3.996 \(\int \cos ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx\)

Optimal. Leaf size=200 \[ -\frac{3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}-\frac{3 (8 A+3 B) \cos ^5(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{112 d}+\frac{3 a^3 (8 A+3 B) \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{9 a^3 (8 A+3 B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{9}{128} a^3 x (8 A+3 B)-\frac{a (8 A+3 B) \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{56 d}-\frac{B \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d} \]

[Out]

(9*a^3*(8*A + 3*B)*x)/128 - (3*a^3*(8*A + 3*B)*Cos[c + d*x]^5)/(80*d) + (9*a^3*(8*A + 3*B)*Cos[c + d*x]*Sin[c
+ d*x])/(128*d) + (3*a^3*(8*A + 3*B)*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (a*(8*A + 3*B)*Cos[c + d*x]^5*(a +
a*Sin[c + d*x])^2)/(56*d) - (B*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3)/(8*d) - (3*(8*A + 3*B)*Cos[c + d*x]^5*(a
^3 + a^3*Sin[c + d*x]))/(112*d)

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Rubi [A]  time = 0.238976, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {2860, 2678, 2669, 2635, 8} \[ -\frac{3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}-\frac{3 (8 A+3 B) \cos ^5(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{112 d}+\frac{3 a^3 (8 A+3 B) \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{9 a^3 (8 A+3 B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{9}{128} a^3 x (8 A+3 B)-\frac{a (8 A+3 B) \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{56 d}-\frac{B \cos ^5(c+d x) (a \sin (c+d x)+a)^3}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]

[Out]

(9*a^3*(8*A + 3*B)*x)/128 - (3*a^3*(8*A + 3*B)*Cos[c + d*x]^5)/(80*d) + (9*a^3*(8*A + 3*B)*Cos[c + d*x]*Sin[c
+ d*x])/(128*d) + (3*a^3*(8*A + 3*B)*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) - (a*(8*A + 3*B)*Cos[c + d*x]^5*(a +
a*Sin[c + d*x])^2)/(56*d) - (B*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3)/(8*d) - (3*(8*A + 3*B)*Cos[c + d*x]^5*(a
^3 + a^3*Sin[c + d*x]))/(112*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 ^4(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx &=-\frac{B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}+\frac{1}{8} (8 A+3 B) \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx\\ &=-\frac{a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac{B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}+\frac{1}{56} (9 a (8 A+3 B)) \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac{a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac{B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac{3 (8 A+3 B) \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}+\frac{1}{16} \left (3 a^2 (8 A+3 B)\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac{3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}-\frac{a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac{B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac{3 (8 A+3 B) \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}+\frac{1}{16} \left (3 a^3 (8 A+3 B)\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}+\frac{3 a^3 (8 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac{a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac{B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac{3 (8 A+3 B) \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}+\frac{1}{64} \left (9 a^3 (8 A+3 B)\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}+\frac{9 a^3 (8 A+3 B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{3 a^3 (8 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac{a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac{B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac{3 (8 A+3 B) \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}+\frac{1}{128} \left (9 a^3 (8 A+3 B)\right ) \int 1 \, dx\\ &=\frac{9}{128} a^3 (8 A+3 B) x-\frac{3 a^3 (8 A+3 B) \cos ^5(c+d x)}{80 d}+\frac{9 a^3 (8 A+3 B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{3 a^3 (8 A+3 B) \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac{a (8 A+3 B) \cos ^5(c+d x) (a+a \sin (c+d x))^2}{56 d}-\frac{B \cos ^5(c+d x) (a+a \sin (c+d x))^3}{8 d}-\frac{3 (8 A+3 B) \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{112 d}\\ \end{align*}

