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

Optimal. Leaf size=91 \[ \frac{a^4 \sin ^9(c+d x)}{9 d}+\frac{a^4 \sin ^8(c+d x)}{2 d}+\frac{6 a^4 \sin ^7(c+d x)}{7 d}+\frac{2 a^4 \sin ^6(c+d x)}{3 d}+\frac{a^4 \sin ^5(c+d x)}{5 d} \]

[Out]

(a^4*Sin[c + d*x]^5)/(5*d) + (2*a^4*Sin[c + d*x]^6)/(3*d) + (6*a^4*Sin[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x]^8
)/(2*d) + (a^4*Sin[c + d*x]^9)/(9*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0849619, antiderivative size = 91, 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 = {2833, 12, 43} \[ \frac{a^4 \sin ^9(c+d x)}{9 d}+\frac{a^4 \sin ^8(c+d x)}{2 d}+\frac{6 a^4 \sin ^7(c+d x)}{7 d}+\frac{2 a^4 \sin ^6(c+d x)}{3 d}+\frac{a^4 \sin ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*Sin[c + d*x]^5)/(5*d) + (2*a^4*Sin[c + d*x]^6)/(3*d) + (6*a^4*Sin[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x]^8
)/(2*d) + (a^4*Sin[c + d*x]^9)/(9*d)

Rule 2833

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

Rule 12

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.812519, size = 100, normalized size = 1.1 \[ \frac{a^4 (52290 \sin (c+d x)-30660 \sin (3 (c+d x))+9828 \sin (5 (c+d x))-1395 \sin (7 (c+d x))+35 \sin (9 (c+d x))-42840 \cos (2 (c+d x))+18900 \cos (4 (c+d x))-4200 \cos (6 (c+d x))+315 \cos (8 (c+d x))+4095)}{80640 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(4095 - 42840*Cos[2*(c + d*x)] + 18900*Cos[4*(c + d*x)] - 4200*Cos[6*(c + d*x)] + 315*Cos[8*(c + d*x)] +
52290*Sin[c + d*x] - 30660*Sin[3*(c + d*x)] + 9828*Sin[5*(c + d*x)] - 1395*Sin[7*(c + d*x)] + 35*Sin[9*(c + d*
x)]))/(80640*d)

________________________________________________________________________________________

Maple [A]  time = 0.023, size = 71, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{9}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{2}}+{\frac{6\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{2\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{3}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/9*a^4*sin(d*x+c)^9+1/2*a^4*sin(d*x+c)^8+6/7*a^4*sin(d*x+c)^7+2/3*a^4*sin(d*x+c)^6+1/5*a^4*sin(d*x+c)^5)

________________________________________________________________________________________

Maxima [A]  time = 1.10056, size = 96, normalized size = 1.05 \begin{align*} \frac{70 \, a^{4} \sin \left (d x + c\right )^{9} + 315 \, a^{4} \sin \left (d x + c\right )^{8} + 540 \, a^{4} \sin \left (d x + c\right )^{7} + 420 \, a^{4} \sin \left (d x + c\right )^{6} + 126 \, a^{4} \sin \left (d x + c\right )^{5}}{630 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/630*(70*a^4*sin(d*x + c)^9 + 315*a^4*sin(d*x + c)^8 + 540*a^4*sin(d*x + c)^7 + 420*a^4*sin(d*x + c)^6 + 126*
a^4*sin(d*x + c)^5)/d

________________________________________________________________________________________

Fricas [A]  time = 1.98013, size = 324, normalized size = 3.56 \begin{align*} \frac{315 \, a^{4} \cos \left (d x + c\right )^{8} - 1680 \, a^{4} \cos \left (d x + c\right )^{6} + 3150 \, a^{4} \cos \left (d x + c\right )^{4} - 2520 \, a^{4} \cos \left (d x + c\right )^{2} + 2 \,{\left (35 \, a^{4} \cos \left (d x + c\right )^{8} - 410 \, a^{4} \cos \left (d x + c\right )^{6} + 1083 \, a^{4} \cos \left (d x + c\right )^{4} - 1076 \, a^{4} \cos \left (d x + c\right )^{2} + 368 \, a^{4}\right )} \sin \left (d x + c\right )}{630 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/630*(315*a^4*cos(d*x + c)^8 - 1680*a^4*cos(d*x + c)^6 + 3150*a^4*cos(d*x + c)^4 - 2520*a^4*cos(d*x + c)^2 +
2*(35*a^4*cos(d*x + c)^8 - 410*a^4*cos(d*x + c)^6 + 1083*a^4*cos(d*x + c)^4 - 1076*a^4*cos(d*x + c)^2 + 368*a^
4)*sin(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 24.9305, size = 97, normalized size = 1.07 \begin{align*} \begin{cases} \frac{a^{4} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac{a^{4} \sin ^{8}{\left (c + d x \right )}}{2 d} + \frac{6 a^{4} \sin ^{7}{\left (c + d x \right )}}{7 d} + \frac{2 a^{4} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac{a^{4} \sin ^{5}{\left (c + d x \right )}}{5 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{4} \sin ^{4}{\left (c \right )} \cos{\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)*sin(d*x+c)**4*(a+a*sin(d*x+c))**4,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.28994, size = 96, normalized size = 1.05 \begin{align*} \frac{70 \, a^{4} \sin \left (d x + c\right )^{9} + 315 \, a^{4} \sin \left (d x + c\right )^{8} + 540 \, a^{4} \sin \left (d x + c\right )^{7} + 420 \, a^{4} \sin \left (d x + c\right )^{6} + 126 \, a^{4} \sin \left (d x + c\right )^{5}}{630 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/630*(70*a^4*sin(d*x + c)^9 + 315*a^4*sin(d*x + c)^8 + 540*a^4*sin(d*x + c)^7 + 420*a^4*sin(d*x + c)^6 + 126*
a^4*sin(d*x + c)^5)/d