3.573 \(\int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=149 \[ \frac{a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac{3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac{3 a x}{256} \]

[Out]

(3*a*x)/256 - (a*Cos[c + d*x]^7)/(7*d) + (a*Cos[c + d*x]^9)/(9*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(256*d) +
(a*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) - (3*a*Cos[c + d*x]^7*Sin[c
+ d*x])/(80*d) - (a*Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*d)

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Rubi [A]  time = 0.206155, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2838, 2565, 14, 2568, 2635, 8} \[ \frac{a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac{3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac{3 a x}{256} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]

[Out]

(3*a*x)/256 - (a*Cos[c + d*x]^7)/(7*d) + (a*Cos[c + d*x]^9)/(9*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(256*d) +
(a*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) - (3*a*Cos[c + d*x]^7*Sin[c
+ d*x])/(80*d) - (a*Cos[c + d*x]^7*Sin[c + d*x]^3)/(10*d)

Rule 2838

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

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 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^6(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+a \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx\\ &=-\frac{a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac{1}{10} (3 a) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac{1}{80} (3 a) \int \cos ^6(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^9(c+d x)}{9 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac{3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac{1}{32} a \int \cos ^4(c+d x) \, dx\\ &=-\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^9(c+d x)}{9 d}+\frac{a \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac{3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac{1}{128} (3 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^9(c+d x)}{9 d}+\frac{3 a \cos (c+d x) \sin (c+d x)}{256 d}+\frac{a \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac{3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac{1}{256} (3 a) \int 1 \, dx\\ &=\frac{3 a x}{256}-\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^9(c+d x)}{9 d}+\frac{3 a \cos (c+d x) \sin (c+d x)}{256 d}+\frac{a \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac{3 a \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{a \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}\\ \end{align*}

Mathematica [A]  time = 0.336643, size = 101, normalized size = 0.68 \[ \frac{a (1260 \sin (2 (c+d x))-2520 \sin (4 (c+d x))-630 \sin (6 (c+d x))+315 \sin (8 (c+d x))+126 \sin (10 (c+d x))-15120 \cos (c+d x)-6720 \cos (3 (c+d x))+1080 \cos (7 (c+d x))+280 \cos (9 (c+d x))+7560 d x)}{645120 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + a*Sin[c + d*x]),x]

[Out]

(a*(7560*d*x - 15120*Cos[c + d*x] - 6720*Cos[3*(c + d*x)] + 1080*Cos[7*(c + d*x)] + 280*Cos[9*(c + d*x)] + 126
0*Sin[2*(c + d*x)] - 2520*Sin[4*(c + d*x)] - 630*Sin[6*(c + d*x)] + 315*Sin[8*(c + d*x)] + 126*Sin[10*(c + d*x
)]))/(645120*d)

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Maple [A]  time = 0.033, size = 116, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{10}}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80}}+{\frac{\sin \left ( dx+c \right ) }{160} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{3\,dx}{256}}+{\frac{3\,c}{256}} \right ) +a \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*sin(d*x+c)^3*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/10*sin(d*x+c)^3*cos(d*x+c)^7-3/80*sin(d*x+c)*cos(d*x+c)^7+1/160*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8
*cos(d*x+c))*sin(d*x+c)+3/256*d*x+3/256*c)+a*(-1/9*sin(d*x+c)^2*cos(d*x+c)^7-2/63*cos(d*x+c)^7))

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Maxima [A]  time = 1.06026, size = 103, normalized size = 0.69 \begin{align*} \frac{10240 \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a + 63 \,{\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{645120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/645120*(10240*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a + 63*(32*sin(2*d*x + 2*c)^5 + 120*d*x + 120*c + 5*sin(
8*d*x + 8*c) - 40*sin(4*d*x + 4*c))*a)/d

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Fricas [A]  time = 1.18549, size = 269, normalized size = 1.81 \begin{align*} \frac{8960 \, a \cos \left (d x + c\right )^{9} - 11520 \, a \cos \left (d x + c\right )^{7} + 945 \, a d x + 63 \,{\left (128 \, a \cos \left (d x + c\right )^{9} - 176 \, a \cos \left (d x + c\right )^{7} + 8 \, a \cos \left (d x + c\right )^{5} + 10 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/80640*(8960*a*cos(d*x + c)^9 - 11520*a*cos(d*x + c)^7 + 945*a*d*x + 63*(128*a*cos(d*x + c)^9 - 176*a*cos(d*x
+ c)^7 + 8*a*cos(d*x + c)^5 + 10*a*cos(d*x + c)^3 + 15*a*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 32.6663, size = 294, normalized size = 1.97 \begin{align*} \begin{cases} \frac{3 a x \sin ^{10}{\left (c + d x \right )}}{256} + \frac{15 a x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac{15 a x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac{15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac{15 a x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac{3 a x \cos ^{10}{\left (c + d x \right )}}{256} + \frac{3 a \sin ^{9}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{256 d} + \frac{7 a \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac{a \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac{7 a \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac{a \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac{3 a \sin{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac{2 a \cos ^{9}{\left (c + d x \right )}}{63 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \sin ^{3}{\left (c \right )} \cos ^{6}{\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)**6*sin(d*x+c)**3*(a+a*sin(d*x+c)),x)

[Out]

Piecewise((3*a*x*sin(c + d*x)**10/256 + 15*a*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 15*a*x*sin(c + d*x)**6*co
s(c + d*x)**4/128 + 15*a*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 15*a*x*sin(c + d*x)**2*cos(c + d*x)**8/256 +
3*a*x*cos(c + d*x)**10/256 + 3*a*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a*sin(c + d*x)**7*cos(c + d*x)**3/(1
28*d) + a*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) - 7*a*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) - a*sin(c + d*x
)**2*cos(c + d*x)**7/(7*d) - 3*a*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 2*a*cos(c + d*x)**9/(63*d), Ne(d, 0)),
(x*(a*sin(c) + a)*sin(c)**3*cos(c)**6, True))

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Giac [A]  time = 1.27674, size = 185, normalized size = 1.24 \begin{align*} \frac{3}{256} \, a x + \frac{a \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac{3 \, a \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac{a \cos \left (3 \, d x + 3 \, c\right )}{96 \, d} - \frac{3 \, a \cos \left (d x + c\right )}{128 \, d} + \frac{a \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{a \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac{a \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac{a \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac{a \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

3/256*a*x + 1/2304*a*cos(9*d*x + 9*c)/d + 3/1792*a*cos(7*d*x + 7*c)/d - 1/96*a*cos(3*d*x + 3*c)/d - 3/128*a*co
s(d*x + c)/d + 1/5120*a*sin(10*d*x + 10*c)/d + 1/2048*a*sin(8*d*x + 8*c)/d - 1/1024*a*sin(6*d*x + 6*c)/d - 1/2
56*a*sin(4*d*x + 4*c)/d + 1/512*a*sin(2*d*x + 2*c)/d