\(\int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx\) [30]

Optimal result

Integrand size = 38, antiderivative size = 236 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx=-\frac {4 a^4 \cos (e+f x) (c-c \sin (e+f x))^{11/2}}{315 c f \sqrt {a+a \sin (e+f x)}}-\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{11/2}}{105 c f}-\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{11/2}}{15 c f}-\frac {4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f} \]

[Out]

-1/15*a^2*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(11/2)/c/f-4/45*a*cos(f*x+e)*(a+a*sin(f*x+e))^(5/
2)*(c-c*sin(f*x+e))^(11/2)/c/f-1/10*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(11/2)/c/f-4/315*a^4*co
s(f*x+e)*(c-c*sin(f*x+e))^(11/2)/c/f/(a+a*sin(f*x+e))^(1/2)-4/105*a^3*cos(f*x+e)*(c-c*sin(f*x+e))^(11/2)*(a+a*
sin(f*x+e))^(1/2)/c/f

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2920, 2819, 2817} \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx=-\frac {4 a^4 \cos (e+f x) (c-c \sin (e+f x))^{11/2}}{315 c f \sqrt {a \sin (e+f x)+a}}-\frac {4 a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{11/2}}{105 c f}-\frac {a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{11/2}}{15 c f}-\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}-\frac {4 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f} \]

[In]

Int[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2),x]

[Out]

(-4*a^4*Cos[e + f*x]*(c - c*Sin[e + f*x])^(11/2))/(315*c*f*Sqrt[a + a*Sin[e + f*x]]) - (4*a^3*Cos[e + f*x]*Sqr
t[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(11/2))/(105*c*f) - (a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2)*(c
- c*Sin[e + f*x])^(11/2))/(15*c*f) - (4*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(11/2)
)/(45*c*f) - (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(11/2))/(10*c*f)

Rule 2817

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

Rule 2819

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

Rule 2920

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{11/2} \, dx}{a c} \\ & = -\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}+\frac {4 \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2} \, dx}{5 c} \\ & = -\frac {4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}+\frac {(8 a) \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{11/2} \, dx}{15 c} \\ & = -\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{11/2}}{15 c f}-\frac {4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}+\frac {\left (4 a^2\right ) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{11/2} \, dx}{15 c} \\ & = -\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{11/2}}{105 c f}-\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{11/2}}{15 c f}-\frac {4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f}+\frac {\left (8 a^3\right ) \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{11/2} \, dx}{105 c} \\ & = -\frac {4 a^4 \cos (e+f x) (c-c \sin (e+f x))^{11/2}}{315 c f \sqrt {a+a \sin (e+f x)}}-\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{11/2}}{105 c f}-\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{11/2}}{15 c f}-\frac {4 a \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{11/2}}{45 c f}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{11/2}}{10 c f} \\ \end{align*}

Mathematica [A] (verified)

Time = 13.97 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.89 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx=\frac {a^3 c^4 (-1+\sin (e+f x))^4 (1+\sin (e+f x))^3 \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} (13230 \cos (2 (e+f x))+7560 \cos (4 (e+f x))+2835 \cos (6 (e+f x))+630 \cos (8 (e+f x))+63 \cos (10 (e+f x))+158760 \sin (e+f x)+35280 \sin (3 (e+f x))+9072 \sin (5 (e+f x))+1620 \sin (7 (e+f x))+140 \sin (9 (e+f x)))}{322560 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \]

[In]

Integrate[Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2),x]

[Out]

(a^3*c^4*(-1 + Sin[e + f*x])^4*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(13230
*Cos[2*(e + f*x)] + 7560*Cos[4*(e + f*x)] + 2835*Cos[6*(e + f*x)] + 630*Cos[8*(e + f*x)] + 63*Cos[10*(e + f*x)
] + 158760*Sin[e + f*x] + 35280*Sin[3*(e + f*x)] + 9072*Sin[5*(e + f*x)] + 1620*Sin[7*(e + f*x)] + 140*Sin[9*(
e + f*x)]))/(322560*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7)

Maple [A] (verified)

Time = 222.41 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.53

method	result	size
default	\(\frac {\sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, a^{3} c^{4} \left (63 \left (\cos ^{9}\left (f x +e \right )\right )+70 \left (\cos ^{7}\left (f x +e \right )\right ) \sin \left (f x +e \right )+80 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )+96 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+128 \cos \left (f x +e \right ) \sin \left (f x +e \right )+256 \tan \left (f x +e \right )-63 \sec \left (f x +e \right )\right )}{630 f}\)	\(126\)

[In]

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(9/2),x,method=_RETURNVERBOSE)

[Out]

1/630/f*(a*(1+sin(f*x+e)))^(1/2)*(-c*(sin(f*x+e)-1))^(1/2)*a^3*c^4*(63*cos(f*x+e)^9+70*cos(f*x+e)^7*sin(f*x+e)
+80*cos(f*x+e)^5*sin(f*x+e)+96*cos(f*x+e)^3*sin(f*x+e)+128*cos(f*x+e)*sin(f*x+e)+256*tan(f*x+e)-63*sec(f*x+e))

