\(\int \cos ^7(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx\) [661]

Optimal result

Integrand size = 25, antiderivative size = 81 \[ \int \cos ^7(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^8(c+d x)}{8 d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {3 a \sin ^5(c+d x)}{5 d}+\frac {3 a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^9(c+d x)}{9 d} \]

[Out]

-1/8*a*cos(d*x+c)^8/d+1/3*a*sin(d*x+c)^3/d-3/5*a*sin(d*x+c)^5/d+3/7*a*sin(d*x+c)^7/d-1/9*a*sin(d*x+c)^9/d

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2913, 2645, 30, 2644, 276} \[ \int \cos ^7(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \sin ^9(c+d x)}{9 d}+\frac {3 a \sin ^7(c+d x)}{7 d}-\frac {3 a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \cos ^8(c+d x)}{8 d} \]

[In]

Int[Cos[c + d*x]^7*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]

[Out]

-1/8*(a*Cos[c + d*x]^8)/d + (a*Sin[c + d*x]^3)/(3*d) - (3*a*Sin[c + d*x]^5)/(5*d) + (3*a*Sin[c + d*x]^7)/(7*d)
- (a*Sin[c + d*x]^9)/(9*d)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2644

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

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 2913

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]),
x_Symbol] :> Dist[a, Int[Cos[e + f*x]^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[Cos[e + f*x]^p*(d*Sin[e +
f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && IntegerQ[n] && ((LtQ[p, 0]
&& NeQ[a^2 - b^2, 0]) || LtQ[0, n, p - 1] || LtQ[p + 1, -n, 2*p + 1])

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.74 \[ \int \cos ^7(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (-1260 \cos ^8(c+d x)+(1606+1389 \cos (2 (c+d x))+330 \cos (4 (c+d x))+35 \cos (6 (c+d x))) \sin ^3(c+d x)\right )}{10080 d} \]

[In]

Integrate[Cos[c + d*x]^7*Sin[c + d*x]*(a + a*Sin[c + d*x]),x]

[Out]

(a*(-1260*Cos[c + d*x]^8 + (1606 + 1389*Cos[2*(c + d*x)] + 330*Cos[4*(c + d*x)] + 35*Cos[6*(c + d*x)])*Sin[c +
d*x]^3))/(10080*d)

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(-\frac {a \left (\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(88\)
default	\(-\frac {a \left (\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {3 \left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(88\)
parallelrisch	\(\frac {a \left (-17640 \cos \left (2 d x +2 c \right )-140 \sin \left (9 d x +9 c \right )-315 \cos \left (8 d x +8 c \right )-900 \sin \left (7 d x +7 c \right )-2016 \sin \left (5 d x +5 c \right )-2520 \cos \left (6 d x +6 c \right )+17640 \sin \left (d x +c \right )-8820 \cos \left (4 d x +4 c \right )+29295\right )}{322560 d}\)	\(94\)
risch	\(\frac {7 a \sin \left (d x +c \right )}{128 d}-\frac {a \sin \left (9 d x +9 c \right )}{2304 d}-\frac {a \cos \left (8 d x +8 c \right )}{1024 d}-\frac {5 a \sin \left (7 d x +7 c \right )}{1792 d}-\frac {a \cos \left (6 d x +6 c \right )}{128 d}-\frac {a \sin \left (5 d x +5 c \right )}{160 d}-\frac {7 a \cos \left (4 d x +4 c \right )}{256 d}-\frac {7 a \cos \left (2 d x +2 c \right )}{128 d}\)	\(119\)
norman	\(\frac {\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {16 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {632 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}-\frac {2848 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 d}+\frac {632 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}-\frac {16 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 a \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {14 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\)	\(273\)

[In]

int(cos(d*x+c)^7*sin(d*x+c)*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-a/d*(1/9*sin(d*x+c)^9+1/8*sin(d*x+c)^8-3/7*sin(d*x+c)^7-1/2*sin(d*x+c)^6+3/5*sin(d*x+c)^5+3/4*sin(d*x+c)^4-1/
3*sin(d*x+c)^3-1/2*sin(d*x+c)^2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90 \[ \int \cos ^7(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {315 \, a \cos \left (d x + c\right )^{8} + 8 \, {\left (35 \, a \cos \left (d x + c\right )^{8} - 5 \, a \cos \left (d x + c\right )^{6} - 6 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{2} - 16 \, a\right )} \sin \left (d x + c\right )}{2520 \, d} \]

