3.247 \(\int (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx\)

Optimal. Leaf size=180 \[ \frac{11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac{a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac{11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac{11 a^3 c^6 \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{55 a^3 c^6 \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{55 a^3 c^6 \sin (e+f x) \cos (e+f x)}{128 f}+\frac{55}{128} a^3 c^6 x \]

[Out]

(55*a^3*c^6*x)/128 + (11*a^3*c^6*Cos[e + f*x]^7)/(56*f) + (55*a^3*c^6*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (55
*a^3*c^6*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + (11*a^3*c^6*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) + (a^3*Cos[e +
f*x]^7*(c^3 - c^3*Sin[e + f*x])^2)/(9*f) + (11*a^3*Cos[e + f*x]^7*(c^6 - c^6*Sin[e + f*x]))/(72*f)

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Rubi [A]  time = 0.205928, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2736, 2678, 2669, 2635, 8} \[ \frac{11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac{a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac{11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac{11 a^3 c^6 \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{55 a^3 c^6 \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{55 a^3 c^6 \sin (e+f x) \cos (e+f x)}{128 f}+\frac{55}{128} a^3 c^6 x \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6,x]

[Out]

(55*a^3*c^6*x)/128 + (11*a^3*c^6*Cos[e + f*x]^7)/(56*f) + (55*a^3*c^6*Cos[e + f*x]*Sin[e + f*x])/(128*f) + (55
*a^3*c^6*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + (11*a^3*c^6*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) + (a^3*Cos[e +
f*x]^7*(c^3 - c^3*Sin[e + f*x])^2)/(9*f) + (11*a^3*Cos[e + f*x]^7*(c^6 - c^6*Sin[e + f*x]))/(72*f)

Rule 2736

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0,
n, m] || LtQ[m, n, 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 (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6 \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^3 \, dx\\ &=\frac{a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac{1}{9} \left (11 a^3 c^4\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=\frac{a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac{11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac{1}{8} \left (11 a^3 c^5\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac{11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac{a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac{11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac{1}{8} \left (11 a^3 c^6\right ) \int \cos ^6(e+f x) \, dx\\ &=\frac{11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac{11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac{a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac{11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac{1}{48} \left (55 a^3 c^6\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac{11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac{55 a^3 c^6 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac{11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac{a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac{11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac{1}{64} \left (55 a^3 c^6\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac{11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac{55 a^3 c^6 \cos (e+f x) \sin (e+f x)}{128 f}+\frac{55 a^3 c^6 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac{11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac{a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac{11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}+\frac{1}{128} \left (55 a^3 c^6\right ) \int 1 \, dx\\ &=\frac{55}{128} a^3 c^6 x+\frac{11 a^3 c^6 \cos ^7(e+f x)}{56 f}+\frac{55 a^3 c^6 \cos (e+f x) \sin (e+f x)}{128 f}+\frac{55 a^3 c^6 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac{11 a^3 c^6 \cos ^5(e+f x) \sin (e+f x)}{48 f}+\frac{a^3 \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{9 f}+\frac{11 a^3 \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{72 f}\\ \end{align*}

Mathematica [A]  time = 2.14343, size = 109, normalized size = 0.61 \[ \frac{a^3 c^6 (18144 \sin (2 (e+f x))+1512 \sin (4 (e+f x))-672 \sin (6 (e+f x))-189 \sin (8 (e+f x))+16632 \cos (e+f x)+9744 \cos (3 (e+f x))+3024 \cos (5 (e+f x))+324 \cos (7 (e+f x))-28 \cos (9 (e+f x))+27720 e+27720 f x)}{64512 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6,x]

[Out]

(a^3*c^6*(27720*e + 27720*f*x + 16632*Cos[e + f*x] + 9744*Cos[3*(e + f*x)] + 3024*Cos[5*(e + f*x)] + 324*Cos[7
*(e + f*x)] - 28*Cos[9*(e + f*x)] + 18144*Sin[2*(e + f*x)] + 1512*Sin[4*(e + f*x)] - 672*Sin[6*(e + f*x)] - 18
9*Sin[8*(e + f*x)]))/(64512*f)

