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

Optimal. Leaf size=181 \[ -\frac{33 a^3 \cos ^7(c+d x)}{560 d}-\frac{11 \cos ^7(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{240 d}+\frac{11 a^3 \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{11 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{33 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{33 a^3 x}{256}-\frac{\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}-\frac{a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{30 d} \]

[Out]

(33*a^3*x)/256 - (33*a^3*Cos[c + d*x]^7)/(560*d) + (33*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (11*a^3*Cos[c
+ d*x]^3*Sin[c + d*x])/(128*d) + (11*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) - (a*Cos[c + d*x]^7*(a + a*Sin[c
+ d*x])^2)/(30*d) - (Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3)/(10*d) - (11*Cos[c + d*x]^7*(a^3 + a^3*Sin[c + d*
x]))/(240*d)

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Rubi [A]  time = 0.203746, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2860, 2678, 2669, 2635, 8} \[ -\frac{33 a^3 \cos ^7(c+d x)}{560 d}-\frac{11 \cos ^7(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{240 d}+\frac{11 a^3 \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{11 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{33 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{33 a^3 x}{256}-\frac{\cos ^7(c+d x) (a \sin (c+d x)+a)^3}{10 d}-\frac{a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{30 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(33*a^3*x)/256 - (33*a^3*Cos[c + d*x]^7)/(560*d) + (33*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (11*a^3*Cos[c
+ d*x]^3*Sin[c + d*x])/(128*d) + (11*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) - (a*Cos[c + d*x]^7*(a + a*Sin[c
+ d*x])^2)/(30*d) - (Cos[c + d*x]^7*(a + a*Sin[c + d*x])^3)/(10*d) - (11*Cos[c + d*x]^7*(a^3 + a^3*Sin[c + d*
x]))/(240*d)

Rule 2860

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[(d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(f*g*(m + p + 1)), x]
+ Dist[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; Fre
eQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 1, 0]

Rule 2678

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

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

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 (c+d x) (a+a \sin (c+d x))^3 \, dx &=-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}+\frac{3}{10} \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 \, dx\\ &=-\frac{a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}+\frac{1}{30} (11 a) \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac{a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac{11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d}+\frac{1}{80} \left (33 a^2\right ) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac{33 a^3 \cos ^7(c+d x)}{560 d}-\frac{a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac{11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d}+\frac{1}{80} \left (33 a^3\right ) \int \cos ^6(c+d x) \, dx\\ &=-\frac{33 a^3 \cos ^7(c+d x)}{560 d}+\frac{11 a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac{a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac{11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d}+\frac{1}{32} \left (11 a^3\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{33 a^3 \cos ^7(c+d x)}{560 d}+\frac{11 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{11 a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac{a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac{11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d}+\frac{1}{128} \left (33 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{33 a^3 \cos ^7(c+d x)}{560 d}+\frac{33 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{11 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{11 a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac{a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac{11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d}+\frac{1}{256} \left (33 a^3\right ) \int 1 \, dx\\ &=\frac{33 a^3 x}{256}-\frac{33 a^3 \cos ^7(c+d x)}{560 d}+\frac{33 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{11 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{11 a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac{a \cos ^7(c+d x) (a+a \sin (c+d x))^2}{30 d}-\frac{\cos ^7(c+d x) (a+a \sin (c+d x))^3}{10 d}-\frac{11 \cos ^7(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{240 d}\\ \end{align*}

Mathematica [A]  time = 0.925399, size = 116, normalized size = 0.64 \[ \frac{a^3 (10500 \sin (2 (c+d x))-5880 \sin (4 (c+d x))-3570 \sin (6 (c+d x))-525 \sin (8 (c+d x))+42 \sin (10 (c+d x))-31920 \cos (c+d x)-16800 \cos (3 (c+d x))-3360 \cos (5 (c+d x))+600 \cos (7 (c+d x))+280 \cos (9 (c+d x))+31500 c+27720 d x)}{215040 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(31500*c + 27720*d*x - 31920*Cos[c + d*x] - 16800*Cos[3*(c + d*x)] - 3360*Cos[5*(c + d*x)] + 600*Cos[7*(c
+ d*x)] + 280*Cos[9*(c + d*x)] + 10500*Sin[2*(c + d*x)] - 5880*Sin[4*(c + d*x)] - 3570*Sin[6*(c + d*x)] - 525
*Sin[8*(c + d*x)] + 42*Sin[10*(c + d*x)]))/(215040*d)

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Maple [A]  time = 0.038, size = 198, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({a}^{3} \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 ) +3\,{a}^{3} \left ( -1/9\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) +3\,{a}^{3} \left ( -1/8\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}+1/48\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) \sin \left ( dx+c \right ) +{\frac{5\,dx}{128}}+{\frac{5\,c}{128}} \right ) -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^3*(-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)+3*a^3*(-1/9*sin(d*x+c)^2*cos(d*x+c)^7-2/63*cos(d*x+c)^7)+3*a^3*(-
1/8*sin(d*x+c)*cos(d*x+c)^7+1/48*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/128*d*x+5/128*c)
-1/7*a^3*cos(d*x+c)^7)