Mathematica [A]  time = 2.18421, size = 183, normalized size = 0.92 \[ -\frac{a^3 \cos (c+d x) \left (16 (373 A+223 B) \cos (2 (c+d x))+32 (41 A+11 B) \cos (4 (c+d x))+\frac{2520 (8 A+3 B) \sin ^{-1}\left (\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right )}{\sqrt{\cos ^2(c+d x)}}-10640 A \sin (c+d x)+560 A \sin (5 (c+d x))-80 A \cos (6 (c+d x))+4576 A-3045 B \sin (c+d x)+1365 B \sin (3 (c+d x))+595 B \sin (5 (c+d x))-35 B \sin (7 (c+d x))-240 B \cos (6 (c+d x))+2976 B\right )}{17920 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^3*(A + B*Sin[c + d*x]),x]

[Out]

-(a^3*Cos[c + d*x]*(4576*A + 2976*B + (2520*(8*A + 3*B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]])/Sqrt[Cos[c + d
*x]^2] + 16*(373*A + 223*B)*Cos[2*(c + d*x)] + 32*(41*A + 11*B)*Cos[4*(c + d*x)] - 80*A*Cos[6*(c + d*x)] - 240
*B*Cos[6*(c + d*x)] - 10640*A*Sin[c + d*x] - 3045*B*Sin[c + d*x] + 1365*B*Sin[3*(c + d*x)] + 560*A*Sin[5*(c +
d*x)] + 595*B*Sin[5*(c + d*x)] - 35*B*Sin[7*(c + d*x)]))/(17920*d)

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Maple [A]  time = 0.07, size = 323, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ({a}^{3}A \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) +B{a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16}}+{\frac{\sin \left ( dx+c \right ) }{64} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{128}}+{\frac{3\,c}{128}} \right ) +3\,{a}^{3}A \left ( -1/6\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+1/24\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +1/16\,dx+c/16 \right ) +3\,B{a}^{3} \left ( -1/7\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) -{\frac{3\,{a}^{3}A \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5}}+3\,B{a}^{3} \left ( -1/6\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+1/24\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +1/16\,dx+c/16 \right ) +{a}^{3}A \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) -{\frac{B{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x)

[Out]

1/d*(a^3*A*(-1/7*sin(d*x+c)^2*cos(d*x+c)^5-2/35*cos(d*x+c)^5)+B*a^3*(-1/8*sin(d*x+c)^3*cos(d*x+c)^5-1/16*sin(d
*x+c)*cos(d*x+c)^5+1/64*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/128*d*x+3/128*c)+3*a^3*A*(-1/6*sin(d*x+c)*c
os(d*x+c)^5+1/24*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+1/16*d*x+1/16*c)+3*B*a^3*(-1/7*sin(d*x+c)^2*cos(d*x+
c)^5-2/35*cos(d*x+c)^5)-3/5*a^3*A*cos(d*x+c)^5+3*B*a^3*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1/24*(cos(d*x+c)^3+3/2*co
s(d*x+c))*sin(d*x+c)+1/16*d*x+1/16*c)+a^3*A*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)-1/5*B
*a^3*cos(d*x+c)^5)

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Maxima [A]  time = 1.10521, size = 313, normalized size = 1.56 \begin{align*} -\frac{21504 \, A a^{3} \cos \left (d x + c\right )^{5} + 7168 \, B a^{3} \cos \left (d x + c\right )^{5} - 1024 \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} A a^{3} - 560 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} A a^{3} - 1120 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 3072 \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} B a^{3} - 560 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{3} - 35 \,{\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{3}}{35840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/35840*(21504*A*a^3*cos(d*x + c)^5 + 7168*B*a^3*cos(d*x + c)^5 - 1024*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*
A*a^3 - 560*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*A*a^3 - 1120*(12*d*x + 12*c + sin(4*d*
x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 - 3072*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*B*a^3 - 560*(4*sin(2*d*x + 2
*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*B*a^3 - 35*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))
*B*a^3)/d

