\(\int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx\) [608]

Optimal result

Integrand size = 27, antiderivative size = 181 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {33 a^3 x}{256}-\frac {33 a^3 \cos ^7(c+d x)}{560 d}+\frac {33 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {11 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {11 a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d} \]

[Out]

33/256*a^3*x-33/560*a^3*cos(d*x+c)^7/d+33/256*a^3*cos(d*x+c)*sin(d*x+c)/d+11/128*a^3*cos(d*x+c)^3*sin(d*x+c)/d
+11/160*a^3*cos(d*x+c)^5*sin(d*x+c)/d-1/30*a*cos(d*x+c)^7*(a+a*sin(d*x+c))^2/d-1/10*cos(d*x+c)^7*(a+a*sin(d*x+
c))^3/d-11/240*cos(d*x+c)^7*(a^3+a^3*sin(d*x+c))/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2939, 2757, 2748, 2715, 8} \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {33 a^3 \cos ^7(c+d x)}{560 d}-\frac {11 \cos ^7(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{240 d}+\frac {11 a^3 \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac {11 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {33 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {33 a^3 x}{256}-\frac {\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}-\frac {a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{30 d} \]

[In]

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

[Out]

(33*a^3*x)/256 - (33*a^3*Cos[c + d*x]^7)/(560*d) + (33*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (11*a^3*Cos[c
+ d*x]^3*Sin[c + d*x])/(128*d) + (11*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) - (a*Cos[c + d*x]^7*(a + a*Sin[c
+ d*x])^2)/(30*d) - (Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3)/(10*d) - (11*Cos[c + d*x]^7*(a^3 + a^3*Sin[c + d*
x]))/(240*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 {\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}+\frac {3}{10} \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 \, dx \\ & = -\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}+\frac {1}{30} (11 a) \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d}+\frac {1}{80} \left (33 a^2\right ) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {33 a^3 \cos ^7(c+d x)}{560 d}-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d}+\frac {1}{80} \left (33 a^3\right ) \int \cos ^6(c+d x) \, dx \\ & = -\frac {33 a^3 \cos ^7(c+d x)}{560 d}+\frac {11 a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d}+\frac {1}{32} \left (11 a^3\right ) \int \cos ^4(c+d x) \, dx \\ & = -\frac {33 a^3 \cos ^7(c+d x)}{560 d}+\frac {11 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {11 a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d}+\frac {1}{128} \left (33 a^3\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {33 a^3 \cos ^7(c+d x)}{560 d}+\frac {33 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {11 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {11 a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d}+\frac {1}{256} \left (33 a^3\right ) \int 1 \, dx \\ & = \frac {33 a^3 x}{256}-\frac {33 a^3 \cos ^7(c+d x)}{560 d}+\frac {33 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {11 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {11 a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac {\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac {11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.64 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (31500 c+27720 d x-31920 \cos (c+d x)-16800 \cos (3 (c+d x))-3360 \cos (5 (c+d x))+600 \cos (7 (c+d x))+280 \cos (9 (c+d x))+10500 \sin (2 (c+d x))-5880 \sin (4 (c+d x))-3570 \sin (6 (c+d x))-525 \sin (8 (c+d x))+42 \sin (10 (c+d x)))}{215040 d} \]

[In]

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

[Out]

(a^3*(31500*c + 27720*d*x - 31920*Cos[c + d*x] - 16800*Cos[3*(c + d*x)] - 3360*Cos[5*(c + d*x)] + 600*Cos[7*(c
+ d*x)] + 280*Cos[9*(c + d*x)] + 10500*Sin[2*(c + d*x)] - 5880*Sin[4*(c + d*x)] - 3570*Sin[6*(c + d*x)] - 525
*Sin[8*(c + d*x)] + 42*Sin[10*(c + d*x)]))/(215040*d)

