3.588 \(\int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=209 \[ -\frac{2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac{4 a^2 \cos ^9(c+d x)}{9 d}-\frac{2 a^2 \cos ^7(c+d x)}{7 d}-\frac{a^2 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac{17 a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac{17 a^2 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{17 a^2 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac{17 a^2 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{17 a^2 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{17 a^2 x}{1024} \]

[Out]

(17*a^2*x)/1024 - (2*a^2*Cos[c + d*x]^7)/(7*d) + (4*a^2*Cos[c + d*x]^9)/(9*d) - (2*a^2*Cos[c + d*x]^11)/(11*d)
+ (17*a^2*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (17*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + (17*a^2*Cos[c
+ d*x]^5*Sin[c + d*x])/(1920*d) - (17*a^2*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (17*a^2*Cos[c + d*x]^7*Sin[c
+ d*x]^3)/(120*d) - (a^2*Cos[c + d*x]^7*Sin[c + d*x]^5)/(12*d)

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Rubi [A]  time = 0.395716, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2873, 2568, 2635, 8, 2565, 270} \[ -\frac{2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac{4 a^2 \cos ^9(c+d x)}{9 d}-\frac{2 a^2 \cos ^7(c+d x)}{7 d}-\frac{a^2 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac{17 a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac{17 a^2 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{17 a^2 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac{17 a^2 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{17 a^2 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{17 a^2 x}{1024} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]

[Out]

(17*a^2*x)/1024 - (2*a^2*Cos[c + d*x]^7)/(7*d) + (4*a^2*Cos[c + d*x]^9)/(9*d) - (2*a^2*Cos[c + d*x]^11)/(11*d)
+ (17*a^2*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (17*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + (17*a^2*Cos[c
+ d*x]^5*Sin[c + d*x])/(1920*d) - (17*a^2*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (17*a^2*Cos[c + d*x]^7*Sin[c
+ d*x]^3)/(120*d) - (a^2*Cos[c + d*x]^7*Sin[c + d*x]^5)/(12*d)

Rule 2873

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

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]

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cos ^6(c+d x) \sin ^4(c+d x)+2 a^2 \cos ^6(c+d x) \sin ^5(c+d x)+a^2 \cos ^6(c+d x) \sin ^6(c+d x)\right ) \, dx\\ &=a^2 \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+a^2 \int \cos ^6(c+d x) \sin ^6(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx\\ &=-\frac{a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}-\frac{a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac{1}{10} \left (3 a^2\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\frac{1}{12} \left (5 a^2\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{3 a^2 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac{a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac{1}{80} \left (3 a^2\right ) \int \cos ^6(c+d x) \, dx+\frac{1}{8} a^2 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{2 a^2 \cos ^7(c+d x)}{7 d}+\frac{4 a^2 \cos ^9(c+d x)}{9 d}-\frac{2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac{17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac{a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac{1}{64} a^2 \int \cos ^6(c+d x) \, dx+\frac{1}{32} a^2 \int \cos ^4(c+d x) \, dx\\ &=-\frac{2 a^2 \cos ^7(c+d x)}{7 d}+\frac{4 a^2 \cos ^9(c+d x)}{9 d}-\frac{2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac{17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac{a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac{1}{384} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{128} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{2 a^2 \cos ^7(c+d x)}{7 d}+\frac{4 a^2 \cos ^9(c+d x)}{9 d}-\frac{2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac{3 a^2 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{17 a^2 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac{17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac{17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac{a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac{1}{512} \left (5 a^2\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{256} \left (3 a^2\right ) \int 1 \, dx\\ &=\frac{3 a^2 x}{256}-\frac{2 a^2 \cos ^7(c+d x)}{7 d}+\frac{4 a^2 \cos ^9(c+d x)}{9 d}-\frac{2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac{17 a^2 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac{17 a^2 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac{17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac{17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac{a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac{\left (5 a^2\right ) \int 1 \, dx}{1024}\\ &=\frac{17 a^2 x}{1024}-\frac{2 a^2 \cos ^7(c+d x)}{7 d}+\frac{4 a^2 \cos ^9(c+d x)}{9 d}-\frac{2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac{17 a^2 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac{17 a^2 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac{17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac{17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac{a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}\\ \end{align*}

