\(\int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx\) [392]

Optimal result

Integrand size = 29, antiderivative size = 203 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {15 a^3 x}{256}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {5 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {5 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d} \]

[Out]

15/256*a^3*x-4/5*a^3*cos(d*x+c)^5/d+9/7*a^3*cos(d*x+c)^7/d-2/3*a^3*cos(d*x+c)^9/d+1/11*a^3*cos(d*x+c)^11/d+15/
256*a^3*cos(d*x+c)*sin(d*x+c)/d+5/128*a^3*cos(d*x+c)^3*sin(d*x+c)/d-5/32*a^3*cos(d*x+c)^5*sin(d*x+c)/d-5/16*a^
3*cos(d*x+c)^5*sin(d*x+c)^3/d-3/10*a^3*cos(d*x+c)^5*sin(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2952, 2648, 2715, 8, 2645, 276} \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \cos ^{11}(c+d x)}{11 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}-\frac {3 a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac {5 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac {5 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac {5 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {15 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {15 a^3 x}{256} \]

[In]

Int[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]

[Out]

(15*a^3*x)/256 - (4*a^3*Cos[c + d*x]^5)/(5*d) + (9*a^3*Cos[c + d*x]^7)/(7*d) - (2*a^3*Cos[c + d*x]^9)/(3*d) +
(a^3*Cos[c + d*x]^11)/(11*d) + (15*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (5*a^3*Cos[c + d*x]^3*Sin[c + d*x]
)/(128*d) - (5*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(32*d) - (5*a^3*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*d) - (3*a^3
*Cos[c + d*x]^5*Sin[c + d*x]^5)/(10*d)

Rule 8

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

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 ^4(c+d x) \sin ^4(c+d x)+3 a^3 \cos ^4(c+d x) \sin ^5(c+d x)+3 a^3 \cos ^4(c+d x) \sin ^6(c+d x)+a^3 \cos ^4(c+d x) \sin ^7(c+d x)\right ) \, dx \\ & = a^3 \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+a^3 \int \cos ^4(c+d x) \sin ^7(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^6(c+d x) \, dx \\ & = -\frac {a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{8} \left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx+\frac {1}{2} \left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int x^4 \left (1-x^2\right )^3 \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {5 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{16} a^3 \int \cos ^4(c+d x) \, dx+\frac {1}{16} \left (9 a^3\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \left (x^4-3 x^6+3 x^8-x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {5 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {5 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{64} \left (3 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{32} \left (3 a^3\right ) \int \cos ^4(c+d x) \, dx \\ & = -\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {5 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {5 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{128} \left (3 a^3\right ) \int 1 \, dx+\frac {1}{128} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {3 a^3 x}{128}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {5 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {5 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{256} \left (9 a^3\right ) \int 1 \, dx \\ & = \frac {15 a^3 x}{256}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {9 a^3 \cos ^7(c+d x)}{7 d}-\frac {2 a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {15 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {5 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {5 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {3 a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 12.20 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.62 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (138600 c+138600 d x-198660 \cos (c+d x)-41580 \cos (3 (c+d x))+27258 \cos (5 (c+d x))+3630 \cos (7 (c+d x))-3850 \cos (9 (c+d x))+210 \cos (11 (c+d x))-13860 \sin (2 (c+d x))-46200 \sin (4 (c+d x))+6930 \sin (6 (c+d x))+5775 \sin (8 (c+d x))-1386 \sin (10 (c+d x)))}{2365440 d} \]

[In]

Integrate[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(138600*c + 138600*d*x - 198660*Cos[c + d*x] - 41580*Cos[3*(c + d*x)] + 27258*Cos[5*(c + d*x)] + 3630*Cos
[7*(c + d*x)] - 3850*Cos[9*(c + d*x)] + 210*Cos[11*(c + d*x)] - 13860*Sin[2*(c + d*x)] - 46200*Sin[4*(c + d*x)
] + 6930*Sin[6*(c + d*x)] + 5775*Sin[8*(c + d*x)] - 1386*Sin[10*(c + d*x)]))/(2365440*d)

