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

Optimal. Leaf size=109 \[ -\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5 a x}{128} \]

[Out]

(5*a*x)/128 - (a*Cos[c + d*x]^7)/(7*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*a*Cos[c + d*x]^3*Sin[c +
d*x])/(192*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)

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Rubi [A]  time = 0.119503, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2838, 2565, 30, 2568, 2635, 8} \[ -\frac{a \cos ^7(c+d x)}{7 d}-\frac{a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5 a x}{128} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a*x)/128 - (a*Cos[c + d*x]^7)/(7*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*a*Cos[c + d*x]^3*Sin[c +
d*x])/(192*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a*Cos[c + d*x]^7*Sin[c + d*x])/(8*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 (c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^6(c+d x) \sin (c+d x) \, dx+a \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx\\ &=-\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{8} a \int \cos ^6(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int x^6 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{48} (5 a) \int \cos ^4(c+d x) \, dx\\ &=-\frac{a \cos ^7(c+d x)}{7 d}+\frac{5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{64} (5 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac{a \cos ^7(c+d x)}{7 d}+\frac{5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac{5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{128} (5 a) \int 1 \, dx\\ &=\frac{5 a x}{128}-\frac{a \cos ^7(c+d x)}{7 d}+\frac{5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac{5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{a \cos ^7(c+d x) \sin (c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.261617, size = 91, normalized size = 0.83 \[ -\frac{a (-336 \sin (2 (c+d x))+168 \sin (4 (c+d x))+112 \sin (6 (c+d x))+21 \sin (8 (c+d x))+1680 \cos (c+d x)+1008 \cos (3 (c+d x))+336 \cos (5 (c+d x))+48 \cos (7 (c+d x))-840 d x)}{21504 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(-840*d*x + 1680*Cos[c + d*x] + 1008*Cos[3*(c + d*x)] + 336*Cos[5*(c + d*x)] + 48*Cos[7*(c + d*x)] - 336*S
in[2*(c + d*x)] + 168*Sin[4*(c + d*x)] + 112*Sin[6*(c + d*x)] + 21*Sin[8*(c + d*x)]))/(21504*d)

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Maple [A]  time = 0.025, size = 78, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8}}+{\frac{\sin \left ( dx+c \right ) }{48} \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{5\,dx}{128}}+{\frac{5\,c}{128}} \right ) -{\frac{a \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)),x)

[Out]

1/d*(a*(-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*cos(d*x+c)^7)

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Maxima [A]  time = 1.03327, size = 85, normalized size = 0.78 \begin{align*} -\frac{3072 \, a \cos \left (d x + c\right )^{7} - 7 \,{\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}{21504 \, 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)),x, algorithm="maxima")

[Out]

-1/21504*(3072*a*cos(d*x + c)^7 - 7*(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)/d

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Fricas [A]  time = 1.19186, size = 200, normalized size = 1.83 \begin{align*} -\frac{384 \, a \cos \left (d x + c\right )^{7} - 105 \, a d x + 7 \,{\left (48 \, 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 )}{2688 \, 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)),x, algorithm="fricas")

[Out]

-1/2688*(384*a*cos(d*x + c)^7 - 105*a*d*x + 7*(48*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 = 11.5849, size = 223, normalized size = 2.05 \begin{align*} \begin{cases} \frac{5 a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac{5 a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac{15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{5 a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac{5 a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac{5 a \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{55 a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac{73 a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac{5 a \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac{a \cos ^{7}{\left (c + d x \right )}}{7 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \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)),x)

[Out]

Piecewise((5*a*x*sin(c + d*x)**8/128 + 5*a*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 15*a*x*sin(c + d*x)**4*cos(c
+ d*x)**4/64 + 5*a*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 5*a*x*cos(c + d*x)**8/128 + 5*a*sin(c + d*x)**7*cos
(c + d*x)/(128*d) + 55*a*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 73*a*sin(c + d*x)**3*cos(c + d*x)**5/(384*d
) - 5*a*sin(c + d*x)*cos(c + d*x)**7/(128*d) - a*cos(c + d*x)**7/(7*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)*co
s(c)**6, True))

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Giac [A]  time = 1.16715, size = 165, normalized size = 1.51 \begin{align*} \frac{5}{128} \, a x - \frac{a \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac{a \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac{3 \, a \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac{5 \, a \cos \left (d x + c\right )}{64 \, d} - \frac{a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac{a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac{a \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{a \sin \left (2 \, d x + 2 \, c\right )}{64 \, 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)),x, algorithm="giac")

[Out]

5/128*a*x - 1/448*a*cos(7*d*x + 7*c)/d - 1/64*a*cos(5*d*x + 5*c)/d - 3/64*a*cos(3*d*x + 3*c)/d - 5/64*a*cos(d*
x + c)/d - 1/1024*a*sin(8*d*x + 8*c)/d - 1/192*a*sin(6*d*x + 6*c)/d - 1/128*a*sin(4*d*x + 4*c)/d + 1/64*a*sin(
2*d*x + 2*c)/d