\(\int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx\) [206]

Optimal result

Integrand size = 27, antiderivative size = 73 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^4(c+d x)}{4 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {a^3 \sin ^6(c+d x)}{2 d}+\frac {a^3 \sin ^7(c+d x)}{7 d} \]

[Out]

1/4*a^3*sin(d*x+c)^4/d+3/5*a^3*sin(d*x+c)^5/d+1/2*a^3*sin(d*x+c)^6/d+1/7*a^3*sin(d*x+c)^7/d

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2912, 12, 45} \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sin ^7(c+d x)}{7 d}+\frac {a^3 \sin ^6(c+d x)}{2 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {a^3 \sin ^4(c+d x)}{4 d} \]

[In]

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

[Out]

(a^3*Sin[c + d*x]^4)/(4*d) + (3*a^3*Sin[c + d*x]^5)/(5*d) + (a^3*Sin[c + d*x]^6)/(2*d) + (a^3*Sin[c + d*x]^7)/
(7*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2912

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3 (a+x)^3}{a^3} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int x^3 (a+x)^3 \, dx,x,a \sin (c+d x)\right )}{a^4 d} \\ & = \frac {\text {Subst}\left (\int \left (a^3 x^3+3 a^2 x^4+3 a x^5+x^6\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d} \\ & = \frac {a^3 \sin ^4(c+d x)}{4 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {a^3 \sin ^6(c+d x)}{2 d}+\frac {a^3 \sin ^7(c+d x)}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 (-350+805 \cos (2 (c+d x))-280 \cos (4 (c+d x))+35 \cos (6 (c+d x))-1015 \sin (c+d x)+525 \sin (3 (c+d x))-119 \sin (5 (c+d x))+5 \sin (7 (c+d x)))}{2240 d} \]

[In]

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

[Out]

-1/2240*(a^3*(-350 + 805*Cos[2*(c + d*x)] - 280*Cos[4*(c + d*x)] + 35*Cos[6*(c + d*x)] - 1015*Sin[c + d*x] + 5
25*Sin[3*(c + d*x)] - 119*Sin[5*(c + d*x)] + 5*Sin[7*(c + d*x)]))/d

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {\frac {a^{3} \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {a^{3} \left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}}{d}\)	\(58\)
default	\(\frac {\frac {a^{3} \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {a^{3} \left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}}{d}\)	\(58\)
parallelrisch	\(\frac {a^{3} \left (560-805 \cos \left (2 d x +2 c \right )-5 \sin \left (7 d x +7 c \right )+119 \sin \left (5 d x +5 c \right )-35 \cos \left (6 d x +6 c \right )+1015 \sin \left (d x +c \right )-525 \sin \left (3 d x +3 c \right )+280 \cos \left (4 d x +4 c \right )\right )}{2240 d}\)	\(85\)
risch	\(\frac {29 a^{3} \sin \left (d x +c \right )}{64 d}-\frac {a^{3} \sin \left (7 d x +7 c \right )}{448 d}-\frac {a^{3} \cos \left (6 d x +6 c \right )}{64 d}+\frac {17 a^{3} \sin \left (5 d x +5 c \right )}{320 d}+\frac {a^{3} \cos \left (4 d x +4 c \right )}{8 d}-\frac {15 a^{3} \sin \left (3 d x +3 c \right )}{64 d}-\frac {23 a^{3} \cos \left (2 d x +2 c \right )}{64 d}\)	\(118\)
norman	\(\frac {\frac {4 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {44 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {44 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {96 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {1984 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {96 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\)	\(151\)

[In]

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

[Out]

1/d*(1/7*a^3*sin(d*x+c)^7+1/2*a^3*sin(d*x+c)^6+3/5*a^3*sin(d*x+c)^5+1/4*a^3*sin(d*x+c)^4)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.34 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {70 \, a^{3} \cos \left (d x + c\right )^{6} - 245 \, a^{3} \cos \left (d x + c\right )^{4} + 280 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, {\left (5 \, a^{3} \cos \left (d x + c\right )^{6} - 36 \, a^{3} \cos \left (d x + c\right )^{4} + 57 \, a^{3} \cos \left (d x + c\right )^{2} - 26 \, a^{3}\right )} \sin \left (d x + c\right )}{140 \, d} \]

[In]

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

[Out]

-1/140*(70*a^3*cos(d*x + c)^6 - 245*a^3*cos(d*x + c)^4 + 280*a^3*cos(d*x + c)^2 + 4*(5*a^3*cos(d*x + c)^6 - 36
*a^3*cos(d*x + c)^4 + 57*a^3*cos(d*x + c)^2 - 26*a^3)*sin(d*x + c))/d

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {a^{3} \sin ^{7}{\left (c + d x \right )}}{7 d} + \frac {a^{3} \sin ^{6}{\left (c + d x \right )}}{2 d} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {a^{3} \sin ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{3}{\left (c \right )} \cos {\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**3*sin(c + d*x)**7/(7*d) + a**3*sin(c + d*x)**6/(2*d) + 3*a**3*sin(c + d*x)**5/(5*d) + a**3*sin(c
+ d*x)**4/(4*d), Ne(d, 0)), (x*(a*sin(c) + a)**3*sin(c)**3*cos(c), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {20 \, a^{3} \sin \left (d x + c\right )^{7} + 70 \, a^{3} \sin \left (d x + c\right )^{6} + 84 \, a^{3} \sin \left (d x + c\right )^{5} + 35 \, a^{3} \sin \left (d x + c\right )^{4}}{140 \, d} \]

[In]

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

[Out]

1/140*(20*a^3*sin(d*x + c)^7 + 70*a^3*sin(d*x + c)^6 + 84*a^3*sin(d*x + c)^5 + 35*a^3*sin(d*x + c)^4)/d

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {20 \, a^{3} \sin \left (d x + c\right )^{7} + 70 \, a^{3} \sin \left (d x + c\right )^{6} + 84 \, a^{3} \sin \left (d x + c\right )^{5} + 35 \, a^{3} \sin \left (d x + c\right )^{4}}{140 \, d} \]

[In]

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

[Out]

1/140*(20*a^3*sin(d*x + c)^7 + 70*a^3*sin(d*x + c)^6 + 84*a^3*sin(d*x + c)^5 + 35*a^3*sin(d*x + c)^4)/d

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78 \[ \int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {\frac {a^3\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a^3\,{\sin \left (c+d\,x\right )}^6}{2}+\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^5}{5}+\frac {a^3\,{\sin \left (c+d\,x\right )}^4}{4}}{d} \]

[In]

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

[Out]

((a^3*sin(c + d*x)^4)/4 + (3*a^3*sin(c + d*x)^5)/5 + (a^3*sin(c + d*x)^6)/2 + (a^3*sin(c + d*x)^7)/7)/d