3.607 \(\int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=183 \[ -\frac{a^3 \cos ^{11}(c+d x)}{11 d}+\frac{5 a^3 \cos ^9(c+d x)}{9 d}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}-\frac{3 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac{19 a^3 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{19 a^3 \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac{19 a^3 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{19 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{19 a^3 x}{256} \]

[Out]

(19*a^3*x)/256 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (5*a^3*Cos[c + d*x]^9)/(9*d) - (a^3*Cos[c + d*x]^11)/(11*d) +
(19*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (19*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (19*a^3*Cos[c + d*
x]^5*Sin[c + d*x])/(480*d) - (19*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - (3*a^3*Cos[c + d*x]^7*Sin[c + d*x]^
3)/(10*d)

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Rubi [A]  time = 0.333503, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2873, 2568, 2635, 8, 2565, 14, 270} \[ -\frac{a^3 \cos ^{11}(c+d x)}{11 d}+\frac{5 a^3 \cos ^9(c+d x)}{9 d}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}-\frac{3 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac{19 a^3 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{19 a^3 \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac{19 a^3 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{19 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{19 a^3 x}{256} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

(19*a^3*x)/256 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (5*a^3*Cos[c + d*x]^9)/(9*d) - (a^3*Cos[c + d*x]^11)/(11*d) +
(19*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (19*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (19*a^3*Cos[c + d*
x]^5*Sin[c + d*x])/(480*d) - (19*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - (3*a^3*Cos[c + d*x]^7*Sin[c + d*x]^
3)/(10*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 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 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 ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cos ^6(c+d x) \sin ^2(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^3(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^4(c+d x)+a^3 \cos ^6(c+d x) \sin ^5(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+a^3 \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx\\ &=-\frac{a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac{1}{8} a^3 \int \cos ^6(c+d x) \, dx+\frac{1}{10} \left (9 a^3\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac{1}{48} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{80} \left (9 a^3\right ) \int \cos ^6(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{5 a^3 \cos ^9(c+d x)}{9 d}-\frac{a^3 \cos ^{11}(c+d x)}{11 d}+\frac{5 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac{19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac{1}{64} \left (5 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{32} \left (3 a^3\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{5 a^3 \cos ^9(c+d x)}{9 d}-\frac{a^3 \cos ^{11}(c+d x)}{11 d}+\frac{5 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{19 a^3 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac{19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac{19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac{1}{128} \left (5 a^3\right ) \int 1 \, dx+\frac{1}{128} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{5 a^3 x}{128}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{5 a^3 \cos ^9(c+d x)}{9 d}-\frac{a^3 \cos ^{11}(c+d x)}{11 d}+\frac{19 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{19 a^3 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac{19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac{19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac{1}{256} \left (9 a^3\right ) \int 1 \, dx\\ &=\frac{19 a^3 x}{256}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{5 a^3 \cos ^9(c+d x)}{9 d}-\frac{a^3 \cos ^{11}(c+d x)}{11 d}+\frac{19 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{19 a^3 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac{19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac{19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}\\ \end{align*}

Mathematica [A]  time = 1.14324, size = 126, normalized size = 0.69 \[ \frac{a^3 (152460 \sin (2 (c+d x))-138600 \sin (4 (c+d x))-57750 \sin (6 (c+d x))+3465 \sin (8 (c+d x))+4158 \sin (10 (c+d x))-568260 \cos (c+d x)-244860 \cos (3 (c+d x))+6930 \cos (5 (c+d x))+40590 \cos (7 (c+d x))+8470 \cos (9 (c+d x))-630 \cos (11 (c+d x))+415800 c+526680 d x)}{7096320 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(415800*c + 526680*d*x - 568260*Cos[c + d*x] - 244860*Cos[3*(c + d*x)] + 6930*Cos[5*(c + d*x)] + 40590*Co
s[7*(c + d*x)] + 8470*Cos[9*(c + d*x)] - 630*Cos[11*(c + d*x)] + 152460*Sin[2*(c + d*x)] - 138600*Sin[4*(c + d
*x)] - 57750*Sin[6*(c + d*x)] + 3465*Sin[8*(c + d*x)] + 4158*Sin[10*(c + d*x)]))/(7096320*d)

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Maple [A]  time = 0.043, size = 236, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{11}}-{\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 ) +3\,{a}^{3} \left ( -1/10\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\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}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{3\,dx}{256}}+{\frac{3\,c}{256}} \right ) +3\,{a}^{3} \left ( -1/9\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) +{a}^{3} \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 ) \right ) } \end{align*}

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))^3,x)