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.65

method	result	size
parallelrisch	\(-\frac {3 \left (-10 d x +\sin \left (2 d x +2 c \right )+\frac {10 \sin \left (4 d x +4 c \right )}{3}-\frac {\sin \left (6 d x +6 c \right )}{2}-\frac {5 \sin \left (8 d x +8 c \right )}{12}-\frac {\cos \left (11 d x +11 c \right )}{66}+\frac {\sin \left (10 d x +10 c \right )}{10}+\frac {43 \cos \left (d x +c \right )}{3}+3 \cos \left (3 d x +3 c \right )-\frac {59 \cos \left (5 d x +5 c \right )}{30}-\frac {11 \cos \left (7 d x +7 c \right )}{42}+\frac {5 \cos \left (9 d x +9 c \right )}{18}+\frac {53248}{3465}\right ) a^{3}}{512 d}\)	\(131\)
risch	\(\frac {a^{3} \cos \left (11 d x +11 c \right )}{11264 d}+\frac {15 a^{3} x}{256}-\frac {43 a^{3} \cos \left (d x +c \right )}{512 d}-\frac {3 a^{3} \sin \left (10 d x +10 c \right )}{5120 d}-\frac {5 a^{3} \cos \left (9 d x +9 c \right )}{3072 d}+\frac {5 a^{3} \sin \left (8 d x +8 c \right )}{2048 d}+\frac {11 a^{3} \cos \left (7 d x +7 c \right )}{7168 d}+\frac {3 a^{3} \sin \left (6 d x +6 c \right )}{1024 d}+\frac {59 a^{3} \cos \left (5 d x +5 c \right )}{5120 d}-\frac {5 a^{3} \sin \left (4 d x +4 c \right )}{256 d}-\frac {9 a^{3} \cos \left (3 d x +3 c \right )}{512 d}-\frac {3 a^{3} \sin \left (2 d x +2 c \right )}{512 d}\)	\(192\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{11}-\frac {2 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{33}-\frac {8 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{231}-\frac {16 \left (\cos ^{5}\left (d x +c \right )\right )}{1155}\right )+3 a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\)	\(288\)
default	\(\frac {a^{3} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{11}-\frac {2 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{33}-\frac {8 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{231}-\frac {16 \left (\cos ^{5}\left (d x +c \right )\right )}{1155}\right )+3 a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\)	\(288\)

[In]

int(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-3/512*(-10*d*x+sin(2*d*x+2*c)+10/3*sin(4*d*x+4*c)-1/2*sin(6*d*x+6*c)-5/12*sin(8*d*x+8*c)-1/66*cos(11*d*x+11*c
)+1/10*sin(10*d*x+10*c)+43/3*cos(d*x+c)+3*cos(3*d*x+3*c)-59/30*cos(5*d*x+5*c)-11/42*cos(7*d*x+7*c)+5/18*cos(9*
d*x+9*c)+53248/3465)*a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.67 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {26880 \, a^{3} \cos \left (d x + c\right )^{11} - 197120 \, a^{3} \cos \left (d x + c\right )^{9} + 380160 \, a^{3} \cos \left (d x + c\right )^{7} - 236544 \, a^{3} \cos \left (d x + c\right )^{5} + 17325 \, a^{3} d x - 231 \, {\left (384 \, a^{3} \cos \left (d x + c\right )^{9} - 1168 \, a^{3} \cos \left (d x + c\right )^{7} + 984 \, a^{3} \cos \left (d x + c\right )^{5} - 50 \, a^{3} \cos \left (d x + c\right )^{3} - 75 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{295680 \, d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/295680*(26880*a^3*cos(d*x + c)^11 - 197120*a^3*cos(d*x + c)^9 + 380160*a^3*cos(d*x + c)^7 - 236544*a^3*cos(d
*x + c)^5 + 17325*a^3*d*x - 231*(384*a^3*cos(d*x + c)^9 - 1168*a^3*cos(d*x + c)^7 + 984*a^3*cos(d*x + c)^5 - 5
0*a^3*cos(d*x + c)^3 - 75*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. 648 vs. \(2 (194) = 388\).

