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

Optimal. Leaf size=73 \[ \frac{a^3 \sin ^7(c+d x)}{7 d}+\frac{a^3 \sin ^6(c+d x)}{2 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{a^3 \sin ^4(c+d x)}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.0738102, antiderivative size = 73, 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} \[ \frac{a^3 \sin ^7(c+d x)}{7 d}+\frac{a^3 \sin ^6(c+d x)}{2 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}+\frac{a^3 \sin ^4(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.322901, size = 80, normalized size = 1.1 \[ -\frac{a^3 (-1015 \sin (c+d x)+525 \sin (3 (c+d x))-119 \sin (5 (c+d x))+5 \sin (7 (c+d x))+805 \cos (2 (c+d x))-280 \cos (4 (c+d x))+35 \cos (6 (c+d x))-350)}{2240 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3*(-350 + 805*Cos[2*(c + d*x)] - 280*Cos[4*(c + d*x)] + 35*Cos[6*(c + d*x)] - 1015*Sin[c + d*x] + 525*Sin[
3*(c + d*x)] - 119*Sin[5*(c + d*x)] + 5*Sin[7*(c + d*x)]))/(2240*d)

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Maple [A]  time = 0.021, size = 58, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{2}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5}}+{\frac{{a}^{3} \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))^3,x)

[Out]

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

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Maxima [A]  time = 0.996887, size = 78, normalized size = 1.07 \begin{align*} \frac{20 \, a^{3} \sin \left (d x + c\right )^{7} + 70 \, a^{3} \sin \left (d x + c\right )^{6} + 84 \, a^{3} \sin \left (d x + c\right )^{5} + 35 \, a^{3} \sin \left (d x + c\right )^{4}}{140 \, 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))^3,x, algorithm="maxima")

[Out]

1/140*(20*a^3*sin(d*x + c)^7 + 70*a^3*sin(d*x + c)^6 + 84*a^3*sin(d*x + c)^5 + 35*a^3*sin(d*x + c)^4)/d

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Fricas [A]  time = 1.71006, size = 244, normalized size = 3.34 \begin{align*} -\frac{70 \, a^{3} \cos \left (d x + c\right )^{6} - 245 \, a^{3} \cos \left (d x + c\right )^{4} + 280 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \,{\left (5 \, a^{3} \cos \left (d x + c\right )^{6} - 36 \, a^{3} \cos \left (d x + c\right )^{4} + 57 \, a^{3} \cos \left (d x + c\right )^{2} - 26 \, a^{3}\right )} \sin \left (d x + c\right )}{140 \, 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))^3,x, algorithm="fricas")

[Out]

-1/140*(70*a^3*cos(d*x + c)^6 - 245*a^3*cos(d*x + c)^4 + 280*a^3*cos(d*x + c)^2 + 4*(5*a^3*cos(d*x + c)^6 - 36
*a^3*cos(d*x + c)^4 + 57*a^3*cos(d*x + c)^2 - 26*a^3)*sin(d*x + c))/d

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Sympy [A]  time = 13.1465, size = 104, normalized size = 1.42 \begin{align*} \begin{cases} \frac{a^{3} \sin ^{7}{\left (c + d x \right )}}{7 d} + \frac{a^{3} \sin ^{6}{\left (c + d x \right )}}{2 d} + \frac{3 a^{3} \sin ^{5}{\left (c + d x \right )}}{5 d} - \frac{a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} - \frac{a^{3} \cos ^{4}{\left (c + d x \right )}}{4 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{3} \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))**3,x)

[Out]

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

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Giac [A]  time = 1.27538, size = 78, normalized size = 1.07 \begin{align*} \frac{20 \, a^{3} \sin \left (d x + c\right )^{7} + 70 \, a^{3} \sin \left (d x + c\right )^{6} + 84 \, a^{3} \sin \left (d x + c\right )^{5} + 35 \, a^{3} \sin \left (d x + c\right )^{4}}{140 \, 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))^3,x, algorithm="giac")

[Out]

1/140*(20*a^3*sin(d*x + c)^7 + 70*a^3*sin(d*x + c)^6 + 84*a^3*sin(d*x + c)^5 + 35*a^3*sin(d*x + c)^4)/d