)

Time = 13.30 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.27 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

(41*a^3*x)/1024 - ((1435*a^3*tan(c/2 + (d*x)/2)^3)/1536 - (36401*a^3*tan(c/2 + (d*x)/2)^5)/2560 + (1263*a^3*ta
n(c/2 + (d*x)/2)^7)/2560 + (184331*a^3*tan(c/2 + (d*x)/2)^9)/3840 - (35387*a^3*tan(c/2 + (d*x)/2)^11)/256 + (3
5387*a^3*tan(c/2 + (d*x)/2)^13)/256 - (184331*a^3*tan(c/2 + (d*x)/2)^15)/3840 - (1263*a^3*tan(c/2 + (d*x)/2)^1
7)/2560 + (36401*a^3*tan(c/2 + (d*x)/2)^19)/2560 - (1435*a^3*tan(c/2 + (d*x)/2)^21)/1536 - (41*a^3*tan(c/2 + (
d*x)/2)^23)/512 + a^3*((41*c)/1024 + (41*d*x)/1024) - a^3*((41*c)/1024 + (41*d*x)/1024 - 92/693) - tan(c/2 + (
d*x)/2)^22*(a^3*((123*c)/256 + (123*d*x)/256) - 12*a^3*((41*c)/1024 + (41*d*x)/1024)) - tan(c/2 + (d*x)/2)^2*(
a^3*((123*c)/256 + (123*d*x)/256 - 368/231) - 12*a^3*((41*c)/1024 + (41*d*x)/1024)) + tan(c/2 + (d*x)/2)^20*(6
6*a^3*((41*c)/1024 + (41*d*x)/1024) - a^3*((1353*c)/512 + (1353*d*x)/512 - 4)) + tan(c/2 + (d*x)/2)^4*(66*a^3*
((41*c)/1024 + (41*d*x)/1024) - a^3*((1353*c)/512 + (1353*d*x)/512 - 100/21)) + tan(c/2 + (d*x)/2)^18*(220*a^3
*((41*c)/1024 + (41*d*x)/1024) - a^3*((2255*c)/256 + (2255*d*x)/256 - 112/3)) + tan(c/2 + (d*x)/2)^6*(220*a^3*
((41*c)/1024 + (41*d*x)/1024) - a^3*((2255*c)/256 + (2255*d*x)/256 + 512/63)) + tan(c/2 + (d*x)/2)^14*(792*a^3
*((41*c)/1024 + (41*d*x)/1024) - a^3*((4059*c)/128 + (4059*d*x)/128 - 128)) + tan(c/2 + (d*x)/2)^10*(792*a^3*(
(41*c)/1024 + (41*d*x)/1024) - a^3*((4059*c)/128 + (4059*d*x)/128 + 160/7)) + tan(c/2 + (d*x)/2)^12*(924*a^3*(
(41*c)/1024 + (41*d*x)/1024) - a^3*((9471*c)/256 + (9471*d*x)/256 - 184/3)) + tan(c/2 + (d*x)/2)^16*(495*a^3*(
(41*c)/1024 + (41*d*x)/1024) - a^3*((20295*c)/1024 + (20295*d*x)/1024 + 36)) + tan(c/2 + (d*x)/2)^8*(495*a^3*(
(41*c)/1024 + (41*d*x)/1024) - a^3*((20295*c)/1024 + (20295*d*x)/1024 - 712/7)) + (41*a^3*tan(c/2 + (d*x)/2))/
512)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^12)