\(\int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx\) [573]

Optimal result

Integrand size = 27, antiderivative size = 149 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 a x}{256}-\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^9(c+d x)}{9 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{256 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d} \]

[Out]

3/256*a*x-1/7*a*cos(d*x+c)^7/d+1/9*a*cos(d*x+c)^9/d+3/256*a*cos(d*x+c)*sin(d*x+c)/d+1/128*a*cos(d*x+c)^3*sin(d
*x+c)/d+1/160*a*cos(d*x+c)^5*sin(d*x+c)/d-3/80*a*cos(d*x+c)^7*sin(d*x+c)/d-1/10*a*cos(d*x+c)^7*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2917, 2645, 14, 2648, 2715, 8} \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac {3 a x}{256} \]

[In]

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

[Out]

(3*a*x)/256 - (a*Cos[c + d*x]^7)/(7*d) + (a*Cos[c + d*x]^9)/(9*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(256*d) +
(a*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) - (3*a*Cos[c + d*x]^7*Sin[c
+ d*x])/(80*d) - (a*Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*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 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 2917

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.68 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (7560 d x-15120 \cos (c+d x)-6720 \cos (3 (c+d x))+1080 \cos (7 (c+d x))+280 \cos (9 (c+d x))+1260 \sin (2 (c+d x))-2520 \sin (4 (c+d x))-630 \sin (6 (c+d x))+315 \sin (8 (c+d x))+126 \sin (10 (c+d x)))}{645120 d} \]

[In]

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

[Out]

(a*(7560*d*x - 15120*Cos[c + d*x] - 6720*Cos[3*(c + d*x)] + 1080*Cos[7*(c + d*x)] + 280*Cos[9*(c + d*x)] + 126
0*Sin[2*(c + d*x)] - 2520*Sin[4*(c + d*x)] - 630*Sin[6*(c + d*x)] + 315*Sin[8*(c + d*x)] + 126*Sin[10*(c + d*x
)]))/(645120*d)

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.72

method	result	size
parallelrisch	\(-\frac {\left (-3 d x +\sin \left (4 d x +4 c \right )+\frac {\sin \left (6 d x +6 c \right )}{4}-\frac {\sin \left (8 d x +8 c \right )}{8}-\frac {\sin \left (10 d x +10 c \right )}{20}+6 \cos \left (d x +c \right )+\frac {8 \cos \left (3 d x +3 c \right )}{3}-\frac {3 \cos \left (7 d x +7 c \right )}{7}-\frac {\cos \left (9 d x +9 c \right )}{9}-\frac {\sin \left (2 d x +2 c \right )}{2}+\frac {512}{63}\right ) a}{256 d}\)	\(107\)
derivativedivides	\(\frac {a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )}{d}\)	\(116\)
default	\(\frac {a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )}{d}\)	\(116\)
risch	\(\frac {3 a x}{256}-\frac {3 a \cos \left (d x +c \right )}{128 d}+\frac {a \sin \left (10 d x +10 c \right )}{5120 d}+\frac {a \cos \left (9 d x +9 c \right )}{2304 d}+\frac {a \sin \left (8 d x +8 c \right )}{2048 d}+\frac {3 a \cos \left (7 d x +7 c \right )}{1792 d}-\frac {a \sin \left (6 d x +6 c \right )}{1024 d}-\frac {a \sin \left (4 d x +4 c \right )}{256 d}-\frac {a \cos \left (3 d x +3 c \right )}{96 d}+\frac {a \sin \left (2 d x +2 c \right )}{512 d}\)	\(138\)

[In]

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

[Out]

-1/256*(-3*d*x+sin(4*d*x+4*c)+1/4*sin(6*d*x+6*c)-1/8*sin(8*d*x+8*c)-1/20*sin(10*d*x+10*c)+6*cos(d*x+c)+8/3*cos
(3*d*x+3*c)-3/7*cos(7*d*x+7*c)-1/9*cos(9*d*x+9*c)-1/2*sin(2*d*x+2*c)+512/63)*a/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.64 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {8960 \, a \cos \left (d x + c\right )^{9} - 11520 \, a \cos \left (d x + c\right )^{7} + 945 \, a d x + 63 \, {\left (128 \, a \cos \left (d x + c\right )^{9} - 176 \, a \cos \left (d x + c\right )^{7} + 8 \, a \cos \left (d x + c\right )^{5} + 10 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]

[In]

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

[Out]

1/80640*(8960*a*cos(d*x + c)^9 - 11520*a*cos(d*x + c)^7 + 945*a*d*x + 63*(128*a*cos(d*x + c)^9 - 176*a*cos(d*x
+ c)^7 + 8*a*cos(d*x + c)^5 + 10*a*cos(d*x + c)^3 + 15*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. 294 vs. \(2 (138) = 276\).

