3.242 \(\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx\)

Optimal. Leaf size=241 \[ \frac{\left (95 a^3 A b+112 a^2 b^2 B+12 a^4 B+80 a A b^3+16 b^4 B\right ) \sin (c+d x)}{30 d}+\frac{\left (12 a^2 B+35 a A b+16 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac{b \left (130 a^2 A b+24 a^3 B+116 a b^2 B+45 A b^3\right ) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left (24 a^2 A b^2+8 a^4 A+16 a^3 b B+12 a b^3 B+3 A b^4\right )+\frac{(4 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^4}{5 d} \]

[Out]

((8*a^4*A + 24*a^2*A*b^2 + 3*A*b^4 + 16*a^3*b*B + 12*a*b^3*B)*x)/8 + ((95*a^3*A*b + 80*a*A*b^3 + 12*a^4*B + 11
2*a^2*b^2*B + 16*b^4*B)*Sin[c + d*x])/(30*d) + (b*(130*a^2*A*b + 45*A*b^3 + 24*a^3*B + 116*a*b^2*B)*Cos[c + d*
x]*Sin[c + d*x])/(120*d) + ((35*a*A*b + 12*a^2*B + 16*b^2*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((5
*A*b + 4*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (B*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)

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Rubi [A]  time = 0.337711, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2753, 2734} \[ \frac{\left (95 a^3 A b+112 a^2 b^2 B+12 a^4 B+80 a A b^3+16 b^4 B\right ) \sin (c+d x)}{30 d}+\frac{\left (12 a^2 B+35 a A b+16 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac{b \left (130 a^2 A b+24 a^3 B+116 a b^2 B+45 A b^3\right ) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left (24 a^2 A b^2+8 a^4 A+16 a^3 b B+12 a b^3 B+3 A b^4\right )+\frac{(4 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^4}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^4*A + 24*a^2*A*b^2 + 3*A*b^4 + 16*a^3*b*B + 12*a*b^3*B)*x)/8 + ((95*a^3*A*b + 80*a*A*b^3 + 12*a^4*B + 11
2*a^2*b^2*B + 16*b^4*B)*Sin[c + d*x])/(30*d) + (b*(130*a^2*A*b + 45*A*b^3 + 24*a^3*B + 116*a*b^2*B)*Cos[c + d*
x]*Sin[c + d*x])/(120*d) + ((35*a*A*b + 12*a^2*B + 16*b^2*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(60*d) + ((5
*A*b + 4*a*B)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(20*d) + (B*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(5*d)

Rule 2753

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin{align*} \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx &=\frac{B (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{5} \int (a+b \cos (c+d x))^3 (5 a A+4 b B+(5 A b+4 a B) \cos (c+d x)) \, dx\\ &=\frac{(5 A b+4 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{B (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{20} \int (a+b \cos (c+d x))^2 \left (20 a^2 A+15 A b^2+28 a b B+\left (35 a A b+12 a^2 B+16 b^2 B\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac{(5 A b+4 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{B (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{60} \int (a+b \cos (c+d x)) \left (60 a^3 A+115 a A b^2+108 a^2 b B+32 b^3 B+\left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{1}{8} \left (8 a^4 A+24 a^2 A b^2+3 A b^4+16 a^3 b B+12 a b^3 B\right ) x+\frac{\left (95 a^3 A b+80 a A b^3+12 a^4 B+112 a^2 b^2 B+16 b^4 B\right ) \sin (c+d x)}{30 d}+\frac{b \left (130 a^2 A b+45 A b^3+24 a^3 B+116 a b^2 B\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac{\left (35 a A b+12 a^2 B+16 b^2 B\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac{(5 A b+4 a B) (a+b \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac{B (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.619624, size = 263, normalized size = 1.09 \[ \frac{60 \left (32 a^3 A b+36 a^2 b^2 B+8 a^4 B+24 a A b^3+5 b^4 B\right ) \sin (c+d x)+120 b \left (6 a^2 A b+4 a^3 B+4 a b^2 B+A b^3\right ) \sin (2 (c+d x))+1440 a^2 A b^2 c+1440 a^2 A b^2 d x+480 a^4 A c+480 a^4 A d x+240 a^2 b^2 B \sin (3 (c+d x))+960 a^3 b B c+960 a^3 b B d x+160 a A b^3 \sin (3 (c+d x))+60 a b^3 B \sin (4 (c+d x))+720 a b^3 B c+720 a b^3 B d x+15 A b^4 \sin (4 (c+d x))+180 A b^4 c+180 A b^4 d x+50 b^4 B \sin (3 (c+d x))+6 b^4 B \sin (5 (c+d x))}{480 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(480*a^4*A*c + 1440*a^2*A*b^2*c + 180*A*b^4*c + 960*a^3*b*B*c + 720*a*b^3*B*c + 480*a^4*A*d*x + 1440*a^2*A*b^2
*d*x + 180*A*b^4*d*x + 960*a^3*b*B*d*x + 720*a*b^3*B*d*x + 60*(32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*
B + 5*b^4*B)*Sin[c + d*x] + 120*b*(6*a^2*A*b + A*b^3 + 4*a^3*B + 4*a*b^2*B)*Sin[2*(c + d*x)] + 160*a*A*b^3*Sin
[3*(c + d*x)] + 240*a^2*b^2*B*Sin[3*(c + d*x)] + 50*b^4*B*Sin[3*(c + d*x)] + 15*A*b^4*Sin[4*(c + d*x)] + 60*a*
b^3*B*Sin[4*(c + d*x)] + 6*b^4*B*Sin[5*(c + d*x)])/(480*d)

