\(\int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx\) [575]

Optimal result

Integrand size = 25, antiderivative size = 109 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a x}{128}-\frac {a \cos ^7(c+d x)}{7 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d} \]

[Out]

5/128*a*x-1/7*a*cos(d*x+c)^7/d+5/128*a*cos(d*x+c)*sin(d*x+c)/d+5/192*a*cos(d*x+c)^3*sin(d*x+c)/d+1/48*a*cos(d*
x+c)^5*sin(d*x+c)/d-1/8*a*cos(d*x+c)^7*sin(d*x+c)/d

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2917, 2645, 30, 2648, 2715, 8} \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5 a x}{128} \]

[In]

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

[Out]

(5*a*x)/128 - (a*Cos[c + d*x]^7)/(7*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*a*Cos[c + d*x]^3*Sin[c +
d*x])/(192*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)

Rule 8

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

Rule 30

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

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 (c+d x) \, dx+a \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx \\ & = -\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{8} a \int \cos ^6(c+d x) \, dx-\frac {a \text {Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{48} (5 a) \int \cos ^4(c+d x) \, dx \\ & = -\frac {a \cos ^7(c+d x)}{7 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{64} (5 a) \int \cos ^2(c+d x) \, dx \\ & = -\frac {a \cos ^7(c+d x)}{7 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{128} (5 a) \int 1 \, dx \\ & = \frac {5 a x}{128}-\frac {a \cos ^7(c+d x)}{7 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a (-840 d x+1680 \cos (c+d x)+1008 \cos (3 (c+d x))+336 \cos (5 (c+d x))+48 \cos (7 (c+d x))-336 \sin (2 (c+d x))+168 \sin (4 (c+d x))+112 \sin (6 (c+d x))+21 \sin (8 (c+d x)))}{21504 d} \]

[In]

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

[Out]

-1/21504*(a*(-840*d*x + 1680*Cos[c + d*x] + 1008*Cos[3*(c + d*x)] + 336*Cos[5*(c + d*x)] + 48*Cos[7*(c + d*x)]
- 336*Sin[2*(c + d*x)] + 168*Sin[4*(c + d*x)] + 112*Sin[6*(c + d*x)] + 21*Sin[8*(c + d*x)]))/d

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d}\)	\(78\)
default	\(\frac {a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d}\)	\(78\)
parallelrisch	\(-\frac {\left (-5 d x +\sin \left (4 d x +4 c \right )+\frac {2 \sin \left (6 d x +6 c \right )}{3}+\frac {\sin \left (8 d x +8 c \right )}{8}+10 \cos \left (d x +c \right )+6 \cos \left (3 d x +3 c \right )+2 \cos \left (5 d x +5 c \right )+\frac {2 \cos \left (7 d x +7 c \right )}{7}-2 \sin \left (2 d x +2 c \right )+\frac {128}{7}\right ) a}{128 d}\)	\(96\)
risch	\(\frac {5 a x}{128}-\frac {5 a \cos \left (d x +c \right )}{64 d}-\frac {a \sin \left (8 d x +8 c \right )}{1024 d}-\frac {a \cos \left (7 d x +7 c \right )}{448 d}-\frac {a \sin \left (6 d x +6 c \right )}{192 d}-\frac {a \cos \left (5 d x +5 c \right )}{64 d}-\frac {a \sin \left (4 d x +4 c \right )}{128 d}-\frac {3 a \cos \left (3 d x +3 c \right )}{64 d}+\frac {a \sin \left (2 d x +2 c \right )}{64 d}\)	\(123\)
norman	\(\frac {\frac {895 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {5 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {5 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {397 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {5 a \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {10 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {5 a x}{128}-\frac {2 a}{7 d}-\frac {2 a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {6 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {895 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {397 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {6 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d}-\frac {2 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {35 a x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {1765 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {35 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {175 a x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {35 a x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {5 a x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {5 a x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {1765 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {10 a \left (\tan ^{8}\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 )^{8}}\)	\(401\)

[In]

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

[Out]

1/d*(a*(-1/8*cos(d*x+c)^7*sin(d*x+c)+1/48*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/128*d*x
+5/128*c)-1/7*a*cos(d*x+c)^7)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.67 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {384 \, a \cos \left (d x + c\right )^{7} - 105 \, a d x + 7 \, {\left (48 \, a \cos \left (d x + c\right )^{7} - 8 \, a \cos \left (d x + c\right )^{5} - 10 \, a \cos \left (d x + c\right )^{3} - 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, d} \]

[In]

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

[Out]

-1/2688*(384*a*cos(d*x + c)^7 - 105*a*d*x + 7*(48*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. 223 vs. \(2 (102) = 204\).

Time = 0.63 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.05 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {5 a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {5 a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {5 a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {5 a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {5 a \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {73 a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {5 a \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {a \cos ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin {\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((5*a*x*sin(c + d*x)**8/128 + 5*a*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 15*a*x*sin(c + d*x)**4*cos(c
+ d*x)**4/64 + 5*a*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 5*a*x*cos(c + d*x)**8/128 + 5*a*sin(c + d*x)**7*cos
(c + d*x)/(128*d) + 55*a*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 73*a*sin(c + d*x)**3*cos(c + d*x)**5/(384*d
) - 5*a*sin(c + d*x)*cos(c + d*x)**7/(128*d) - a*cos(c + d*x)**7/(7*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)*co
s(c)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.58 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3072 \, a \cos \left (d x + c\right )^{7} - 7 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{21504 \, d} \]

[In]

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

[Out]

-1/21504*(3072*a*cos(d*x + c)^7 - 7*(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)/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.12 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=\frac {5}{128} \, a x - \frac {a \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {a \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {3 \, a \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {5 \, a \cos \left (d x + c\right )}{64 \, d} - \frac {a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]

[In]

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

[Out]

5/128*a*x - 1/448*a*cos(7*d*x + 7*c)/d - 1/64*a*cos(5*d*x + 5*c)/d - 3/64*a*cos(3*d*x + 3*c)/d - 5/64*a*cos(d*
x + c)/d - 1/1024*a*sin(8*d*x + 8*c)/d - 1/192*a*sin(6*d*x + 6*c)/d - 1/128*a*sin(4*d*x + 4*c)/d + 1/64*a*sin(
2*d*x + 2*c)/d

Mupad [B] (verification not implemented)

Time = 10.78 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a\,\left (210\,\cos \left (c+d\,x\right )+126\,\cos \left (3\,c+3\,d\,x\right )+42\,\cos \left (5\,c+5\,d\,x\right )+6\,\cos \left (7\,c+7\,d\,x\right )-42\,\sin \left (2\,c+2\,d\,x\right )+21\,\sin \left (4\,c+4\,d\,x\right )+14\,\sin \left (6\,c+6\,d\,x\right )+\frac {21\,\sin \left (8\,c+8\,d\,x\right )}{8}-105\,d\,x\right )}{2688\,d} \]

[In]

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

[Out]

-(a*(210*cos(c + d*x) + 126*cos(3*c + 3*d*x) + 42*cos(5*c + 5*d*x) + 6*cos(7*c + 7*d*x) - 42*sin(2*c + 2*d*x)
+ 21*sin(4*c + 4*d*x) + 14*sin(6*c + 6*d*x) + (21*sin(8*c + 8*d*x))/8 - 105*d*x))/(2688*d)