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

Optimal. Leaf size=196 \[ -\frac{a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}-\frac{(9 A+2 B) \cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{72 d}+\frac{a^2 (9 A+2 B) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a^2 (9 A+2 B) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a^2 (9 A+2 B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} a^2 x (9 A+2 B)-\frac{B \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d} \]

[Out]

(5*a^2*(9*A + 2*B)*x)/128 - (a^2*(9*A + 2*B)*Cos[c + d*x]^7)/(56*d) + (5*a^2*(9*A + 2*B)*Cos[c + d*x]*Sin[c +
d*x])/(128*d) + (5*a^2*(9*A + 2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a^2*(9*A + 2*B)*Cos[c + d*x]^5*Sin[
c + d*x])/(48*d) - (B*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(9*d) - ((9*A + 2*B)*Cos[c + d*x]^7*(a^2 + a^2*Si
n[c + d*x]))/(72*d)

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Rubi [A]  time = 0.211018, antiderivative size = 196, 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} \[ -\frac{a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}-\frac{(9 A+2 B) \cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{72 d}+\frac{a^2 (9 A+2 B) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 a^2 (9 A+2 B) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 a^2 (9 A+2 B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} a^2 x (9 A+2 B)-\frac{B \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a^2*(9*A + 2*B)*x)/128 - (a^2*(9*A + 2*B)*Cos[c + d*x]^7)/(56*d) + (5*a^2*(9*A + 2*B)*Cos[c + d*x]*Sin[c +
d*x])/(128*d) + (5*a^2*(9*A + 2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a^2*(9*A + 2*B)*Cos[c + d*x]^5*Sin[
c + d*x])/(48*d) - (B*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(9*d) - ((9*A + 2*B)*Cos[c + d*x]^7*(a^2 + a^2*Si
n[c + d*x]))/(72*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) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}+\frac{1}{9} (9 A+2 B) \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac{(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d}+\frac{1}{8} (a (9 A+2 B)) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac{a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac{(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d}+\frac{1}{8} \left (a^2 (9 A+2 B)\right ) \int \cos ^6(c+d x) \, dx\\ &=-\frac{a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}+\frac{a^2 (9 A+2 B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac{(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d}+\frac{1}{48} \left (5 a^2 (9 A+2 B)\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}+\frac{5 a^2 (9 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{a^2 (9 A+2 B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac{(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d}+\frac{1}{64} \left (5 a^2 (9 A+2 B)\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}+\frac{5 a^2 (9 A+2 B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{5 a^2 (9 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{a^2 (9 A+2 B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac{(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d}+\frac{1}{128} \left (5 a^2 (9 A+2 B)\right ) \int 1 \, dx\\ &=\frac{5}{128} a^2 (9 A+2 B) x-\frac{a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}+\frac{5 a^2 (9 A+2 B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{5 a^2 (9 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{a^2 (9 A+2 B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac{(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d}\\ \end{align*}

Mathematica [A]  time = 5.05663, size = 216, normalized size = 1.1 \[ -\frac{a^2 \cos (c+d x) \left (32 (135 A+86 B) \cos (2 (c+d x))+16 (108 A+59 B) \cos (4 (c+d x))+\frac{2520 (9 A+2 B) \sin ^{-1}\left (\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right )}{\sqrt{\cos ^2(c+d x)}}-13671 A \sin (c+d x)-2457 A \sin (3 (c+d x))-63 A \sin (5 (c+d x))+63 A \sin (7 (c+d x))+288 A \cos (6 (c+d x))+2880 A-2478 B \sin (c+d x)+462 B \sin (3 (c+d x))+546 B \sin (5 (c+d x))+126 B \sin (7 (c+d x))+64 B \cos (6 (c+d x))-28 B \cos (8 (c+d x))+1900 B\right )}{32256 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*Cos[c + d*x]*(2880*A + 1900*B + (2520*(9*A + 2*B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]])/Sqrt[Cos[c + d
*x]^2] + 32*(135*A + 86*B)*Cos[2*(c + d*x)] + 16*(108*A + 59*B)*Cos[4*(c + d*x)] + 288*A*Cos[6*(c + d*x)] + 64
*B*Cos[6*(c + d*x)] - 28*B*Cos[8*(c + d*x)] - 13671*A*Sin[c + d*x] - 2478*B*Sin[c + d*x] - 2457*A*Sin[3*(c + d
*x)] + 462*B*Sin[3*(c + d*x)] - 63*A*Sin[5*(c + d*x)] + 546*B*Sin[5*(c + d*x)] + 63*A*Sin[7*(c + d*x)] + 126*B
*Sin[7*(c + d*x)]))/(32256*d)

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Maple [A]  time = 0.069, size = 245, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({a}^{2}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 ) +B{a}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) -{\frac{2\,{a}^{2}A \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7}}+2\,B{a}^{2} \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 ) +{a}^{2}A \left ({\frac{\sin \left ( dx+c \right ) }{6} \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}{16}}+{\frac{5\,c}{16}} \right ) -{\frac{B{a}^{2} \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*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x)

[Out]

1/d*(a^2*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)+B*a^2*(-1/9*sin(d*x+c)^2*cos(d*x+c)^7-2/63*cos(d*x+c)^7)-2/7*a^2*A*cos(d*x+c)^7+2*B*a^2*(-1/8*si
n(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)+a^2*A
*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)-1/7*B*a^2*cos(d*x+c)^7)