Time = 1.97 (sec) , antiderivative size = 648, normalized size of antiderivative = 3.19 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {9 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {45 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {3 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {45 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {3 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {45 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {9 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {45 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {9 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {3 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {9 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {21 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {3 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} - \frac {a^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {11 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {6 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {21 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {11 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {8 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{105 d} - \frac {12 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {9 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {16 a^{3} \cos ^{11}{\left (c + d x \right )}}{1155 d} - \frac {8 a^{3} \cos ^{9}{\left (c + d x \right )}}{105 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{4}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)**4*(a+a*sin(d*x+c))**3,x)

[Out]

Piecewise((9*a**3*x*sin(c + d*x)**10/256 + 45*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 3*a**3*x*sin(c + d*
x)**8/128 + 45*a**3*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 3*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 45*a
**3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 9*a**3*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 45*a**3*x*sin(c + d*
x)**2*cos(c + d*x)**8/256 + 3*a**3*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 9*a**3*x*cos(c + d*x)**10/256 + 3*a*
*3*x*cos(c + d*x)**8/128 + 9*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 21*a**3*sin(c + d*x)**7*cos(c + d*x)*
*3/(128*d) + 3*a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) - a**3*sin(c + d*x)**6*cos(c + d*x)**5/(5*d) - 3*a**3
*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 11*a**3*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) - 6*a**3*sin(c + d*x
)**4*cos(c + d*x)**7/(35*d) - 3*a**3*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 21*a**3*sin(c + d*x)**3*cos(c + d
*x)**7/(128*d) - 11*a**3*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 8*a**3*sin(c + d*x)**2*cos(c + d*x)**9/(105
*d) - 12*a**3*sin(c + d*x)**2*cos(c + d*x)**7/(35*d) - 9*a**3*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 3*a**3*si
n(c + d*x)*cos(c + d*x)**7/(128*d) - 16*a**3*cos(c + d*x)**11/(1155*d) - 8*a**3*cos(c + d*x)**9/(105*d), Ne(d,
0)), (x*(a*sin(c) + a)**3*sin(c)**4*cos(c)**4, True))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.83 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {2048 \, {\left (105 \, \cos \left (d x + c\right )^{11} - 385 \, \cos \left (d x + c\right )^{9} + 495 \, \cos \left (d x + c\right )^{7} - 231 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 22528 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 693 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 2310 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{2365440 \, d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/2365440*(2048*(105*cos(d*x + c)^11 - 385*cos(d*x + c)^9 + 495*cos(d*x + c)^7 - 231*cos(d*x + c)^5)*a^3 - 225
28*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*a^3 - 693*(32*sin(2*d*x + 2*c)^5 - 120*d*x - 12
0*c - 5*sin(8*d*x + 8*c) + 40*sin(4*d*x + 4*c))*a^3 + 2310*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4
*c))*a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.70 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.94 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {15}{256} \, a^{3} x + \frac {a^{3} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} - \frac {5 \, a^{3} \cos \left (9 \, d x + 9 \, c\right )}{3072 \, d} + \frac {11 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {59 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{5120 \, d} - \frac {9 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{512 \, d} - \frac {43 \, a^{3} \cos \left (d x + c\right )}{512 \, d} - \frac {3 \, a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {5 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {3 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {5 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {3 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