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Maxima [A]  time = 1.08366, size = 190, normalized size = 1.05 \begin{align*} -\frac{30720 \, a^{3} \cos \left (d x + c\right )^{7} - 10240 \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 21 \,{\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^{3} - 210 \,{\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{215040 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/215040*(30720*a^3*cos(d*x + c)^7 - 10240*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^3 - 21*(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^3 - 210*(64*sin(2*d*x + 2*c)^3 + 120*d*x +
120*c - 3*sin(8*d*x + 8*c) - 24*sin(4*d*x + 4*c))*a^3)/d

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Fricas [A]  time = 1.23707, size = 296, normalized size = 1.64 \begin{align*} \frac{8960 \, a^{3} \cos \left (d x + c\right )^{9} - 15360 \, a^{3} \cos \left (d x + c\right )^{7} + 3465 \, a^{3} d x + 21 \,{\left (128 \, a^{3} \cos \left (d x + c\right )^{9} - 656 \, a^{3} \cos \left (d x + c\right )^{7} + 88 \, a^{3} \cos \left (d x + c\right )^{5} + 110 \, a^{3} \cos \left (d x + c\right )^{3} + 165 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{26880 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/26880*(8960*a^3*cos(d*x + c)^9 - 15360*a^3*cos(d*x + c)^7 + 3465*a^3*d*x + 21*(128*a^3*cos(d*x + c)^9 - 656*
a^3*cos(d*x + c)^7 + 88*a^3*cos(d*x + c)^5 + 110*a^3*cos(d*x + c)^3 + 165*a^3*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 34.385, size = 542, normalized size = 2.99 \begin{align*} \begin{cases} \frac{3 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac{15 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac{15 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac{15 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac{15 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac{15 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac{45 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac{15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac{3 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac{15 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac{3 a^{3} \sin ^{9}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{256 d} + \frac{7 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac{15 a^{3} \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac{55 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac{7 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac{73 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac{3 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac{3 a^{3} \sin{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac{15 a^{3} \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac{2 a^{3} \cos ^{9}{\left (c + d x \right )}}{21 d} - \frac{a^{3} \cos ^{7}{\left (c + d x \right )}}{7 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{3} \sin{\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)*(a+a*sin(d*x+c))**3,x)

[Out]

Piecewise((3*a**3*x*sin(c + d*x)**10/256 + 15*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 15*a**3*x*sin(c + d
*x)**8/128 + 15*a**3*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 15*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 15
*a**3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 45*a**3*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*a**3*x*sin(c +
d*x)**2*cos(c + d*x)**8/256 + 15*a**3*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 3*a**3*x*cos(c + d*x)**10/256 +
15*a**3*x*cos(c + d*x)**8/128 + 3*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a**3*sin(c + d*x)**7*cos(c + d
*x)**3/(128*d) + 15*a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) + a**3*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) +
55*a**3*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) - 7*a**3*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) + 73*a**3*sin
(c + d*x)**3*cos(c + d*x)**5/(128*d) - 3*a**3*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 3*a**3*sin(c + d*x)*cos(
c + d*x)**9/(256*d) - 15*a**3*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 2*a**3*cos(c + d*x)**9/(21*d) - a**3*cos(
c + d*x)**7/(7*d), Ne(d, 0)), (x*(a*sin(c) + a)**3*sin(c)*cos(c)**6, True))

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Giac [A]  time = 1.2254, size = 235, normalized size = 1.3 \begin{align*} \frac{33}{256} \, a^{3} x + \frac{a^{3} \cos \left (9 \, d x + 9 \, c\right )}{768 \, d} + \frac{5 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac{a^{3} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac{5 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac{19 \, a^{3} \cos \left (d x + c\right )}{128 \, d} + \frac{a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac{5 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac{17 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac{7 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac{25 \, a^{3} \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)*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

33/256*a^3*x + 1/768*a^3*cos(9*d*x + 9*c)/d + 5/1792*a^3*cos(7*d*x + 7*c)/d - 1/64*a^3*cos(5*d*x + 5*c)/d - 5/
64*a^3*cos(3*d*x + 3*c)/d - 19/128*a^3*cos(d*x + c)/d + 1/5120*a^3*sin(10*d*x + 10*c)/d - 5/2048*a^3*sin(8*d*x
+ 8*c)/d - 17/1024*a^3*sin(6*d*x + 6*c)/d - 7/256*a^3*sin(4*d*x + 4*c)/d + 25/512*a^3*sin(2*d*x + 2*c)/d