Time = 1.26 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.97 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {3 a x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {15 a x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {3 a x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {3 a \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {7 a \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {3 a \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {2 a \cos ^{9}{\left (c + d x \right )}}{63 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{3}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((3*a*x*sin(c + d*x)**10/256 + 15*a*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 15*a*x*sin(c + d*x)**6*co
s(c + d*x)**4/128 + 15*a*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 15*a*x*sin(c + d*x)**2*cos(c + d*x)**8/256 +
3*a*x*cos(c + d*x)**10/256 + 3*a*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a*sin(c + d*x)**7*cos(c + d*x)**3/(1
28*d) + a*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) - 7*a*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) - a*sin(c + d*x
)**2*cos(c + d*x)**7/(7*d) - 3*a*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 2*a*cos(c + d*x)**9/(63*d), Ne(d, 0)),
(x*(a*sin(c) + a)*sin(c)**3*cos(c)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.51 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {10240 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a + 63 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{645120 \, d} \]

[In]

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

[Out]

1/645120*(10240*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a + 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))*a)/d

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3}{256} \, a x + \frac {a \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {3 \, a \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{96 \, d} - \frac {3 \, a \cos \left (d x + c\right )}{128 \, d} + \frac {a \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {a \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {a \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

[In]

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

[Out]

3/256*a*x + 1/2304*a*cos(9*d*x + 9*c)/d + 3/1792*a*cos(7*d*x + 7*c)/d - 1/96*a*cos(3*d*x + 3*c)/d - 3/128*a*co
s(d*x + c)/d + 1/5120*a*sin(10*d*x + 10*c)/d + 1/2048*a*sin(8*d*x + 8*c)/d - 1/1024*a*sin(6*d*x + 6*c)/d - 1/2
56*a*sin(4*d*x + 4*c)/d + 1/512*a*sin(2*d*x + 2*c)/d

Mupad [B] (verification not implemented)

Time = 14.31 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.78 \[ \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3\,a\,x}{256}+\frac {\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}+\left (\frac {a\,\left (42525\,c+42525\,d\,x-322560\right )}{80640}-\frac {135\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-\frac {867\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}+\left (\frac {a\,\left (113400\,c+113400\,d\,x+215040\right )}{80640}-\frac {45\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {519\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\left (\frac {a\,\left (198450\,c+198450\,d\,x-1075200\right )}{80640}-\frac {315\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {1879\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\left (\frac {a\,\left (238140\,c+238140\,d\,x-645120\right )}{80640}-\frac {189\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {1879\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {519\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\left (\frac {a\,\left (113400\,c+113400\,d\,x-829440\right )}{80640}-\frac {45\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {867\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}+\left (\frac {a\,\left (42525\,c+42525\,d\,x+92160\right )}{80640}-\frac {135\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}+\left (\frac {a\,\left (9450\,c+9450\,d\,x-51200\right )}{80640}-\frac {15\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {a\,\left (945\,c+945\,d\,x-5120\right )}{80640}-\frac {3\,a\,\left (c+d\,x\right )}{256}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]

[In]

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

[Out]

(3*a*x)/256 + ((a*(945*c + 945*d*x - 5120))/80640 - (3*a*tan(c/2 + (d*x)/2))/128 - (3*a*(c + d*x))/256 + tan(c
/2 + (d*x)/2)^2*((a*(9450*c + 9450*d*x - 51200))/80640 - (15*a*(c + d*x))/128) + tan(c/2 + (d*x)/2)^4*((a*(425
25*c + 42525*d*x + 92160))/80640 - (135*a*(c + d*x))/256) + tan(c/2 + (d*x)/2)^16*((a*(42525*c + 42525*d*x - 3
22560))/80640 - (135*a*(c + d*x))/256) + tan(c/2 + (d*x)/2)^14*((a*(113400*c + 113400*d*x + 215040))/80640 - (
45*a*(c + d*x))/32) + tan(c/2 + (d*x)/2)^6*((a*(113400*c + 113400*d*x - 829440))/80640 - (45*a*(c + d*x))/32)
+ tan(c/2 + (d*x)/2)^10*((a*(238140*c + 238140*d*x - 645120))/80640 - (189*a*(c + d*x))/64) + tan(c/2 + (d*x)/
2)^12*((a*(198450*c + 198450*d*x - 1075200))/80640 - (315*a*(c + d*x))/128) - (29*a*tan(c/2 + (d*x)/2)^3)/128
+ (867*a*tan(c/2 + (d*x)/2)^5)/160 - (519*a*tan(c/2 + (d*x)/2)^7)/32 + (1879*a*tan(c/2 + (d*x)/2)^9)/64 - (187
9*a*tan(c/2 + (d*x)/2)^11)/64 + (519*a*tan(c/2 + (d*x)/2)^13)/32 - (867*a*tan(c/2 + (d*x)/2)^15)/160 + (29*a*t
an(c/2 + (d*x)/2)^17)/128 + (3*a*tan(c/2 + (d*x)/2)^19)/128)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^10)