3.218 \(\int \cos (c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^4 \, dx\)

Optimal. Leaf size=88 \[ \frac{a^4 \sin ^8(c+d x)}{8 d}+\frac{4 a^4 \sin ^7(c+d x)}{7 d}+\frac{a^4 \sin ^6(c+d x)}{d}+\frac{4 a^4 \sin ^5(c+d x)}{5 d}+\frac{a^4 \sin ^4(c+d x)}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.0809407, antiderivative size = 88, 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} \[ \frac{a^4 \sin ^8(c+d x)}{8 d}+\frac{4 a^4 \sin ^7(c+d x)}{7 d}+\frac{a^4 \sin ^6(c+d x)}{d}+\frac{4 a^4 \sin ^5(c+d x)}{5 d}+\frac{a^4 \sin ^4(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^4,x]

[Out]

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

Mathematica [A]  time = 0.529192, size = 90, normalized size = 1.02 \[ \frac{a^4 (87360 \sin (c+d x)-47040 \sin (3 (c+d x))+12096 \sin (5 (c+d x))-960 \sin (7 (c+d x))-69720 \cos (2 (c+d x))+26460 \cos (4 (c+d x))-4200 \cos (6 (c+d x))+105 \cos (8 (c+d x))+36400)}{107520 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]*Sin[c + d*x]^3*(a + a*Sin[c + d*x])^4,x]

[Out]

(a^4*(36400 - 69720*Cos[2*(c + d*x)] + 26460*Cos[4*(c + d*x)] - 4200*Cos[6*(c + d*x)] + 105*Cos[8*(c + d*x)] +
87360*Sin[c + d*x] - 47040*Sin[3*(c + d*x)] + 12096*Sin[5*(c + d*x)] - 960*Sin[7*(c + d*x)]))/(107520*d)

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Maple [A]  time = 0.021, size = 70, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{8}}+{\frac{4\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7}}+{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{6}+{\frac{4\,{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{{a}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*sin(d*x+c)^3*(a+a*sin(d*x+c))^4,x)

[Out]

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

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Maxima [A]  time = 1.11293, size = 96, normalized size = 1.09 \begin{align*} \frac{35 \, a^{4} \sin \left (d x + c\right )^{8} + 160 \, a^{4} \sin \left (d x + c\right )^{7} + 280 \, a^{4} \sin \left (d x + c\right )^{6} + 224 \, a^{4} \sin \left (d x + c\right )^{5} + 70 \, a^{4} \sin \left (d x + c\right )^{4}}{280 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^3*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/280*(35*a^4*sin(d*x + c)^8 + 160*a^4*sin(d*x + c)^7 + 280*a^4*sin(d*x + c)^6 + 224*a^4*sin(d*x + c)^5 + 70*a
^4*sin(d*x + c)^4)/d

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Fricas [A]  time = 2.00981, size = 281, normalized size = 3.19 \begin{align*} \frac{35 \, a^{4} \cos \left (d x + c\right )^{8} - 420 \, a^{4} \cos \left (d x + c\right )^{6} + 1120 \, a^{4} \cos \left (d x + c\right )^{4} - 1120 \, a^{4} \cos \left (d x + c\right )^{2} - 32 \,{\left (5 \, a^{4} \cos \left (d x + c\right )^{6} - 22 \, a^{4} \cos \left (d x + c\right )^{4} + 29 \, a^{4} \cos \left (d x + c\right )^{2} - 12 \, a^{4}\right )} \sin \left (d x + c\right )}{280 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^3*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/280*(35*a^4*cos(d*x + c)^8 - 420*a^4*cos(d*x + c)^6 + 1120*a^4*cos(d*x + c)^4 - 1120*a^4*cos(d*x + c)^2 - 32
*(5*a^4*cos(d*x + c)^6 - 22*a^4*cos(d*x + c)^4 + 29*a^4*cos(d*x + c)^2 - 12*a^4)*sin(d*x + c))/d

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Sympy [A]  time = 18.5593, size = 119, normalized size = 1.35 \begin{align*} \begin{cases} \frac{a^{4} \sin ^{8}{\left (c + d x \right )}}{8 d} + \frac{4 a^{4} \sin ^{7}{\left (c + d x \right )}}{7 d} + \frac{a^{4} \sin ^{6}{\left (c + d x \right )}}{d} + \frac{4 a^{4} \sin ^{5}{\left (c + d x \right )}}{5 d} - \frac{a^{4} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} - \frac{a^{4} \cos ^{4}{\left (c + d x \right )}}{4 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{4} \sin ^{3}{\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)**3*(a+a*sin(d*x+c))**4,x)

[Out]

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

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Giac [A]  time = 1.22331, size = 96, normalized size = 1.09 \begin{align*} \frac{35 \, a^{4} \sin \left (d x + c\right )^{8} + 160 \, a^{4} \sin \left (d x + c\right )^{7} + 280 \, a^{4} \sin \left (d x + c\right )^{6} + 224 \, a^{4} \sin \left (d x + c\right )^{5} + 70 \, a^{4} \sin \left (d x + c\right )^{4}}{280 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^3*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

1/280*(35*a^4*sin(d*x + c)^8 + 160*a^4*sin(d*x + c)^7 + 280*a^4*sin(d*x + c)^6 + 224*a^4*sin(d*x + c)^5 + 70*a
^4*sin(d*x + c)^4)/d