3.496 \(\int \cos ^5(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=81 \[ \frac{a \sin ^9(c+d x)}{9 d}-\frac{2 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \cos ^8(c+d x)}{8 d}-\frac{a \cos ^6(c+d x)}{6 d} \]

[Out]

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

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Rubi [A]  time = 0.126958, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2834, 2565, 14, 2564, 270} \[ \frac{a \sin ^9(c+d x)}{9 d}-\frac{2 a \sin ^7(c+d x)}{7 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{a \cos ^8(c+d x)}{8 d}-\frac{a \cos ^6(c+d x)}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]

[Out]

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

Rule 2834

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])

Rule 2565

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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2564

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 270

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]

Rubi steps

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

Mathematica [A]  time = 0.354229, size = 97, normalized size = 1.2 \[ \frac{a (7560 \sin (c+d x)-1680 \sin (3 (c+d x))-1008 \sin (5 (c+d x))+180 \sin (7 (c+d x))+140 \sin (9 (c+d x))-7560 \cos (2 (c+d x))-1260 \cos (4 (c+d x))+840 \cos (6 (c+d x))+315 \cos (8 (c+d x)))}{322560 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]

[Out]

(a*(-7560*Cos[2*(c + d*x)] - 1260*Cos[4*(c + d*x)] + 840*Cos[6*(c + d*x)] + 315*Cos[8*(c + d*x)] + 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)]))/(322560
*d)

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Maple [A]  time = 0.033, size = 102, normalized size = 1.3 \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 ) +a \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{8}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{24}} \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)^3*(a+a*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))+a*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6))

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Maxima [A]  time = 1.16918, size = 97, normalized size = 1.2 \begin{align*} \frac{280 \, a \sin \left (d x + c\right )^{9} + 315 \, a \sin \left (d x + c\right )^{8} - 720 \, a \sin \left (d x + c\right )^{7} - 840 \, a \sin \left (d x + c\right )^{6} + 504 \, a \sin \left (d x + c\right )^{5} + 630 \, a \sin \left (d x + c\right )^{4}}{2520 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^3*(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 - 720*a*sin(d*x + c)^7 - 840*a*sin(d*x + c)^6 + 504*a*sin(
d*x + c)^5 + 630*a*sin(d*x + c)^4)/d

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Fricas [A]  time = 1.14556, size = 223, normalized size = 2.75 \begin{align*} \frac{315 \, a \cos \left (d x + c\right )^{8} - 420 \, a \cos \left (d x + c\right )^{6} + 8 \,{\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 )}{2520 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/2520*(315*a*cos(d*x + c)^8 - 420*a*cos(d*x + c)^6 + 8*(35*a*cos(d*x + c)^8 - 50*a*cos(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 = 21.0753, size = 136, normalized size = 1.68 \begin{align*} \begin{cases} \frac{8 a \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac{a \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac{4 a \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac{a \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 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 ^{4}{\left (c + d x \right )}}{4 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \sin ^{3}{\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)**3*(a+a*sin(d*x+c)),x)

[Out]

Piecewise((8*a*sin(c + d*x)**9/(315*d) + a*sin(c + d*x)**8/(24*d) + 4*a*sin(c + d*x)**7*cos(c + d*x)**2/(35*d)
+ a*sin(c + d*x)**6*cos(c + d*x)**2/(6*d) + a*sin(c + d*x)**5*cos(c + d*x)**4/(5*d) + a*sin(c + d*x)**4*cos(c
+ d*x)**4/(4*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**3*cos(c)**5, True))

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Giac [A]  time = 1.30362, size = 180, normalized size = 2.22 \begin{align*} \frac{a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac{a \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac{a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac{3 \, a \cos \left (2 \, d x + 2 \, c\right )}{128 \, 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)^3*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/1024*a*cos(8*d*x + 8*c)/d + 1/384*a*cos(6*d*x + 6*c)/d - 1/256*a*cos(4*d*x + 4*c)/d - 3/128*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*sin(d*x + c)/d