3.512 \(\int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=109 \[ \frac{(a \sin (c+d x)+a)^9}{9 a^7 d}-\frac{3 (a \sin (c+d x)+a)^8}{4 a^6 d}+\frac{13 (a \sin (c+d x)+a)^7}{7 a^5 d}-\frac{2 (a \sin (c+d x)+a)^6}{a^4 d}+\frac{4 (a \sin (c+d x)+a)^5}{5 a^3 d} \]

[Out]

(4*(a + a*Sin[c + d*x])^5)/(5*a^3*d) - (2*(a + a*Sin[c + d*x])^6)/(a^4*d) + (13*(a + a*Sin[c + d*x])^7)/(7*a^5
*d) - (3*(a + a*Sin[c + d*x])^8)/(4*a^6*d) + (a + a*Sin[c + d*x])^9/(9*a^7*d)

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Rubi [A]  time = 0.125813, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ \frac{(a \sin (c+d x)+a)^9}{9 a^7 d}-\frac{3 (a \sin (c+d x)+a)^8}{4 a^6 d}+\frac{13 (a \sin (c+d x)+a)^7}{7 a^5 d}-\frac{2 (a \sin (c+d x)+a)^6}{a^4 d}+\frac{4 (a \sin (c+d x)+a)^5}{5 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]

[Out]

(4*(a + a*Sin[c + d*x])^5)/(5*a^3*d) - (2*(a + a*Sin[c + d*x])^6)/(a^4*d) + (13*(a + a*Sin[c + d*x])^7)/(7*a^5
*d) - (3*(a + a*Sin[c + d*x])^8)/(4*a^6*d) + (a + a*Sin[c + d*x])^9/(9*a^7*d)

Rule 2836

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 + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \cos ^5(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 x^2 (a+x)^4}{a^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int (a-x)^2 x^2 (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^4 (a+x)^4-12 a^3 (a+x)^5+13 a^2 (a+x)^6-6 a (a+x)^7+(a+x)^8\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{4 (a+a \sin (c+d x))^5}{5 a^3 d}-\frac{2 (a+a \sin (c+d x))^6}{a^4 d}+\frac{13 (a+a \sin (c+d x))^7}{7 a^5 d}-\frac{3 (a+a \sin (c+d x))^8}{4 a^6 d}+\frac{(a+a \sin (c+d x))^9}{9 a^7 d}\\ \end{align*}

Mathematica [A]  time = 0.731221, size = 99, normalized size = 0.91 \[ -\frac{a^2 (-16380 \sin (c+d x)+1680 \sin (3 (c+d x))+2016 \sin (5 (c+d x))+270 \sin (7 (c+d x))-70 \sin (9 (c+d x))+7560 \cos (2 (c+d x))+1260 \cos (4 (c+d x))-840 \cos (6 (c+d x))-315 \cos (8 (c+d x)))}{161280 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]

[Out]

-(a^2*(7560*Cos[2*(c + d*x)] + 1260*Cos[4*(c + d*x)] - 840*Cos[6*(c + d*x)] - 315*Cos[8*(c + d*x)] - 16380*Sin
[c + d*x] + 1680*Sin[3*(c + d*x)] + 2016*Sin[5*(c + d*x)] + 270*Sin[7*(c + d*x)] - 70*Sin[9*(c + d*x)]))/(1612
80*d)

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Maple [A]  time = 0.037, size = 156, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({a}^{2} \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 ) +2\,{a}^{2} \left ( -1/8\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}-1/24\, \left ( \cos \left ( dx+c \right ) \right ) ^{6} \right ) +{a}^{2} \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7}}+{\frac{\sin \left ( dx+c \right ) }{35} \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 ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*sin(d*x+c)^2*(a+a*sin(d*x+c))^2,x)

[Out]

1/d*(a^2*(-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))+2*a^2*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6)+a^2*(-1/7*sin(d*x+c)*cos(d*x+c)^6+1/35*
(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)))