Mathematica [A]  time = 1.34498, size = 136, normalized size = 0.65 \[ \frac{a^2 (55440 \sin (2 (c+d x))-162855 \sin (4 (c+d x))-27720 \sin (6 (c+d x))+24255 \sin (8 (c+d x))+5544 \sin (10 (c+d x))-1155 \sin (12 (c+d x))-554400 \cos (c+d x)-184800 \cos (3 (c+d x))+55440 \cos (5 (c+d x))+39600 \cos (7 (c+d x))-6160 \cos (9 (c+d x))-5040 \cos (11 (c+d x))+166320 c+471240 d x)}{28385280 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]

[Out]

(a^2*(166320*c + 471240*d*x - 554400*Cos[c + d*x] - 184800*Cos[3*(c + d*x)] + 55440*Cos[5*(c + d*x)] + 39600*C
os[7*(c + d*x)] - 6160*Cos[9*(c + d*x)] - 5040*Cos[11*(c + d*x)] + 55440*Sin[2*(c + d*x)] - 162855*Sin[4*(c +
d*x)] - 27720*Sin[6*(c + d*x)] + 24255*Sin[8*(c + d*x)] + 5544*Sin[10*(c + d*x)] - 1155*Sin[12*(c + d*x)]))/(2
8385280*d)

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Maple [A]  time = 0.04, size = 238, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{12}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{24}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{64}}+{\frac{\sin \left ( dx+c \right ) }{384} \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}{1024}}+{\frac{5\,c}{1024}} \right ) +2\,{a}^{2} \left ( -1/11\, \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{99}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{693}} \right ) +{a}^{2} \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 ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x)

[Out]

1/d*(a^2*(-1/12*sin(d*x+c)^5*cos(d*x+c)^7-1/24*sin(d*x+c)^3*cos(d*x+c)^7-1/64*sin(d*x+c)*cos(d*x+c)^7+1/384*(c
os(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/1024*d*x+5/1024*c)+2*a^2*(-1/11*sin(d*x+c)^4*cos(d*
x+c)^7-4/99*sin(d*x+c)^2*cos(d*x+c)^7-8/693*cos(d*x+c)^7)+a^2*(-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))

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Maxima [A]  time = 1.04498, size = 186, normalized size = 0.89 \begin{align*} -\frac{81920 \,{\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{2} - 2772 \,{\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^{2} - 1155 \,{\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{28385280 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/28385280*(81920*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^2 - 2772*(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^2 - 1155*(4*sin(4*d*x + 4*c)^3 + 120*d*x
+ 120*c + 9*sin(8*d*x + 8*c) - 48*sin(4*d*x + 4*c))*a^2)/d

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Fricas [A]  time = 1.28049, size = 389, normalized size = 1.86 \begin{align*} -\frac{645120 \, a^{2} \cos \left (d x + c\right )^{11} - 1576960 \, a^{2} \cos \left (d x + c\right )^{9} + 1013760 \, a^{2} \cos \left (d x + c\right )^{7} - 58905 \, a^{2} d x + 231 \,{\left (1280 \, a^{2} \cos \left (d x + c\right )^{11} - 4736 \, a^{2} \cos \left (d x + c\right )^{9} + 4272 \, a^{2} \cos \left (d x + c\right )^{7} - 136 \, a^{2} \cos \left (d x + c\right )^{5} - 170 \, a^{2} \cos \left (d x + c\right )^{3} - 255 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/3548160*(645120*a^2*cos(d*x + c)^11 - 1576960*a^2*cos(d*x + c)^9 + 1013760*a^2*cos(d*x + c)^7 - 58905*a^2*d
*x + 231*(1280*a^2*cos(d*x + c)^11 - 4736*a^2*cos(d*x + c)^9 + 4272*a^2*cos(d*x + c)^7 - 136*a^2*cos(d*x + c)^
5 - 170*a^2*cos(d*x + c)^3 - 255*a^2*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 89.7845, size = 656, normalized size = 3.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**4*(a+a*sin(d*x+c))**2,x)