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.67

method	result	size
parallelrisch	\(-\frac {a^{3} \left (-27720 d x +31920 \cos \left (d x +c \right )-10500 \sin \left (2 d x +2 c \right )+5880 \sin \left (4 d x +4 c \right )+3570 \sin \left (6 d x +6 c \right )-600 \cos \left (7 d x +7 c \right )+525 \sin \left (8 d x +8 c \right )-280 \cos \left (9 d x +9 c \right )+3360 \cos \left (5 d x +5 c \right )+16800 \cos \left (3 d x +3 c \right )-42 \sin \left (10 d x +10 c \right )+51200\right )}{215040 d}\)	\(122\)
risch	\(\frac {33 a^{3} x}{256}-\frac {19 a^{3} \cos \left (d x +c \right )}{128 d}+\frac {a^{3} \sin \left (10 d x +10 c \right )}{5120 d}+\frac {a^{3} \cos \left (9 d x +9 c \right )}{768 d}-\frac {5 a^{3} \sin \left (8 d x +8 c \right )}{2048 d}+\frac {5 a^{3} \cos \left (7 d x +7 c \right )}{1792 d}-\frac {17 a^{3} \sin \left (6 d x +6 c \right )}{1024 d}-\frac {a^{3} \cos \left (5 d x +5 c \right )}{64 d}-\frac {7 a^{3} \sin \left (4 d x +4 c \right )}{256 d}-\frac {5 a^{3} \cos \left (3 d x +3 c \right )}{64 d}+\frac {25 a^{3} \sin \left (2 d x +2 c \right )}{512 d}\)	\(175\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+3 a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d}\)	\(198\)
default	\(\frac {a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+3 a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d}\)	\(198\)

[In]

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

[Out]

-1/215040*a^3*(-27720*d*x+31920*cos(d*x+c)-10500*sin(2*d*x+2*c)+5880*sin(4*d*x+4*c)+3570*sin(6*d*x+6*c)-600*co
s(7*d*x+7*c)+525*sin(8*d*x+8*c)-280*cos(9*d*x+9*c)+3360*cos(5*d*x+5*c)+16800*cos(3*d*x+3*c)-42*sin(10*d*x+10*c
)+51200)/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.61 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {8960 \, a^{3} \cos \left (d x + c\right )^{9} - 15360 \, a^{3} \cos \left (d x + c\right )^{7} + 3465 \, a^{3} d x + 21 \, {\left (128 \, a^{3} \cos \left (d x + c\right )^{9} - 656 \, a^{3} \cos \left (d x + c\right )^{7} + 88 \, a^{3} \cos \left (d x + c\right )^{5} + 110 \, a^{3} \cos \left (d x + c\right )^{3} + 165 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{26880 \, d} \]

[In]

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

[Out]

1/26880*(8960*a^3*cos(d*x + c)^9 - 15360*a^3*cos(d*x + c)^7 + 3465*a^3*d*x + 21*(128*a^3*cos(d*x + c)^9 - 656*
a^3*cos(d*x + c)^7 + 88*a^3*cos(d*x + c)^5 + 110*a^3*cos(d*x + c)^3 + 165*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. 542 vs. \(2 (170) = 340\).

Time = 1.30 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.99 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {3 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {15 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {15 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {45 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {15 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {55 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {7 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac {73 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {3 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {15 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {2 a^{3} \cos ^{9}{\left (c + d x \right )}}{21 d} - \frac {a^{3} \cos ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin {\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.78 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {30720 \, a^{3} \cos \left (d x + c\right )^{7} - 10240 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 21 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 210 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{215040 \, d} \]

[In]

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

[Out]

-1/215040*(30720*a^3*cos(d*x + c)^7 - 10240*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^3 - 21*(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 - 210*(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^3)/d

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.96 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {33}{256} \, a^{3} x + \frac {a^{3} \cos \left (9 \, d x + 9 \, c\right )}{768 \, d} + \frac {5 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a^{3} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {5 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {19 \, a^{3} \cos \left (d x + c\right )}{128 \, d} + \frac {a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac {5 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {17 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {7 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {25 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

[In]

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

[Out]

33/256*a^3*x + 1/768*a^3*cos(9*d*x + 9*c)/d + 5/1792*a^3*cos(7*d*x + 7*c)/d - 1/64*a^3*cos(5*d*x + 5*c)/d - 5/
64*a^3*cos(3*d*x + 3*c)/d - 19/128*a^3*cos(d*x + c)/d + 1/5120*a^3*sin(10*d*x + 10*c)/d - 5/2048*a^3*sin(8*d*x
+ 8*c)/d - 17/1024*a^3*sin(6*d*x + 6*c)/d - 7/256*a^3*sin(4*d*x + 4*c)/d + 25/512*a^3*sin(2*d*x + 2*c)/d

Mupad [B] (verification not implemented)