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Maxima [A]  time = 1.03578, size = 281, normalized size = 1.43 \begin{align*} -\frac{18432 \, A a^{2} \cos \left (d x + c\right )^{7} + 9216 \, B a^{2} \cos \left (d x + c\right )^{7} - 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 a^{2} + 336 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 1024 \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} B a^{2} - 42 \,{\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 )} B a^{2}}{64512 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64512*(18432*A*a^2*cos(d*x + c)^7 + 9216*B*a^2*cos(d*x + c)^7 - 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*a^2 + 336*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x +
4*c) - 48*sin(2*d*x + 2*c))*A*a^2 - 1024*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*B*a^2 - 42*(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))*B*a^2)/d

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Fricas [A]  time = 2.02562, size = 343, normalized size = 1.75 \begin{align*} \frac{896 \, B a^{2} \cos \left (d x + c\right )^{9} - 2304 \,{\left (A + B\right )} a^{2} \cos \left (d x + c\right )^{7} + 315 \,{\left (9 \, A + 2 \, B\right )} a^{2} d x - 21 \,{\left (48 \,{\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{7} - 8 \,{\left (9 \, A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} - 10 \,{\left (9 \, A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} - 15 \,{\left (9 \, A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8064*(896*B*a^2*cos(d*x + c)^9 - 2304*(A + B)*a^2*cos(d*x + c)^7 + 315*(9*A + 2*B)*a^2*d*x - 21*(48*(A + 2*B
)*a^2*cos(d*x + c)^7 - 8*(9*A + 2*B)*a^2*cos(d*x + c)^5 - 10*(9*A + 2*B)*a^2*cos(d*x + c)^3 - 15*(9*A + 2*B)*a
^2*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 26.0963, size = 719, normalized size = 3.67 \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*(a+a*sin(d*x+c))**2*(A+B*sin(d*x+c)),x)

[Out]

Piecewise((5*A*a**2*x*sin(c + d*x)**8/128 + 5*A*a**2*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 5*A*a**2*x*sin(c +
d*x)**6/16 + 15*A*a**2*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*A*a**2*x*sin(c + d*x)**4*cos(c + d*x)**2/16
+ 5*A*a**2*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 15*A*a**2*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*A*a**2*x*
cos(c + d*x)**8/128 + 5*A*a**2*x*cos(c + d*x)**6/16 + 5*A*a**2*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 55*A*a**
2*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 5*A*a**2*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 73*A*a**2*sin(c + d
*x)**3*cos(c + d*x)**5/(384*d) + 5*A*a**2*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) - 5*A*a**2*sin(c + d*x)*cos(c
+ d*x)**7/(128*d) + 11*A*a**2*sin(c + d*x)*cos(c + d*x)**5/(16*d) - 2*A*a**2*cos(c + d*x)**7/(7*d) + 5*B*a**2*
x*sin(c + d*x)**8/64 + 5*B*a**2*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + 15*B*a**2*x*sin(c + d*x)**4*cos(c + d*x
)**4/32 + 5*B*a**2*x*sin(c + d*x)**2*cos(c + d*x)**6/16 + 5*B*a**2*x*cos(c + d*x)**8/64 + 5*B*a**2*sin(c + d*x
)**7*cos(c + d*x)/(64*d) + 55*B*a**2*sin(c + d*x)**5*cos(c + d*x)**3/(192*d) + 73*B*a**2*sin(c + d*x)**3*cos(c
+ d*x)**5/(192*d) - B*a**2*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 5*B*a**2*sin(c + d*x)*cos(c + d*x)**7/(64*
d) - 2*B*a**2*cos(c + d*x)**9/(63*d) - B*a**2*cos(c + d*x)**7/(7*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*sin(c) +
a)**2*cos(c)**6, True))

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Giac [A]  time = 1.35269, size = 317, normalized size = 1.62 \begin{align*} \frac{B a^{2} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac{B a^{2} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac{5}{128} \,{\left (9 \, A a^{2} + 2 \, B a^{2}\right )} x - \frac{{\left (8 \, A a^{2} + B a^{2}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac{{\left (2 \, A a^{2} + B a^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac{{\left (18 \, A a^{2} + 11 \, B a^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac{{\left (20 \, A a^{2} + 13 \, B a^{2}\right )} \cos \left (d x + c\right )}{128 \, d} - \frac{{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac{{\left (5 \, A a^{2} - 2 \, B a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{{\left (8 \, A a^{2} + B a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2304*B*a^2*cos(9*d*x + 9*c)/d - 1/96*B*a^2*sin(6*d*x + 6*c)/d + 5/128*(9*A*a^2 + 2*B*a^2)*x - 1/1792*(8*A*a^
2 + B*a^2)*cos(7*d*x + 7*c)/d - 1/64*(2*A*a^2 + B*a^2)*cos(5*d*x + 5*c)/d - 1/192*(18*A*a^2 + 11*B*a^2)*cos(3*
d*x + 3*c)/d - 1/128*(20*A*a^2 + 13*B*a^2)*cos(d*x + c)/d - 1/1024*(A*a^2 + 2*B*a^2)*sin(8*d*x + 8*c)/d + 1/12
8*(5*A*a^2 - 2*B*a^2)*sin(4*d*x + 4*c)/d + 1/32*(8*A*a^2 + B*a^2)*sin(2*d*x + 2*c)/d