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Maxima [A]  time = 1.11049, size = 131, normalized size = 1.2 \begin{align*} \frac{140 \, a^{2} \sin \left (d x + c\right )^{9} + 315 \, a^{2} \sin \left (d x + c\right )^{8} - 180 \, a^{2} \sin \left (d x + c\right )^{7} - 840 \, a^{2} \sin \left (d x + c\right )^{6} - 252 \, a^{2} \sin \left (d x + c\right )^{5} + 630 \, a^{2} \sin \left (d x + c\right )^{4} + 420 \, a^{2} \sin \left (d x + c\right )^{3}}{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)^2*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/1260*(140*a^2*sin(d*x + c)^9 + 315*a^2*sin(d*x + c)^8 - 180*a^2*sin(d*x + c)^7 - 840*a^2*sin(d*x + c)^6 - 25
2*a^2*sin(d*x + c)^5 + 630*a^2*sin(d*x + c)^4 + 420*a^2*sin(d*x + c)^3)/d

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Fricas [A]  time = 1.15399, size = 246, normalized size = 2.26 \begin{align*} \frac{315 \, a^{2} \cos \left (d x + c\right )^{8} - 420 \, a^{2} \cos \left (d x + c\right )^{6} + 4 \,{\left (35 \, a^{2} \cos \left (d x + c\right )^{8} - 95 \, a^{2} \cos \left (d x + c\right )^{6} + 12 \, a^{2} \cos \left (d x + c\right )^{4} + 16 \, a^{2} \cos \left (d x + c\right )^{2} + 32 \, a^{2}\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)^2*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/1260*(315*a^2*cos(d*x + c)^8 - 420*a^2*cos(d*x + c)^6 + 4*(35*a^2*cos(d*x + c)^8 - 95*a^2*cos(d*x + c)^6 + 1
2*a^2*cos(d*x + c)^4 + 16*a^2*cos(d*x + c)^2 + 32*a^2)*sin(d*x + c))/d

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Sympy [A]  time = 20.8176, size = 214, normalized size = 1.96 \begin{align*} \begin{cases} \frac{8 a^{2} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac{a^{2} \sin ^{8}{\left (c + d x \right )}}{12 d} + \frac{4 a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac{8 a^{2} \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac{a^{2} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac{4 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac{a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{2 d} + \frac{a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{2} \sin ^{2}{\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)**2*(a+a*sin(d*x+c))**2,x)

[Out]

Piecewise((8*a**2*sin(c + d*x)**9/(315*d) + a**2*sin(c + d*x)**8/(12*d) + 4*a**2*sin(c + d*x)**7*cos(c + d*x)*
*2/(35*d) + 8*a**2*sin(c + d*x)**7/(105*d) + a**2*sin(c + d*x)**6*cos(c + d*x)**2/(3*d) + a**2*sin(c + d*x)**5
*cos(c + d*x)**4/(5*d) + 4*a**2*sin(c + d*x)**5*cos(c + d*x)**2/(15*d) + a**2*sin(c + d*x)**4*cos(c + d*x)**4/
(2*d) + a**2*sin(c + d*x)**3*cos(c + d*x)**4/(3*d), Ne(d, 0)), (x*(a*sin(c) + a)**2*sin(c)**2*cos(c)**5, True)
)

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Giac [A]  time = 1.22668, size = 204, normalized size = 1.87 \begin{align*} \frac{a^{2} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} + \frac{a^{2} \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac{a^{2} \cos \left (4 \, d x + 4 \, c\right )}{128 \, d} - \frac{3 \, a^{2} \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{a^{2} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac{3 \, a^{2} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac{a^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac{a^{2} \sin \left (3 \, d x + 3 \, c\right )}{96 \, d} + \frac{13 \, a^{2} \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)^2*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/512*a^2*cos(8*d*x + 8*c)/d + 1/192*a^2*cos(6*d*x + 6*c)/d - 1/128*a^2*cos(4*d*x + 4*c)/d - 3/64*a^2*cos(2*d*
x + 2*c)/d + 1/2304*a^2*sin(9*d*x + 9*c)/d - 3/1792*a^2*sin(7*d*x + 7*c)/d - 1/80*a^2*sin(5*d*x + 5*c)/d - 1/9
6*a^2*sin(3*d*x + 3*c)/d + 13/128*a^2*sin(d*x + c)/d