Time = 13.41 (sec) , antiderivative size = 572, normalized size of antiderivative = 3.16 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {33\,a^3\,x}{256}-\frac {\frac {333\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}-\frac {577\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {705\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}-\frac {2749\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {2749\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}-\frac {333\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {577\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}+\frac {705\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}-\frac {33\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+a^3\,\left (\frac {33\,c}{256}+\frac {33\,d\,x}{256}\right )-a^3\,\left (\frac {33\,c}{256}+\frac {33\,d\,x}{256}-\frac {10}{21}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (10\,a^3\,\left (\frac {33\,c}{256}+\frac {33\,d\,x}{256}\right )-a^3\,\left (\frac {165\,c}{128}+\frac {165\,d\,x}{128}-2\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (10\,a^3\,\left (\frac {33\,c}{256}+\frac {33\,d\,x}{256}\right )-a^3\,\left (\frac {165\,c}{128}+\frac {165\,d\,x}{128}-\frac {58}{21}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (120\,a^3\,\left (\frac {33\,c}{256}+\frac {33\,d\,x}{256}\right )-a^3\,\left (\frac {495\,c}{32}+\frac {495\,d\,x}{32}-8\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (120\,a^3\,\left (\frac {33\,c}{256}+\frac {33\,d\,x}{256}\right )-a^3\,\left (\frac {495\,c}{32}+\frac {495\,d\,x}{32}-\frac {344}{7}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (45\,a^3\,\left (\frac {33\,c}{256}+\frac {33\,d\,x}{256}\right )-a^3\,\left (\frac {1485\,c}{256}+\frac {1485\,d\,x}{256}-18\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (45\,a^3\,\left (\frac {33\,c}{256}+\frac {33\,d\,x}{256}\right )-a^3\,\left (\frac {1485\,c}{256}+\frac {1485\,d\,x}{256}-\frac {24}{7}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (252\,a^3\,\left (\frac {33\,c}{256}+\frac {33\,d\,x}{256}\right )-a^3\,\left (\frac {2079\,c}{64}+\frac {2079\,d\,x}{64}-60\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (210\,a^3\,\left (\frac {33\,c}{256}+\frac {33\,d\,x}{256}\right )-a^3\,\left (\frac {3465\,c}{128}+\frac {3465\,d\,x}{128}-28\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (210\,a^3\,\left (\frac {33\,c}{256}+\frac {33\,d\,x}{256}\right )-a^3\,\left (\frac {3465\,c}{128}+\frac {3465\,d\,x}{128}-72\right )\right )+\frac {33\,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 )}^{10}} \]

[In]

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

[Out]

(33*a^3*x)/256 - ((333*a^3*tan(c/2 + (d*x)/2)^7)/32 - (577*a^3*tan(c/2 + (d*x)/2)^5)/160 - (705*a^3*tan(c/2 +
(d*x)/2)^3)/128 - (2749*a^3*tan(c/2 + (d*x)/2)^9)/64 + (2749*a^3*tan(c/2 + (d*x)/2)^11)/64 - (333*a^3*tan(c/2
+ (d*x)/2)^13)/32 + (577*a^3*tan(c/2 + (d*x)/2)^15)/160 + (705*a^3*tan(c/2 + (d*x)/2)^17)/128 - (33*a^3*tan(c/
2 + (d*x)/2)^19)/128 + a^3*((33*c)/256 + (33*d*x)/256) - a^3*((33*c)/256 + (33*d*x)/256 - 10/21) + tan(c/2 + (
d*x)/2)^18*(10*a^3*((33*c)/256 + (33*d*x)/256) - a^3*((165*c)/128 + (165*d*x)/128 - 2)) + tan(c/2 + (d*x)/2)^2
*(10*a^3*((33*c)/256 + (33*d*x)/256) - a^3*((165*c)/128 + (165*d*x)/128 - 58/21)) + tan(c/2 + (d*x)/2)^14*(120
*a^3*((33*c)/256 + (33*d*x)/256) - a^3*((495*c)/32 + (495*d*x)/32 - 8)) + tan(c/2 + (d*x)/2)^6*(120*a^3*((33*c
)/256 + (33*d*x)/256) - a^3*((495*c)/32 + (495*d*x)/32 - 344/7)) + tan(c/2 + (d*x)/2)^16*(45*a^3*((33*c)/256 +
(33*d*x)/256) - a^3*((1485*c)/256 + (1485*d*x)/256 - 18)) + tan(c/2 + (d*x)/2)^4*(45*a^3*((33*c)/256 + (33*d*
x)/256) - a^3*((1485*c)/256 + (1485*d*x)/256 - 24/7)) + tan(c/2 + (d*x)/2)^10*(252*a^3*((33*c)/256 + (33*d*x)/
256) - a^3*((2079*c)/64 + (2079*d*x)/64 - 60)) + tan(c/2 + (d*x)/2)^8*(210*a^3*((33*c)/256 + (33*d*x)/256) - a
^3*((3465*c)/128 + (3465*d*x)/128 - 28)) + tan(c/2 + (d*x)/2)^12*(210*a^3*((33*c)/256 + (33*d*x)/256) - a^3*((
3465*c)/128 + (3465*d*x)/128 - 72)) + (33*a^3*tan(c/2 + (d*x)/2))/128)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^10)