[Out]

1/d*(a^3*(-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)+3*a^3*(-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)+3*a^3*(-1/9*sin(d*x+c)^2*cos(d*x+c)^7-2/63*cos(d*x+c)^7)+a^3*(-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))

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Maxima [A]  time = 1.16468, size = 221, normalized size = 1.21 \begin{align*} -\frac{10240 \,{\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^{3} - 337920 \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 2079 \,{\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^{3} - 2310 \,{\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^{3}}{7096320 \, d} \end{align*}

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))^3,x, algorithm="maxima")

[Out]

-1/7096320*(10240*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^3 - 337920*(7*cos(d*x + c)^9
- 9*cos(d*x + c)^7)*a^3 - 2079*(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^3 - 2310*(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^3)/d

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Fricas [A]  time = 1.29344, size = 347, normalized size = 1.9 \begin{align*} -\frac{80640 \, a^{3} \cos \left (d x + c\right )^{11} - 492800 \, a^{3} \cos \left (d x + c\right )^{9} + 506880 \, a^{3} \cos \left (d x + c\right )^{7} - 65835 \, a^{3} d x - 231 \,{\left (1152 \, a^{3} \cos \left (d x + c\right )^{9} - 2064 \, a^{3} \cos \left (d x + c\right )^{7} + 152 \, a^{3} \cos \left (d x + c\right )^{5} + 190 \, a^{3} \cos \left (d x + c\right )^{3} + 285 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, d} \end{align*}

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))^3,x, algorithm="fricas")

[Out]

-1/887040*(80640*a^3*cos(d*x + c)^11 - 492800*a^3*cos(d*x + c)^9 + 506880*a^3*cos(d*x + c)^7 - 65835*a^3*d*x -
231*(1152*a^3*cos(d*x + c)^9 - 2064*a^3*cos(d*x + c)^7 + 152*a^3*cos(d*x + c)^5 + 190*a^3*cos(d*x + c)^3 + 28
5*a^3*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 59.4676, size = 597, normalized size = 3.26 \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)**2*(a+a*sin(d*x+c))**3,x)

[Out]

Piecewise((9*a**3*x*sin(c + d*x)**10/256 + 45*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 5*a**3*x*sin(c + d*
x)**8/128 + 45*a**3*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 5*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 45*a
**3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 15*a**3*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 45*a**3*x*sin(c + d
*x)**2*cos(c + d*x)**8/256 + 5*a**3*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 9*a**3*x*cos(c + d*x)**10/256 + 5*a
**3*x*cos(c + d*x)**8/128 + 9*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 21*a**3*sin(c + d*x)**7*cos(c + d*x)
**3/(128*d) + 5*a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 3*a**3*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 55
*a**3*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) - a**3*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 21*a**3*sin(c + d
*x)**3*cos(c + d*x)**7/(128*d) + 73*a**3*sin(c + d*x)**3*cos(c + d*x)**5/(384*d) - 4*a**3*sin(c + d*x)**2*cos(
c + d*x)**9/(63*d) - 3*a**3*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 9*a**3*sin(c + d*x)*cos(c + d*x)**9/(256*d
) - 5*a**3*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 8*a**3*cos(c + d*x)**11/(693*d) - 2*a**3*cos(c + d*x)**9/(21
*d), Ne(d, 0)), (x*(a*sin(c) + a)**3*sin(c)**2*cos(c)**6, True))

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Giac [A]  time = 1.27088, size = 258, normalized size = 1.41 \begin{align*} \frac{19}{256} \, a^{3} x - \frac{a^{3} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac{11 \, a^{3} \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac{41 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac{a^{3} \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} - \frac{53 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac{41 \, a^{3} \cos \left (d x + c\right )}{512 \, d} + \frac{3 \, a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac{25 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac{5 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac{11 \, a^{3} \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)^2*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

19/256*a^3*x - 1/11264*a^3*cos(11*d*x + 11*c)/d + 11/9216*a^3*cos(9*d*x + 9*c)/d + 41/7168*a^3*cos(7*d*x + 7*c
)/d + 1/1024*a^3*cos(5*d*x + 5*c)/d - 53/1536*a^3*cos(3*d*x + 3*c)/d - 41/512*a^3*cos(d*x + c)/d + 3/5120*a^3*
sin(10*d*x + 10*c)/d + 1/2048*a^3*sin(8*d*x + 8*c)/d - 25/3072*a^3*sin(6*d*x + 6*c)/d - 5/256*a^3*sin(4*d*x +
4*c)/d + 11/512*a^3*sin(2*d*x + 2*c)/d