3.1199 \(\int \cos ^5(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x)) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.120472, antiderivative size = 97, normalized size of antiderivative = 1., 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 = {2837, 12, 766} \[ \frac{a \sin ^9(c+d x)}{9 d}-\frac{2 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{b \sin ^{10}(c+d x)}{10 d}-\frac{b \sin ^8(c+d x)}{4 d}+\frac{b \sin ^6(c+d x)}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2837

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*x)/b)^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]

Rule 12

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

Rule 766

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]

Rubi steps

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

Mathematica [A]  time = 0.30239, size = 94, normalized size = 0.97 \[ \frac{7560 a \sin (c+d x)-1680 a \sin (3 (c+d x))-1008 a \sin (5 (c+d x))+180 a \sin (7 (c+d x))+140 a \sin (9 (c+d x))-3150 b \cos (2 (c+d x))+525 b \cos (6 (c+d x))-63 b \cos (10 (c+d x))}{322560 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.033, size = 120, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{9}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{21}}+{\frac{\sin \left ( dx+c \right ) }{105} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) +b \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{10}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{20}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{60}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(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))+b*(-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))

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Maxima [A]  time = 0.99289, size = 97, normalized size = 1. \begin{align*} \frac{126 \, b \sin \left (d x + c\right )^{10} + 140 \, a \sin \left (d x + c\right )^{9} - 315 \, b \sin \left (d x + c\right )^{8} - 360 \, a \sin \left (d x + c\right )^{7} + 210 \, b \sin \left (d x + c\right )^{6} + 252 \, a \sin \left (d x + c\right )^{5}}{1260 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.02855, size = 257, normalized size = 2.65 \begin{align*} -\frac{126 \, b \cos \left (d x + c\right )^{10} - 315 \, b \cos \left (d x + c\right )^{8} + 210 \, b \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1260*(126*b*cos(d*x + c)^10 - 315*b*cos(d*x + c)^8 + 210*b*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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Sympy [A]  time = 33.5514, size = 136, normalized size = 1.4 \begin{align*} \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{b \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} - \frac{b \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{12 d} - \frac{b \cos ^{10}{\left (c + d x \right )}}{60 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\right ) \sin ^{4}{\left (c \right )} \cos ^{5}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*sin(d*x+c)**4*(a+b*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) - b*sin(c + d*x)**4*cos(c + d*x)**6/(6*d) - b*sin(c + d*x)**2*cos(c + d*x)**8/(12*d) - b*cos(c
+ d*x)**10/(60*d), Ne(d, 0)), (x*(a + b*sin(c))*sin(c)**4*cos(c)**5, True))

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Giac [A]  time = 1.21775, size = 159, normalized size = 1.64 \begin{align*} -\frac{b \cos \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{5 \, b \cos \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac{5 \, b \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5120*b*cos(10*d*x + 10*c)/d + 5/3072*b*cos(6*d*x + 6*c)/d - 5/512*b*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