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Maple [A]  time = 0.023, size = 297, normalized size = 1.7 \begin{align*}{\frac{1}{f} \left ( -{\frac{{c}^{6}{a}^{3}\cos \left ( fx+e \right ) }{9} \left ({\frac{128}{35}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{8}+{\frac{8\, \left ( \sin \left ( fx+e \right ) \right ) ^{6}}{7}}+{\frac{48\, \left ( \sin \left ( fx+e \right ) \right ) ^{4}}{35}}+{\frac{64\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{35}} \right ) }-3\,{c}^{6}{a}^{3} \left ( -1/8\, \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{7}+7/6\, \left ( \sin \left ( fx+e \right ) \right ) ^{5}+{\frac{35\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{24}}+{\frac{35\,\sin \left ( fx+e \right ) }{16}} \right ) \cos \left ( fx+e \right ) +{\frac{35\,fx}{128}}+{\frac{35\,e}{128}} \right ) +8\,{c}^{6}{a}^{3} \left ( -1/6\, \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{5}+5/4\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}+{\frac{15\,\sin \left ( fx+e \right ) }{8}} \right ) \cos \left ( fx+e \right ) +{\frac{5\,fx}{16}}+{\frac{5\,e}{16}} \right ) +{\frac{6\,{c}^{6}{a}^{3}\cos \left ( fx+e \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) }-6\,{c}^{6}{a}^{3} \left ( -1/4\, \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+3/2\,\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) +3/8\,fx+3/8\,e \right ) -{\frac{8\,{c}^{6}{a}^{3} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}+3\,{c}^{6}{a}^{3}\cos \left ( fx+e \right ) +{c}^{6}{a}^{3} \left ( fx+e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^6,x)

[Out]

1/f*(-1/9*c^6*a^3*(128/35+sin(f*x+e)^8+8/7*sin(f*x+e)^6+48/35*sin(f*x+e)^4+64/35*sin(f*x+e)^2)*cos(f*x+e)-3*c^
6*a^3*(-1/8*(sin(f*x+e)^7+7/6*sin(f*x+e)^5+35/24*sin(f*x+e)^3+35/16*sin(f*x+e))*cos(f*x+e)+35/128*f*x+35/128*e
)+8*c^6*a^3*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)+6/5*c^6*a^3*(8/3
+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-6*c^6*a^3*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/
8*e)-8/3*c^6*a^3*(2+sin(f*x+e)^2)*cos(f*x+e)+3*c^6*a^3*cos(f*x+e)+c^6*a^3*(f*x+e))

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Maxima [A]  time = 1.22534, size = 406, normalized size = 2.26 \begin{align*} -\frac{1024 \,{\left (35 \, \cos \left (f x + e\right )^{9} - 180 \, \cos \left (f x + e\right )^{7} + 378 \, \cos \left (f x + e\right )^{5} - 420 \, \cos \left (f x + e\right )^{3} + 315 \, \cos \left (f x + e\right )\right )} a^{3} c^{6} - 129024 \,{\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{3} c^{6} - 860160 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{3} c^{6} + 315 \,{\left (128 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 840 \, f x + 840 \, e + 3 \, \sin \left (8 \, f x + 8 \, e\right ) + 168 \, \sin \left (4 \, f x + 4 \, e\right ) - 768 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{6} - 13440 \,{\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{6} + 60480 \,{\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{3} c^{6} - 322560 \,{\left (f x + e\right )} a^{3} c^{6} - 967680 \, a^{3} c^{6} \cos \left (f x + e\right )}{322560 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^6,x, algorithm="maxima")

[Out]

-1/322560*(1024*(35*cos(f*x + e)^9 - 180*cos(f*x + e)^7 + 378*cos(f*x + e)^5 - 420*cos(f*x + e)^3 + 315*cos(f*
x + e))*a^3*c^6 - 129024*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^3*c^6 - 860160*(cos(f*x +
e)^3 - 3*cos(f*x + e))*a^3*c^6 + 315*(128*sin(2*f*x + 2*e)^3 + 840*f*x + 840*e + 3*sin(8*f*x + 8*e) + 168*sin(
4*f*x + 4*e) - 768*sin(2*f*x + 2*e))*a^3*c^6 - 13440*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*sin(4*f*x + 4*e
) - 48*sin(2*f*x + 2*e))*a^3*c^6 + 60480*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^3*c^6 - 322
560*(f*x + e)*a^3*c^6 - 967680*a^3*c^6*cos(f*x + e))/f

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Fricas [A]  time = 1.47667, size = 297, normalized size = 1.65 \begin{align*} -\frac{896 \, a^{3} c^{6} \cos \left (f x + e\right )^{9} - 4608 \, a^{3} c^{6} \cos \left (f x + e\right )^{7} - 3465 \, a^{3} c^{6} f x + 21 \,{\left (144 \, a^{3} c^{6} \cos \left (f x + e\right )^{7} - 88 \, a^{3} c^{6} \cos \left (f x + e\right )^{5} - 110 \, a^{3} c^{6} \cos \left (f x + e\right )^{3} - 165 \, a^{3} c^{6} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8064 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^6,x, algorithm="fricas")

[Out]