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Fricas [A]  time = 1.91957, size = 342, normalized size = 1.71 \begin{align*} \frac{640 \,{\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{7} - 3584 \,{\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{5} + 315 \,{\left (8 \, A + 3 \, B\right )} a^{3} d x + 35 \,{\left (16 \, B a^{3} \cos \left (d x + c\right )^{7} - 8 \,{\left (8 \, A + 11 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} + 6 \,{\left (8 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 9 \,{\left (8 \, A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/4480*(640*(A + 3*B)*a^3*cos(d*x + c)^7 - 3584*(A + B)*a^3*cos(d*x + c)^5 + 315*(8*A + 3*B)*a^3*d*x + 35*(16*
B*a^3*cos(d*x + c)^7 - 8*(8*A + 11*B)*a^3*cos(d*x + c)^5 + 6*(8*A + 3*B)*a^3*cos(d*x + c)^3 + 9*(8*A + 3*B)*a^
3*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 16.3624, size = 823, normalized size = 4.12 \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)**4*(a+a*sin(d*x+c))**3*(A+B*sin(d*x+c)),x)

[Out]

Piecewise((3*A*a**3*x*sin(c + d*x)**6/16 + 9*A*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*A*a**3*x*sin(c +
d*x)**4/8 + 9*A*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*A*a**3*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*A
*a**3*x*cos(c + d*x)**6/16 + 3*A*a**3*x*cos(c + d*x)**4/8 + 3*A*a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) + A*a
**3*sin(c + d*x)**3*cos(c + d*x)**3/(2*d) + 3*A*a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d) - A*a**3*sin(c + d*x)*
*2*cos(c + d*x)**5/(5*d) - 3*A*a**3*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 5*A*a**3*sin(c + d*x)*cos(c + d*x)**
3/(8*d) - 2*A*a**3*cos(c + d*x)**7/(35*d) - 3*A*a**3*cos(c + d*x)**5/(5*d) + 3*B*a**3*x*sin(c + d*x)**8/128 +
3*B*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 3*B*a**3*x*sin(c + d*x)**6/16 + 9*B*a**3*x*sin(c + d*x)**4*cos
(c + d*x)**4/64 + 9*B*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**6/3
2 + 9*B*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*B*a**3*x*cos(c + d*x)**8/128 + 3*B*a**3*x*cos(c + d*x)**
6/16 + 3*B*a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 11*B*a**3*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) + 3*B
*a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) - 11*B*a**3*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) + B*a**3*sin(c +
d*x)**3*cos(c + d*x)**3/(2*d) - 3*B*a**3*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 3*B*a**3*sin(c + d*x)*cos(c
+ d*x)**7/(128*d) - 3*B*a**3*sin(c + d*x)*cos(c + d*x)**5/(16*d) - 6*B*a**3*cos(c + d*x)**7/(35*d) - B*a**3*co
s(c + d*x)**5/(5*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*sin(c) + a)**3*cos(c)**4, True))

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Giac [A]  time = 1.33106, size = 293, normalized size = 1.46 \begin{align*} \frac{B a^{3} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac{9}{128} \,{\left (8 \, A a^{3} + 3 \, B a^{3}\right )} x + \frac{{\left (A a^{3} + 3 \, B a^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac{{\left (11 \, A a^{3} + B a^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac{{\left (13 \, A a^{3} + 7 \, B a^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac{{\left (27 \, A a^{3} + 17 \, B a^{3}\right )} \cos \left (d x + c\right )}{64 \, d} - \frac{{\left (A a^{3} + B a^{3}\right )} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} - \frac{{\left (2 \, A a^{3} + 7 \, B a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{{\left (19 \, A a^{3} + 3 \, B a^{3}\right )} \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)^4*(a+a*sin(d*x+c))^3*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

1/1024*B*a^3*sin(8*d*x + 8*c)/d + 9/128*(8*A*a^3 + 3*B*a^3)*x + 1/448*(A*a^3 + 3*B*a^3)*cos(7*d*x + 7*c)/d - 1
/320*(11*A*a^3 + B*a^3)*cos(5*d*x + 5*c)/d - 1/64*(13*A*a^3 + 7*B*a^3)*cos(3*d*x + 3*c)/d - 1/64*(27*A*a^3 + 1
7*B*a^3)*cos(d*x + c)/d - 1/64*(A*a^3 + B*a^3)*sin(6*d*x + 6*c)/d - 1/128*(2*A*a^3 + 7*B*a^3)*sin(4*d*x + 4*c)
/d + 1/64*(19*A*a^3 + 3*B*a^3)*sin(2*d*x + 2*c)/d