\(\int \cos ^5(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x)) \, dx\) [1198]

Optimal result

Integrand size = 27, antiderivative size = 97 \[ \int \cos ^5(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {a \sin ^6(c+d x)}{6 d}+\frac {b \sin ^7(c+d x)}{7 d}-\frac {a \sin ^8(c+d x)}{4 d}-\frac {2 b \sin ^9(c+d x)}{9 d}+\frac {a \sin ^{10}(c+d x)}{10 d}+\frac {b \sin ^{11}(c+d x)}{11 d} \]

[Out]

1/6*a*sin(d*x+c)^6/d+1/7*b*sin(d*x+c)^7/d-1/4*a*sin(d*x+c)^8/d-2/9*b*sin(d*x+c)^9/d+1/10*a*sin(d*x+c)^10/d+1/1
1*b*sin(d*x+c)^11/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 97, 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 = {2916, 12, 780} \[ \int \cos ^5(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {a \sin ^{10}(c+d x)}{10 d}-\frac {a \sin ^8(c+d x)}{4 d}+\frac {a \sin ^6(c+d x)}{6 d}+\frac {b \sin ^{11}(c+d x)}{11 d}-\frac {2 b \sin ^9(c+d x)}{9 d}+\frac {b \sin ^7(c+d x)}{7 d} \]

[In]

Int[Cos[c + d*x]^5*Sin[c + d*x]^5*(a + b*Sin[c + d*x]),x]

[Out]

(a*Sin[c + d*x]^6)/(6*d) + (b*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^8)/(4*d) - (2*b*Sin[c + d*x]^9)/(9*d) +
(a*Sin[c + d*x]^10)/(10*d) + (b*Sin[c + d*x]^11)/(11*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 780

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rule 2916

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^5 (a+x) \left (b^2-x^2\right )^2}{b^5} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int x^5 (a+x) \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^{10} d} \\ & = \frac {\text {Subst}\left (\int \left (a b^4 x^5+b^4 x^6-2 a b^2 x^7-2 b^2 x^8+a x^9+x^{10}\right ) \, dx,x,b \sin (c+d x)\right )}{b^{10} d} \\ & = \frac {a \sin ^6(c+d x)}{6 d}+\frac {b \sin ^7(c+d x)}{7 d}-\frac {a \sin ^8(c+d x)}{4 d}-\frac {2 b \sin ^9(c+d x)}{9 d}+\frac {a \sin ^{10}(c+d x)}{10 d}+\frac {b \sin ^{11}(c+d x)}{11 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08 \[ \int \cos ^5(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {-34650 a \cos (2 (c+d x))+5775 a \cos (6 (c+d x))-693 a \cos (10 (c+d x))+34650 b \sin (c+d x)-11550 b \sin (3 (c+d x))-3465 b \sin (5 (c+d x))+2475 b \sin (7 (c+d x))+385 b \sin (9 (c+d x))-315 b \sin (11 (c+d x))}{3548160 d} \]

[In]

Integrate[Cos[c + d*x]^5*Sin[c + d*x]^5*(a + b*Sin[c + d*x]),x]

[Out]

(-34650*a*Cos[2*(c + d*x)] + 5775*a*Cos[6*(c + d*x)] - 693*a*Cos[10*(c + d*x)] + 34650*b*Sin[c + d*x] - 11550*
b*Sin[3*(c + d*x)] - 3465*b*Sin[5*(c + d*x)] + 2475*b*Sin[7*(c + d*x)] + 385*b*Sin[9*(c + d*x)] - 315*b*Sin[11
*(c + d*x)])/(3548160*d)

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {\frac {b \left (\sin ^{11}\left (d x +c \right )\right )}{11}+\frac {a \left (\sin ^{10}\left (d x +c \right )\right )}{10}-\frac {2 b \left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {a \left (\sin ^{8}\left (d x +c \right )\right )}{4}+\frac {b \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {a \left (\sin ^{6}\left (d x +c \right )\right )}{6}}{d}\)	\(72\)
default	\(\frac {\frac {b \left (\sin ^{11}\left (d x +c \right )\right )}{11}+\frac {a \left (\sin ^{10}\left (d x +c \right )\right )}{10}-\frac {2 b \left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {a \left (\sin ^{8}\left (d x +c \right )\right )}{4}+\frac {b \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {a \left (\sin ^{6}\left (d x +c \right )\right )}{6}}{d}\)	\(72\)
parallelrisch	\(-\frac {\left (-10+\cos \left (3 d x +3 c \right )-6 \cos \left (2 d x +2 c \right )+15 \cos \left (d x +c \right )\right ) \left (315 b \sin \left (5 d x +5 c \right )+1505 b \sin \left (3 d x +3 c \right )+693 \cos \left (4 d x +4 c \right ) a +4158 a \cos \left (2 d x +2 c \right )+1830 b \sin \left (d x +c \right )+4389 a \right ) \left (\cos \left (3 d x +3 c \right )+6 \cos \left (2 d x +2 c \right )+15 \cos \left (d x +c \right )+10\right )}{887040 d}\)	\(127\)
risch	\(\frac {5 b \sin \left (d x +c \right )}{512 d}-\frac {b \sin \left (11 d x +11 c \right )}{11264 d}-\frac {a \cos \left (10 d x +10 c \right )}{5120 d}+\frac {b \sin \left (9 d x +9 c \right )}{9216 d}+\frac {5 b \sin \left (7 d x +7 c \right )}{7168 d}+\frac {5 a \cos \left (6 d x +6 c \right )}{3072 d}-\frac {b \sin \left (5 d x +5 c \right )}{1024 d}-\frac {5 b \sin \left (3 d x +3 c \right )}{1536 d}-\frac {5 a \cos \left (2 d x +2 c \right )}{512 d}\)	\(134\)