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Maple [A]  time = 0.04, size = 258, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{B{b}^{4}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+A{b}^{4} \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 ) +4\,Ba{b}^{3} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{\frac{4\,Aa{b}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,B{a}^{2}{b}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +6\,A{a}^{2}{b}^{2} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +4\,B{a}^{3}b \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +4\,A{a}^{3}b\sin \left ( dx+c \right ) +{a}^{4}B\sin \left ( dx+c \right ) +A{a}^{4} \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/5*B*b^4*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+A*b^4*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x
+c)+3/8*d*x+3/8*c)+4*B*a*b^3*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4/3*A*a*b^3*(2+cos(d
*x+c)^2)*sin(d*x+c)+2*B*a^2*b^2*(2+cos(d*x+c)^2)*sin(d*x+c)+6*A*a^2*b^2*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2
*c)+4*B*a^3*b*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+4*A*a^3*b*sin(d*x+c)+a^4*B*sin(d*x+c)+A*a^4*(d*x+c))

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Maxima [A]  time = 1.15258, size = 332, normalized size = 1.38 \begin{align*} \frac{480 \,{\left (d x + c\right )} A a^{4} + 480 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} b + 720 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b^{2} - 960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} b^{2} - 640 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b^{3} + 60 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{3} + 15 \,{\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 b^{4} + 32 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B b^{4} + 480 \, B a^{4} \sin \left (d x + c\right ) + 1920 \, A a^{3} b \sin \left (d x + c\right )}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(480*(d*x + c)*A*a^4 + 480*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^3*b + 720*(2*d*x + 2*c + sin(2*d*x + 2*c
))*A*a^2*b^2 - 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^2*b^2 - 640*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a*b^3
+ 60*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a*b^3 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) +
8*sin(2*d*x + 2*c))*A*b^4 + 32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*b^4 + 480*B*a^4*sin
(d*x + c) + 1920*A*a^3*b*sin(d*x + c))/d

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Fricas [A]  time = 1.5214, size = 478, normalized size = 1.98 \begin{align*} \frac{15 \,{\left (8 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 12 \, B a b^{3} + 3 \, A b^{4}\right )} d x +{\left (24 \, B b^{4} \cos \left (d x + c\right )^{4} + 120 \, B a^{4} + 480 \, A a^{3} b + 480 \, B a^{2} b^{2} + 320 \, A a b^{3} + 64 \, B b^{4} + 30 \,{\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{3} + 16 \,{\left (15 \, B a^{2} b^{2} + 10 \, A a b^{3} + 2 \, B b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left (16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 12 \, B a b^{3} + 3 \, A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(15*(8*A*a^4 + 16*B*a^3*b + 24*A*a^2*b^2 + 12*B*a*b^3 + 3*A*b^4)*d*x + (24*B*b^4*cos(d*x + c)^4 + 120*B*
a^4 + 480*A*a^3*b + 480*B*a^2*b^2 + 320*A*a*b^3 + 64*B*b^4 + 30*(4*B*a*b^3 + A*b^4)*cos(d*x + c)^3 + 16*(15*B*
a^2*b^2 + 10*A*a*b^3 + 2*B*b^4)*cos(d*x + c)^2 + 15*(16*B*a^3*b + 24*A*a^2*b^2 + 12*B*a*b^3 + 3*A*b^4)*cos(d*x
+ c))*sin(d*x + c))/d

