3.495 \(\int \cos ^5(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=97 \[ \frac {a \sin ^{10}(c+d x)}{10 d}+\frac {a \sin ^9(c+d x)}{9 d}-\frac {a \sin ^8(c+d x)}{4 d}-\frac {2 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^6(c+d x)}{6 d}+\frac {a \sin ^5(c+d x)}{5 d} \]

[Out]

1/5*a*sin(d*x+c)^5/d+1/6*a*sin(d*x+c)^6/d-2/7*a*sin(d*x+c)^7/d-1/4*a*sin(d*x+c)^8/d+1/9*a*sin(d*x+c)^9/d+1/10*
a*sin(d*x+c)^10/d

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Rubi [A]  time = 0.08, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ \frac {a \sin ^{10}(c+d x)}{10 d}+\frac {a \sin ^9(c+d x)}{9 d}-\frac {a \sin ^8(c+d x)}{4 d}-\frac {2 a \sin ^7(c+d x)}{7 d}+\frac {a \sin ^6(c+d x)}{6 d}+\frac {a \sin ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]

[Out]

(a*Sin[c + d*x]^5)/(5*d) + (a*Sin[c + d*x]^6)/(6*d) - (2*a*Sin[c + d*x]^7)/(7*d) - (a*Sin[c + d*x]^8)/(4*d) +
(a*Sin[c + d*x]^9)/(9*d) + (a*Sin[c + d*x]^10)/(10*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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2836

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 + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rubi steps

\begin {align*} \int \cos ^5(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 x^4 (a+x)^3}{a^4} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int (a-x)^2 x^4 (a+x)^3 \, dx,x,a \sin (c+d x)\right )}{a^9 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^5 x^4+a^4 x^5-2 a^3 x^6-2 a^2 x^7+a x^8+x^9\right ) \, dx,x,a \sin (c+d x)\right )}{a^9 d}\\ &=\frac {a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^6(c+d x)}{6 d}-\frac {2 a \sin ^7(c+d x)}{7 d}-\frac {a \sin ^8(c+d x)}{4 d}+\frac {a \sin ^9(c+d x)}{9 d}+\frac {a \sin ^{10}(c+d x)}{10 d}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 87, normalized size = 0.90 \[ -\frac {a (-7560 \sin (c+d x)+1680 \sin (3 (c+d x))+1008 \sin (5 (c+d x))-180 \sin (7 (c+d x))-140 \sin (9 (c+d x))+3150 \cos (2 (c+d x))-525 \cos (6 (c+d x))+63 \cos (10 (c+d x)))}{322560 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*Sin[c + d*x]^4*(a + a*Sin[c + d*x]),x]

[Out]

-1/322560*(a*(3150*Cos[2*(c + d*x)] - 525*Cos[6*(c + d*x)] + 63*Cos[10*(c + d*x)] - 7560*Sin[c + d*x] + 1680*S
in[3*(c + d*x)] + 1008*Sin[5*(c + d*x)] - 180*Sin[7*(c + d*x)] - 140*Sin[9*(c + d*x)]))/d

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fricas [A]  time = 0.56, size = 95, normalized size = 0.98 \[ -\frac {126 \, a \cos \left (d x + c\right )^{10} - 315 \, a \cos \left (d x + c\right )^{8} + 210 \, a \cos \left (d x + c\right )^{6} - 4 \, {\left (35 \, a \cos \left (d x + c\right )^{8} - 50 \, a \cos \left (d x + c\right )^{6} + 3 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right )}{1260 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/1260*(126*a*cos(d*x + c)^10 - 315*a*cos(d*x + c)^8 + 210*a*cos(d*x + c)^6 - 4*(35*a*cos(d*x + c)^8 - 50*a*c
os(d*x + c)^6 + 3*a*cos(d*x + c)^4 + 4*a*cos(d*x + c)^2 + 8*a)*sin(d*x + c))/d

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giac [A]  time = 0.26, size = 118, normalized size = 1.22 \[ -\frac {a \cos \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {5 \, a \cos \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {5 \, a \cos \left (2 \, d x + 2 \, c\right )}{512 \, d} + \frac {a \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {a \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {a \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {3 \, a \sin \left (d x + c\right )}{128 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^4*(a+a*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 + 1/2304*a*sin(9*d*x
+ 9*c)/d + 1/1792*a*sin(7*d*x + 7*c)/d - 1/320*a*sin(5*d*x + 5*c)/d - 1/192*a*sin(3*d*x + 3*c)/d + 3/128*a*si
n(d*x + c)/d

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maple [A]  time = 0.24, size = 120, normalized size = 1.24 \[ \frac {a \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{20}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{60}\right )+a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*sin(d*x+c)^4*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/10*sin(d*x+c)^4*cos(d*x+c)^6-1/20*sin(d*x+c)^2*cos(d*x+c)^6-1/60*cos(d*x+c)^6)+a*(-1/9*sin(d*x+c)^3
*cos(d*x+c)^6-1/21*sin(d*x+c)*cos(d*x+c)^6+1/105*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)))

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maxima [A]  time = 0.50, size = 72, normalized size = 0.74 \[ \frac {126 \, a \sin \left (d x + c\right )^{10} + 140 \, a \sin \left (d x + c\right )^{9} - 315 \, a \sin \left (d x + c\right )^{8} - 360 \, a \sin \left (d x + c\right )^{7} + 210 \, a \sin \left (d x + c\right )^{6} + 252 \, a \sin \left (d x + c\right )^{5}}{1260 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^4*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/1260*(126*a*sin(d*x + c)^10 + 140*a*sin(d*x + c)^9 - 315*a*sin(d*x + c)^8 - 360*a*sin(d*x + c)^7 + 210*a*sin
(d*x + c)^6 + 252*a*sin(d*x + c)^5)/d

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mupad [B]  time = 8.70, size = 71, normalized size = 0.73 \[ \frac {\frac {a\,{\sin \left (c+d\,x\right )}^{10}}{10}+\frac {a\,{\sin \left (c+d\,x\right )}^9}{9}-\frac {a\,{\sin \left (c+d\,x\right )}^8}{4}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5*sin(c + d*x)^4*(a + a*sin(c + d*x)),x)

[Out]

((a*sin(c + d*x)^5)/5 + (a*sin(c + d*x)^6)/6 - (2*a*sin(c + d*x)^7)/7 - (a*sin(c + d*x)^8)/4 + (a*sin(c + d*x)
^9)/9 + (a*sin(c + d*x)^10)/10)/d

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sympy [A]  time = 27.04, size = 136, normalized size = 1.40 \[ \begin {cases} \frac {8 a \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 a \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} - \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{12 d} - \frac {a \cos ^{10}{\left (c + d x \right )}}{60 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \sin ^{4}{\relax (c )} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*sin(d*x+c)**4*(a+a*sin(d*x+c)),x)

[Out]

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

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