[Out]

Piecewise((5*a**2*x*sin(c + d*x)**12/1024 + 15*a**2*x*sin(c + d*x)**10*cos(c + d*x)**2/512 + 3*a**2*x*sin(c +
d*x)**10/256 + 75*a**2*x*sin(c + d*x)**8*cos(c + d*x)**4/1024 + 15*a**2*x*sin(c + d*x)**8*cos(c + d*x)**2/256
+ 25*a**2*x*sin(c + d*x)**6*cos(c + d*x)**6/256 + 15*a**2*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 75*a**2*x*si
n(c + d*x)**4*cos(c + d*x)**8/1024 + 15*a**2*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 15*a**2*x*sin(c + d*x)**2
*cos(c + d*x)**10/512 + 15*a**2*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 5*a**2*x*cos(c + d*x)**12/1024 + 3*a**
2*x*cos(c + d*x)**10/256 + 5*a**2*sin(c + d*x)**11*cos(c + d*x)/(1024*d) + 85*a**2*sin(c + d*x)**9*cos(c + d*x
)**3/(3072*d) + 3*a**2*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 33*a**2*sin(c + d*x)**7*cos(c + d*x)**5/(512*d)
+ 7*a**2*sin(c + d*x)**7*cos(c + d*x)**3/(128*d) - 33*a**2*sin(c + d*x)**5*cos(c + d*x)**7/(512*d) + a**2*sin(
c + d*x)**5*cos(c + d*x)**5/(10*d) - 2*a**2*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 85*a**2*sin(c + d*x)**3*co
s(c + d*x)**9/(3072*d) - 7*a**2*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) - 8*a**2*sin(c + d*x)**2*cos(c + d*x)*
*9/(63*d) - 5*a**2*sin(c + d*x)*cos(c + d*x)**11/(1024*d) - 3*a**2*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 16*a
**2*cos(c + d*x)**11/(693*d), Ne(d, 0)), (x*(a*sin(c) + a)**2*sin(c)**4*cos(c)**6, True))

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Giac [A]  time = 1.25871, size = 281, normalized size = 1.34 \begin{align*} \frac{17}{1024} \, a^{2} x - \frac{a^{2} \cos \left (11 \, d x + 11 \, c\right )}{5632 \, d} - \frac{a^{2} \cos \left (9 \, d x + 9 \, c\right )}{4608 \, d} + \frac{5 \, a^{2} \cos \left (7 \, d x + 7 \, c\right )}{3584 \, d} + \frac{a^{2} \cos \left (5 \, d x + 5 \, c\right )}{512 \, d} - \frac{5 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{768 \, d} - \frac{5 \, a^{2} \cos \left (d x + c\right )}{256 \, d} - \frac{a^{2} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} + \frac{a^{2} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{7 \, a^{2} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac{a^{2} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac{47 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac{a^{2} \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)^4*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

17/1024*a^2*x - 1/5632*a^2*cos(11*d*x + 11*c)/d - 1/4608*a^2*cos(9*d*x + 9*c)/d + 5/3584*a^2*cos(7*d*x + 7*c)/
d + 1/512*a^2*cos(5*d*x + 5*c)/d - 5/768*a^2*cos(3*d*x + 3*c)/d - 5/256*a^2*cos(d*x + c)/d - 1/24576*a^2*sin(1
2*d*x + 12*c)/d + 1/5120*a^2*sin(10*d*x + 10*c)/d + 7/8192*a^2*sin(8*d*x + 8*c)/d - 1/1024*a^2*sin(6*d*x + 6*c
)/d - 47/8192*a^2*sin(4*d*x + 4*c)/d + 1/512*a^2*sin(2*d*x + 2*c)/d