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Sympy [A]  time = 3.67264, size = 580, normalized size = 2.41 \begin{align*} \begin{cases} A a^{4} x + \frac{4 A a^{3} b \sin{\left (c + d x \right )}}{d} + 3 A a^{2} b^{2} x \sin ^{2}{\left (c + d x \right )} + 3 A a^{2} b^{2} x \cos ^{2}{\left (c + d x \right )} + \frac{3 A a^{2} b^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} + \frac{8 A a b^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{4 A a b^{3} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 A b^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 A b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 A b^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{3 A b^{4} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{5 A b^{4} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac{B a^{4} \sin{\left (c + d x \right )}}{d} + 2 B a^{3} b x \sin ^{2}{\left (c + d x \right )} + 2 B a^{3} b x \cos ^{2}{\left (c + d x \right )} + \frac{2 B a^{3} b \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} + \frac{4 B a^{2} b^{2} \sin ^{3}{\left (c + d x \right )}}{d} + \frac{6 B a^{2} b^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 B a b^{3} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 B a b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \frac{3 B a b^{3} x \cos ^{4}{\left (c + d x \right )}}{2} + \frac{3 B a b^{3} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{5 B a b^{3} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac{8 B b^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 B b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{B b^{4} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (A + B \cos{\left (c \right )}\right ) \left (a + b \cos{\left (c \right )}\right )^{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))**4*(A+B*cos(d*x+c)),x)

[Out]

Piecewise((A*a**4*x + 4*A*a**3*b*sin(c + d*x)/d + 3*A*a**2*b**2*x*sin(c + d*x)**2 + 3*A*a**2*b**2*x*cos(c + d*
x)**2 + 3*A*a**2*b**2*sin(c + d*x)*cos(c + d*x)/d + 8*A*a*b**3*sin(c + d*x)**3/(3*d) + 4*A*a*b**3*sin(c + d*x)
*cos(c + d*x)**2/d + 3*A*b**4*x*sin(c + d*x)**4/8 + 3*A*b**4*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + 3*A*b**4*x*
cos(c + d*x)**4/8 + 3*A*b**4*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 5*A*b**4*sin(c + d*x)*cos(c + d*x)**3/(8*d)
+ B*a**4*sin(c + d*x)/d + 2*B*a**3*b*x*sin(c + d*x)**2 + 2*B*a**3*b*x*cos(c + d*x)**2 + 2*B*a**3*b*sin(c + d*x
)*cos(c + d*x)/d + 4*B*a**2*b**2*sin(c + d*x)**3/d + 6*B*a**2*b**2*sin(c + d*x)*cos(c + d*x)**2/d + 3*B*a*b**3
*x*sin(c + d*x)**4/2 + 3*B*a*b**3*x*sin(c + d*x)**2*cos(c + d*x)**2 + 3*B*a*b**3*x*cos(c + d*x)**4/2 + 3*B*a*b
**3*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 5*B*a*b**3*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 8*B*b**4*sin(c + d*x)
**5/(15*d) + 4*B*b**4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + B*b**4*sin(c + d*x)*cos(c + d*x)**4/d, Ne(d, 0))
, (x*(A + B*cos(c))*(a + b*cos(c))**4, True))

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Giac [A]  time = 1.42656, size = 286, normalized size = 1.19 \begin{align*} \frac{B b^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{1}{8} \,{\left (8 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 12 \, B a b^{3} + 3 \, A b^{4}\right )} x + \frac{{\left (4 \, B a b^{3} + A b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{{\left (24 \, B a^{2} b^{2} + 16 \, A a b^{3} + 5 \, B b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{{\left (4 \, B a^{3} b + 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)),x, algorithm="giac")

[Out]

1/80*B*b^4*sin(5*d*x + 5*c)/d + 1/8*(8*A*a^4 + 16*B*a^3*b + 24*A*a^2*b^2 + 12*B*a*b^3 + 3*A*b^4)*x + 1/32*(4*B
*a*b^3 + A*b^4)*sin(4*d*x + 4*c)/d + 1/48*(24*B*a^2*b^2 + 16*A*a*b^3 + 5*B*b^4)*sin(3*d*x + 3*c)/d + 1/4*(4*B*
a^3*b + 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*sin(2*d*x + 2*c)/d + 1/8*(8*B*a^4 + 32*A*a^3*b + 36*B*a^2*b^2 + 24*A*
a*b^3 + 5*B*b^4)*sin(d*x + c)/d