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.61 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx=\frac {{\left (63 \, a^{3} c^{4} \cos \left (f x + e\right )^{10} - 63 \, a^{3} c^{4} + 2 \, {\left (35 \, a^{3} c^{4} \cos \left (f x + e\right )^{8} + 40 \, a^{3} c^{4} \cos \left (f x + e\right )^{6} + 48 \, a^{3} c^{4} \cos \left (f x + e\right )^{4} + 64 \, a^{3} c^{4} \cos \left (f x + e\right )^{2} + 128 \, a^{3} c^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{630 \, f \cos \left (f x + e\right )} \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

1/630*(63*a^3*c^4*cos(f*x + e)^10 - 63*a^3*c^4 + 2*(35*a^3*c^4*cos(f*x + e)^8 + 40*a^3*c^4*cos(f*x + e)^6 + 48
*a^3*c^4*cos(f*x + e)^4 + 64*a^3*c^4*cos(f*x + e)^2 + 128*a^3*c^4)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt
(-c*sin(f*x + e) + c)/(f*cos(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**(7/2)*(c-c*sin(f*x+e))**(9/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(9/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(7/2)*(-c*sin(f*x + e) + c)^(9/2)*cos(f*x + e)^2, x)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.07 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx=\frac {256 \, {\left (126 \, a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{20} - 560 \, a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{18} + 945 \, a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{16} - 720 \, a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} + 210 \, a^{3} c^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12}\right )} \sqrt {a} \sqrt {c}}{315 \, f} \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(9/2),x, algorithm="giac")

[Out]

256/315*(126*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2
*f*x + 1/2*e)^20 - 560*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/
4*pi + 1/2*f*x + 1/2*e)^18 + 945*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e
))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^16 - 720*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f
*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^14 + 210*a^3*c^4*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*
pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^12)*sqrt(a)*sqrt(c)/f

Mupad [B] (verification not implemented)

Time = 14.22 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.96 \[ \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx=\frac {{\mathrm {e}}^{-e\,10{}\mathrm {i}-f\,x\,10{}\mathrm {i}}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {63\,a^3\,c^4\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{64\,f}+\frac {21\,a^3\,c^4\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{256\,f}+\frac {3\,a^3\,c^4\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{64\,f}+\frac {9\,a^3\,c^4\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\cos \left (6\,e+6\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{512\,f}+\frac {a^3\,c^4\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\cos \left (8\,e+8\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{256\,f}+\frac {a^3\,c^4\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\cos \left (10\,e+10\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{2560\,f}+\frac {7\,a^3\,c^4\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{32\,f}+\frac {9\,a^3\,c^4\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{160\,f}+\frac {9\,a^3\,c^4\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\sin \left (7\,e+7\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{896\,f}+\frac {a^3\,c^4\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\sin \left (9\,e+9\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{1152\,f}\right )}{2\,\cos \left (e+f\,x\right )} \]

[In]

int(cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(9/2),x)

[Out]

(exp(- e*10i - f*x*10i)*(c - c*sin(e + f*x))^(1/2)*((63*a^3*c^4*exp(e*10i + f*x*10i)*sin(e + f*x)*(a + a*sin(e
+ f*x))^(1/2))/(64*f) + (21*a^3*c^4*exp(e*10i + f*x*10i)*cos(2*e + 2*f*x)*(a + a*sin(e + f*x))^(1/2))/(256*f)
+ (3*a^3*c^4*exp(e*10i + f*x*10i)*cos(4*e + 4*f*x)*(a + a*sin(e + f*x))^(1/2))/(64*f) + (9*a^3*c^4*exp(e*10i
+ f*x*10i)*cos(6*e + 6*f*x)*(a + a*sin(e + f*x))^(1/2))/(512*f) + (a^3*c^4*exp(e*10i + f*x*10i)*cos(8*e + 8*f*
x)*(a + a*sin(e + f*x))^(1/2))/(256*f) + (a^3*c^4*exp(e*10i + f*x*10i)*cos(10*e + 10*f*x)*(a + a*sin(e + f*x))
^(1/2))/(2560*f) + (7*a^3*c^4*exp(e*10i + f*x*10i)*sin(3*e + 3*f*x)*(a + a*sin(e + f*x))^(1/2))/(32*f) + (9*a^
3*c^4*exp(e*10i + f*x*10i)*sin(5*e + 5*f*x)*(a + a*sin(e + f*x))^(1/2))/(160*f) + (9*a^3*c^4*exp(e*10i + f*x*1
0i)*sin(7*e + 7*f*x)*(a + a*sin(e + f*x))^(1/2))/(896*f) + (a^3*c^4*exp(e*10i + f*x*10i)*sin(9*e + 9*f*x)*(a +
a*sin(e + f*x))^(1/2))/(1152*f)))/(2*cos(e + f*x))