3.574 \(\int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=125 \[ \frac {a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5 a x}{128} \]

[Out]

5/128*a*x-1/7*a*cos(d*x+c)^7/d+1/9*a*cos(d*x+c)^9/d+5/128*a*cos(d*x+c)*sin(d*x+c)/d+5/192*a*cos(d*x+c)^3*sin(d
*x+c)/d+1/48*a*cos(d*x+c)^5*sin(d*x+c)/d-1/8*a*cos(d*x+c)^7*sin(d*x+c)/d

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Rubi [A]  time = 0.15, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2838, 2568, 2635, 8, 2565, 14} \[ \frac {a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 a \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5 a x}{128} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

(5*a*x)/128 - (a*Cos[c + d*x]^7)/(7*d) + (a*Cos[c + d*x]^9)/(9*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 8

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

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 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 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 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]

Rubi steps

\begin {align*} \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+a \int \cos ^6(c+d x) \sin ^3(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 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{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 \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 {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 {a \cos ^9(c+d x)}{9 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 {a \cos ^9(c+d x)}{9 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*}

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Mathematica [A]  time = 0.32, size = 91, normalized size = 0.73 \[ \frac {a (1008 \sin (2 (c+d x))-504 \sin (4 (c+d x))-336 \sin (6 (c+d x))-63 \sin (8 (c+d x))-1512 \cos (c+d x)-672 \cos (3 (c+d x))+108 \cos (7 (c+d x))+28 \cos (9 (c+d x))+2520 d x)}{64512 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

(a*(2520*d*x - 1512*Cos[c + d*x] - 672*Cos[3*(c + d*x)] + 108*Cos[7*(c + d*x)] + 28*Cos[9*(c + d*x)] + 1008*Si
n[2*(c + d*x)] - 504*Sin[4*(c + d*x)] - 336*Sin[6*(c + d*x)] - 63*Sin[8*(c + d*x)]))/(64512*d)

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fricas [A]  time = 0.76, size = 84, normalized size = 0.67 \[ \frac {896 \, a \cos \left (d x + c\right )^{9} - 1152 \, a \cos \left (d x + c\right )^{7} + 315 \, a d x - 21 \, {\left (48 \, a \cos \left (d x + c\right )^{7} - 8 \, a \cos \left (d x + c\right )^{5} - 10 \, a \cos \left (d x + c\right )^{3} - 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/8064*(896*a*cos(d*x + c)^9 - 1152*a*cos(d*x + c)^7 + 315*a*d*x - 21*(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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giac [A]  time = 0.26, size = 122, normalized size = 0.98 \[ \frac {5}{128} \, a x + \frac {a \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {3 \, a \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{96 \, d} - \frac {3 \, a \cos \left (d x + c\right )}{128 \, d} - \frac {a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

5/128*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 - 1/1024*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

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maple [A]  time = 0.24, size = 98, normalized size = 0.78 \[ \frac {a \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+a \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c)),x)

[Out]

1/d*(a*(-1/9*sin(d*x+c)^2*cos(d*x+c)^7-2/63*cos(d*x+c)^7)+a*(-1/8*cos(d*x+c)^7*sin(d*x+c)+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))

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maxima [A]  time = 0.50, size = 76, normalized size = 0.61 \[ \frac {1024 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a + 21 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{64512 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/64512*(1024*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a + 21*(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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mupad [B]  time = 12.36, size = 386, normalized size = 3.09 \[ \frac {5\,a\,x}{128}+\frac {\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}-\frac {191\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{96}+\left (\frac {a\,\left (11340\,c+11340\,d\,x-32256\right )}{8064}-\frac {45\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {83\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\left (\frac {a\,\left (26460\,c+26460\,d\,x+53760\right )}{8064}-\frac {105\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {145\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}+\left (\frac {a\,\left (39690\,c+39690\,d\,x-161280\right )}{8064}-\frac {315\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\left (\frac {a\,\left (39690\,c+39690\,d\,x+96768\right )}{8064}-\frac {315\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {145\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\left (\frac {a\,\left (26460\,c+26460\,d\,x-96768\right )}{8064}-\frac {105\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {83\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}+\left (\frac {a\,\left (11340\,c+11340\,d\,x+13824\right )}{8064}-\frac {45\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {191\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\left (\frac {a\,\left (2835\,c+2835\,d\,x-4608\right )}{8064}-\frac {45\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {a\,\left (315\,c+315\,d\,x-512\right )}{8064}-\frac {5\,a\,\left (c+d\,x\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6*sin(c + d*x)^2*(a + a*sin(c + d*x)),x)

[Out]

(5*a*x)/128 + ((a*(315*c + 315*d*x - 512))/8064 - (5*a*tan(c/2 + (d*x)/2))/64 - (5*a*(c + d*x))/128 + tan(c/2
+ (d*x)/2)^2*((a*(2835*c + 2835*d*x - 4608))/8064 - (45*a*(c + d*x))/128) + tan(c/2 + (d*x)/2)^4*((a*(11340*c
+ 11340*d*x + 13824))/8064 - (45*a*(c + d*x))/32) + tan(c/2 + (d*x)/2)^14*((a*(11340*c + 11340*d*x - 32256))/8
064 - (45*a*(c + d*x))/32) + tan(c/2 + (d*x)/2)^12*((a*(26460*c + 26460*d*x + 53760))/8064 - (105*a*(c + d*x))
/32) + tan(c/2 + (d*x)/2)^6*((a*(26460*c + 26460*d*x - 96768))/8064 - (105*a*(c + d*x))/32) + tan(c/2 + (d*x)/
2)^8*((a*(39690*c + 39690*d*x + 96768))/8064 - (315*a*(c + d*x))/64) + tan(c/2 + (d*x)/2)^10*((a*(39690*c + 39
690*d*x - 161280))/8064 - (315*a*(c + d*x))/64) + (191*a*tan(c/2 + (d*x)/2)^3)/96 - (83*a*tan(c/2 + (d*x)/2)^5
)/32 + (145*a*tan(c/2 + (d*x)/2)^7)/32 - (145*a*tan(c/2 + (d*x)/2)^11)/32 + (83*a*tan(c/2 + (d*x)/2)^13)/32 -
(191*a*tan(c/2 + (d*x)/2)^15)/96 + (5*a*tan(c/2 + (d*x)/2)^17)/64)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^9)

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sympy [A]  time = 18.97, size = 248, normalized size = 1.98 \[ \begin {cases} \frac {5 a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {5 a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {5 a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {5 a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {5 a \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {73 a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {5 a \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {2 a \cos ^{9}{\left (c + d x \right )}}{63 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \sin ^{2}{\relax (c )} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**2*(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
) - a*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 5*a*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 2*a*cos(c + d*x)**9/(
63*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**2*cos(c)**6, True))

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