-1/8064*(896*a^3*c^6*cos(f*x + e)^9 - 4608*a^3*c^6*cos(f*x + e)^7 - 3465*a^3*c^6*f*x + 21*(144*a^3*c^6*cos(f*x
+ e)^7 - 88*a^3*c^6*cos(f*x + e)^5 - 110*a^3*c^6*cos(f*x + e)^3 - 165*a^3*c^6*cos(f*x + e))*sin(f*x + e))/f

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Sympy [A]  time = 47.784, size = 838, normalized size = 4.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**3*(c-c*sin(f*x+e))**6,x)

[Out]

Piecewise((-105*a**3*c**6*x*sin(e + f*x)**8/128 - 105*a**3*c**6*x*sin(e + f*x)**6*cos(e + f*x)**2/32 + 5*a**3*
c**6*x*sin(e + f*x)**6/2 - 315*a**3*c**6*x*sin(e + f*x)**4*cos(e + f*x)**4/64 + 15*a**3*c**6*x*sin(e + f*x)**4
*cos(e + f*x)**2/2 - 9*a**3*c**6*x*sin(e + f*x)**4/4 - 105*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**6/32 + 15
*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**4/2 - 9*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**2/2 - 105*a**3*c*
*6*x*cos(e + f*x)**8/128 + 5*a**3*c**6*x*cos(e + f*x)**6/2 - 9*a**3*c**6*x*cos(e + f*x)**4/4 + a**3*c**6*x - a
**3*c**6*sin(e + f*x)**8*cos(e + f*x)/f + 279*a**3*c**6*sin(e + f*x)**7*cos(e + f*x)/(128*f) - 8*a**3*c**6*sin
(e + f*x)**6*cos(e + f*x)**3/(3*f) + 511*a**3*c**6*sin(e + f*x)**5*cos(e + f*x)**3/(128*f) - 11*a**3*c**6*sin(
e + f*x)**5*cos(e + f*x)/(2*f) - 16*a**3*c**6*sin(e + f*x)**4*cos(e + f*x)**5/(5*f) + 6*a**3*c**6*sin(e + f*x)
**4*cos(e + f*x)/f + 385*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)**5/(128*f) - 20*a**3*c**6*sin(e + f*x)**3*cos(
e + f*x)**3/(3*f) + 15*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)/(4*f) - 64*a**3*c**6*sin(e + f*x)**2*cos(e + f*x
)**7/(35*f) + 8*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**3/f - 8*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)/f + 105
*a**3*c**6*sin(e + f*x)*cos(e + f*x)**7/(128*f) - 5*a**3*c**6*sin(e + f*x)*cos(e + f*x)**5/(2*f) + 9*a**3*c**6
*sin(e + f*x)*cos(e + f*x)**3/(4*f) - 128*a**3*c**6*cos(e + f*x)**9/(315*f) + 16*a**3*c**6*cos(e + f*x)**5/(5*
f) - 16*a**3*c**6*cos(e + f*x)**3/(3*f) + 3*a**3*c**6*cos(e + f*x)/f, Ne(f, 0)), (x*(a*sin(e) + a)**3*(-c*sin(
e) + c)**6, True))

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Giac [A]  time = 2.06995, size = 265, normalized size = 1.47 \begin{align*} \frac{55}{128} \, a^{3} c^{6} x - \frac{a^{3} c^{6} \cos \left (9 \, f x + 9 \, e\right )}{2304 \, f} + \frac{9 \, a^{3} c^{6} \cos \left (7 \, f x + 7 \, e\right )}{1792 \, f} + \frac{3 \, a^{3} c^{6} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac{29 \, a^{3} c^{6} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} + \frac{33 \, a^{3} c^{6} \cos \left (f x + e\right )}{128 \, f} - \frac{3 \, a^{3} c^{6} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} - \frac{a^{3} c^{6} \sin \left (6 \, f x + 6 \, e\right )}{96 \, f} + \frac{3 \, a^{3} c^{6} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac{9 \, a^{3} c^{6} \sin \left (2 \, f x + 2 \, e\right )}{32 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(c-c*sin(f*x+e))^6,x, algorithm="giac")

[Out]

55/128*a^3*c^6*x - 1/2304*a^3*c^6*cos(9*f*x + 9*e)/f + 9/1792*a^3*c^6*cos(7*f*x + 7*e)/f + 3/64*a^3*c^6*cos(5*
f*x + 5*e)/f + 29/192*a^3*c^6*cos(3*f*x + 3*e)/f + 33/128*a^3*c^6*cos(f*x + e)/f - 3/1024*a^3*c^6*sin(8*f*x +
8*e)/f - 1/96*a^3*c^6*sin(6*f*x + 6*e)/f + 3/128*a^3*c^6*sin(4*f*x + 4*e)/f + 9/32*a^3*c^6*sin(2*f*x + 2*e)/f