15/256*a^3*x + 1/11264*a^3*cos(11*d*x + 11*c)/d - 5/3072*a^3*cos(9*d*x + 9*c)/d + 11/7168*a^3*cos(7*d*x + 7*c)
/d + 59/5120*a^3*cos(5*d*x + 5*c)/d - 9/512*a^3*cos(3*d*x + 3*c)/d - 43/512*a^3*cos(d*x + c)/d - 3/5120*a^3*si
n(10*d*x + 10*c)/d + 5/2048*a^3*sin(8*d*x + 8*c)/d + 3/1024*a^3*sin(6*d*x + 6*c)/d - 5/256*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 = 13.18 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.49 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {15\,a^3\,x}{256}-\frac {\frac {15\,a^3\,\left (c+d\,x\right )}{256}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-\frac {231\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640}-\frac {242\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}+\frac {3987\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {3987\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\frac {242\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\frac {231\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{640}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{4}-\frac {15\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{128}-\frac {a^3\,\left (17325\,c+17325\,d\,x-53248\right )}{295680}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {165\,a^3\,\left (c+d\,x\right )}{256}-\frac {a^3\,\left (190575\,c+190575\,d\,x-585728\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {825\,a^3\,\left (c+d\,x\right )}{256}-\frac {a^3\,\left (952875\,c+952875\,d\,x-2928640\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {2475\,a^3\,\left (c+d\,x\right )}{256}-\frac {a^3\,\left (2858625\,c+2858625\,d\,x+675840\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {2475\,a^3\,\left (c+d\,x\right )}{128}-\frac {a^3\,\left (5717250\,c+5717250\,d\,x-3379200\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {2475\,a^3\,\left (c+d\,x\right )}{256}-\frac {a^3\,\left (2858625\,c+2858625\,d\,x-9461760\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {2475\,a^3\,\left (c+d\,x\right )}{128}-\frac {a^3\,\left (5717250\,c+5717250\,d\,x-14192640\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {3465\,a^3\,\left (c+d\,x\right )}{128}-\frac {a^3\,\left (8004150\,c+8004150\,d\,x+16084992\right )}{295680}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {3465\,a^3\,\left (c+d\,x\right )}{128}-\frac {a^3\,\left (8004150\,c+8004150\,d\,x-40685568\right )}{295680}\right )+\frac {15\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{11}} \]

[In]

int(cos(c + d*x)^4*sin(c + d*x)^4*(a + a*sin(c + d*x))^3,x)

[Out]

(15*a^3*x)/256 - ((15*a^3*(c + d*x))/256 + (5*a^3*tan(c/2 + (d*x)/2)^3)/4 - (231*a^3*tan(c/2 + (d*x)/2)^5)/640
- (242*a^3*tan(c/2 + (d*x)/2)^7)/5 + (3987*a^3*tan(c/2 + (d*x)/2)^9)/64 - (3987*a^3*tan(c/2 + (d*x)/2)^13)/64
+ (242*a^3*tan(c/2 + (d*x)/2)^15)/5 + (231*a^3*tan(c/2 + (d*x)/2)^17)/640 - (5*a^3*tan(c/2 + (d*x)/2)^19)/4 -
(15*a^3*tan(c/2 + (d*x)/2)^21)/128 - (a^3*(17325*c + 17325*d*x - 53248))/295680 + tan(c/2 + (d*x)/2)^2*((165*
a^3*(c + d*x))/256 - (a^3*(190575*c + 190575*d*x - 585728))/295680) + tan(c/2 + (d*x)/2)^4*((825*a^3*(c + d*x)
)/256 - (a^3*(952875*c + 952875*d*x - 2928640))/295680) + tan(c/2 + (d*x)/2)^6*((2475*a^3*(c + d*x))/256 - (a^
3*(2858625*c + 2858625*d*x + 675840))/295680) + tan(c/2 + (d*x)/2)^8*((2475*a^3*(c + d*x))/128 - (a^3*(5717250
*c + 5717250*d*x - 3379200))/295680) + tan(c/2 + (d*x)/2)^16*((2475*a^3*(c + d*x))/256 - (a^3*(2858625*c + 285
8625*d*x - 9461760))/295680) + tan(c/2 + (d*x)/2)^14*((2475*a^3*(c + d*x))/128 - (a^3*(5717250*c + 5717250*d*x
- 14192640))/295680) + tan(c/2 + (d*x)/2)^12*((3465*a^3*(c + d*x))/128 - (a^3*(8004150*c + 8004150*d*x + 1608
4992))/295680) + tan(c/2 + (d*x)/2)^10*((3465*a^3*(c + d*x))/128 - (a^3*(8004150*c + 8004150*d*x - 40685568))/
295680) + (15*a^3*tan(c/2 + (d*x)/2))/128)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^11)