[In]

integrate(cos(d*x+c)^7*sin(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/2520*(315*a*cos(d*x + c)^8 + 8*(35*a*cos(d*x + c)^8 - 5*a*cos(d*x + c)^6 - 6*a*cos(d*x + c)^4 - 8*a*cos(d*x
+ c)^2 - 16*a)*sin(d*x + c))/d

Sympy [A] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.41 \[ \int \cos ^7(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {16 a \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {8 a \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {2 a \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {a \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {a \cos ^{8}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin {\left (c \right )} \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**7*sin(d*x+c)*(a+a*sin(d*x+c)),x)

[Out]

Piecewise((16*a*sin(c + d*x)**9/(315*d) + 8*a*sin(c + d*x)**7*cos(c + d*x)**2/(35*d) + 2*a*sin(c + d*x)**5*cos
(c + d*x)**4/(5*d) + a*sin(c + d*x)**3*cos(c + d*x)**6/(3*d) - a*cos(c + d*x)**8/(8*d), Ne(d, 0)), (x*(a*sin(c
) + a)*sin(c)*cos(c)**7, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.16 \[ \int \cos ^7(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {280 \, a \sin \left (d x + c\right )^{9} + 315 \, a \sin \left (d x + c\right )^{8} - 1080 \, a \sin \left (d x + c\right )^{7} - 1260 \, a \sin \left (d x + c\right )^{6} + 1512 \, a \sin \left (d x + c\right )^{5} + 1890 \, a \sin \left (d x + c\right )^{4} - 840 \, a \sin \left (d x + c\right )^{3} - 1260 \, a \sin \left (d x + c\right )^{2}}{2520 \, d} \]

[In]

integrate(cos(d*x+c)^7*sin(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/2520*(280*a*sin(d*x + c)^9 + 315*a*sin(d*x + c)^8 - 1080*a*sin(d*x + c)^7 - 1260*a*sin(d*x + c)^6 + 1512*a*
sin(d*x + c)^5 + 1890*a*sin(d*x + c)^4 - 840*a*sin(d*x + c)^3 - 1260*a*sin(d*x + c)^2)/d

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.46 \[ \int \cos ^7(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \cos \left (6 \, d x + 6 \, c\right )}{128 \, d} - \frac {7 \, a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {7 \, a \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac {a \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {5 \, a \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \sin \left (5 \, d x + 5 \, c\right )}{160 \, d} + \frac {7 \, a \sin \left (d x + c\right )}{128 \, d} \]

[In]

integrate(cos(d*x+c)^7*sin(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/1024*a*cos(8*d*x + 8*c)/d - 1/128*a*cos(6*d*x + 6*c)/d - 7/256*a*cos(4*d*x + 4*c)/d - 7/128*a*cos(2*d*x + 2
*c)/d - 1/2304*a*sin(9*d*x + 9*c)/d - 5/1792*a*sin(7*d*x + 7*c)/d - 1/160*a*sin(5*d*x + 5*c)/d + 7/128*a*sin(d
*x + c)/d

Mupad [B] (verification not implemented)

Time = 10.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.15 \[ \int \cos ^7(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=\frac {-\frac {a\,{\sin \left (c+d\,x\right )}^9}{9}-\frac {a\,{\sin \left (c+d\,x\right )}^8}{8}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{2}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^4}{4}+\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {a\,{\sin \left (c+d\,x\right )}^2}{2}}{d} \]

[In]

int(cos(c + d*x)^7*sin(c + d*x)*(a + a*sin(c + d*x)),x)

[Out]

((a*sin(c + d*x)^2)/2 + (a*sin(c + d*x)^3)/3 - (3*a*sin(c + d*x)^4)/4 - (3*a*sin(c + d*x)^5)/5 + (a*sin(c + d*
x)^6)/2 + (3*a*sin(c + d*x)^7)/7 - (a*sin(c + d*x)^8)/8 - (a*sin(c + d*x)^9)/9)/d