[In]

int(cos(d*x+c)^5*sin(d*x+c)^5*(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/11*b*sin(d*x+c)^11+1/10*a*sin(d*x+c)^10-2/9*b*sin(d*x+c)^9-1/4*a*sin(d*x+c)^8+1/7*b*sin(d*x+c)^7+1/6*a*
sin(d*x+c)^6)

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.09 \[ \int \cos ^5(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {1386 \, a \cos \left (d x + c\right )^{10} - 3465 \, a \cos \left (d x + c\right )^{8} + 2310 \, a \cos \left (d x + c\right )^{6} + 20 \, {\left (63 \, b \cos \left (d x + c\right )^{10} - 161 \, b \cos \left (d x + c\right )^{8} + 113 \, b \cos \left (d x + c\right )^{6} - 3 \, b \cos \left (d x + c\right )^{4} - 4 \, b \cos \left (d x + c\right )^{2} - 8 \, b\right )} \sin \left (d x + c\right )}{13860 \, d} \]

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/13860*(1386*a*cos(d*x + c)^10 - 3465*a*cos(d*x + c)^8 + 2310*a*cos(d*x + c)^6 + 20*(63*b*cos(d*x + c)^10 -
161*b*cos(d*x + c)^8 + 113*b*cos(d*x + c)^6 - 3*b*cos(d*x + c)^4 - 4*b*cos(d*x + c)^2 - 8*b)*sin(d*x + c))/d

Sympy [A] (verification not implemented)

Time = 1.92 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.40 \[ \int \cos ^5(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x)) \, dx=\begin {cases} - \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{12 d} - \frac {a \cos ^{10}{\left (c + d x \right )}}{60 d} + \frac {8 b \sin ^{11}{\left (c + d x \right )}}{693 d} + \frac {4 b \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{63 d} + \frac {b \sin ^{7}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right ) \sin ^{5}{\left (c \right )} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**5*sin(d*x+c)**5*(a+b*sin(d*x+c)),x)

[Out]

Piecewise((-a*sin(c + d*x)**4*cos(c + d*x)**6/(6*d) - a*sin(c + d*x)**2*cos(c + d*x)**8/(12*d) - a*cos(c + d*x
)**10/(60*d) + 8*b*sin(c + d*x)**11/(693*d) + 4*b*sin(c + d*x)**9*cos(c + d*x)**2/(63*d) + b*sin(c + d*x)**7*c
os(c + d*x)**4/(7*d), Ne(d, 0)), (x*(a + b*sin(c))*sin(c)**5*cos(c)**5, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.74 \[ \int \cos ^5(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {1260 \, b \sin \left (d x + c\right )^{11} + 1386 \, a \sin \left (d x + c\right )^{10} - 3080 \, b \sin \left (d x + c\right )^{9} - 3465 \, a \sin \left (d x + c\right )^{8} + 1980 \, b \sin \left (d x + c\right )^{7} + 2310 \, a \sin \left (d x + c\right )^{6}}{13860 \, d} \]

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/13860*(1260*b*sin(d*x + c)^11 + 1386*a*sin(d*x + c)^10 - 3080*b*sin(d*x + c)^9 - 3465*a*sin(d*x + c)^8 + 198
0*b*sin(d*x + c)^7 + 2310*a*sin(d*x + c)^6)/d

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.37 \[ \int \cos ^5(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {a \cos \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {5 \, a \cos \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {5 \, a \cos \left (2 \, d x + 2 \, c\right )}{512 \, d} - \frac {b \sin \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {b \sin \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac {5 \, b \sin \left (7 \, d x + 7 \, c\right )}{7168 \, d} - \frac {b \sin \left (5 \, d x + 5 \, c\right )}{1024 \, d} - \frac {5 \, b \sin \left (3 \, d x + 3 \, c\right )}{1536 \, d} + \frac {5 \, b \sin \left (d x + c\right )}{512 \, d} \]

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/5120*a*cos(10*d*x + 10*c)/d + 5/3072*a*cos(6*d*x + 6*c)/d - 5/512*a*cos(2*d*x + 2*c)/d - 1/11264*b*sin(11*d
*x + 11*c)/d + 1/9216*b*sin(9*d*x + 9*c)/d + 5/7168*b*sin(7*d*x + 7*c)/d - 1/1024*b*sin(5*d*x + 5*c)/d - 5/153
6*b*sin(3*d*x + 3*c)/d + 5/512*b*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.73 \[ \int \cos ^5(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {\frac {b\,{\sin \left (c+d\,x\right )}^{11}}{11}+\frac {a\,{\sin \left (c+d\,x\right )}^{10}}{10}-\frac {2\,b\,{\sin \left (c+d\,x\right )}^9}{9}-\frac {a\,{\sin \left (c+d\,x\right )}^8}{4}+\frac {b\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}}{d} \]

[In]

int(cos(c + d*x)^5*sin(c + d*x)^5*(a + b*sin(c + d*x)),x)

[Out]

((a*sin(c + d*x)^6)/6 - (a*sin(c + d*x)^8)/4 + (a*sin(c + d*x)^10)/10 + (b*sin(c + d*x)^7)/7 - (2*b*sin(c + d*
x)^9)/9 + (b*sin(c + d*x)^11)/11)/d