Integrand size = 28, antiderivative size = 161 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=-\frac {4096 c^2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{405 f}+\frac {1024 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{81 f}-\frac {128 \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{27 f}+\frac {32 \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{81 c f}+\frac {2 \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{81 c^2 f} \]
-4096/15*c^2*sec(f*x+e)^5*(c-c*sin(f*x+e))^(5/2)/a^3/f+1024/3*c*sec(f*x+e) ^5*(c-c*sin(f*x+e))^(7/2)/a^3/f-128*sec(f*x+e)^5*(c-c*sin(f*x+e))^(9/2)/a^ 3/f+32/3*sec(f*x+e)^5*(c-c*sin(f*x+e))^(11/2)/a^3/c/f+2/3*sec(f*x+e)^5*(c- c*sin(f*x+e))^(13/2)/a^3/c^2/f
Time = 8.23 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.75 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=\frac {c^4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} (-5649+2740 \cos (2 (e+f x))+5 \cos (4 (e+f x))-7800 \sin (e+f x)+200 \sin (3 (e+f x)))}{1620 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3} \]
(c^4*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(-5649 + 2740*Cos[2*(e + f*x)] + 5*Cos[4*(e + f*x)] - 7800*Sin[e + f*x] + 200*Si n[3*(e + f*x)]))/(1620*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3)
Time = 0.96 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3215, 3042, 3153, 3042, 3153, 3042, 3153, 3042, 3153, 3042, 3152}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{9/2}}{(a \sin (e+f x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{9/2}}{(a \sin (e+f x)+a)^3}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle \frac {\int \sec ^6(e+f x) (c-c \sin (e+f x))^{15/2}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c-c \sin (e+f x))^{15/2}}{\cos (e+f x)^6}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {\frac {16}{3} c \int \sec ^6(e+f x) (c-c \sin (e+f x))^{13/2}dx+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {16}{3} c \int \frac {(c-c \sin (e+f x))^{13/2}}{\cos (e+f x)^6}dx+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {\frac {16}{3} c \left (12 c \int \sec ^6(e+f x) (c-c \sin (e+f x))^{11/2}dx+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {16}{3} c \left (12 c \int \frac {(c-c \sin (e+f x))^{11/2}}{\cos (e+f x)^6}dx+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {\frac {16}{3} c \left (12 c \left (-8 c \int \sec ^6(e+f x) (c-c \sin (e+f x))^{9/2}dx-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {16}{3} c \left (12 c \left (-8 c \int \frac {(c-c \sin (e+f x))^{9/2}}{\cos (e+f x)^6}dx-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3153 |
| \(\displaystyle \frac {\frac {16}{3} c \left (12 c \left (-8 c \left (-\frac {4}{3} c \int \sec ^6(e+f x) (c-c \sin (e+f x))^{7/2}dx-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 f}\right )-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {16}{3} c \left (12 c \left (-8 c \left (-\frac {4}{3} c \int \frac {(c-c \sin (e+f x))^{7/2}}{\cos (e+f x)^6}dx-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 f}\right )-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3152 |
| \(\displaystyle \frac {\frac {16}{3} c \left (12 c \left (-8 c \left (\frac {8 c^2 \sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{15 f}-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{7/2}}{3 f}\right )-\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{9/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{11/2}}{f}\right )+\frac {2 c \sec ^5(e+f x) (c-c \sin (e+f x))^{13/2}}{3 f}}{a^3 c^3}\) |
((2*c*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(13/2))/(3*f) + (16*c*((2*c*Sec[ e + f*x]^5*(c - c*Sin[e + f*x])^(11/2))/f + 12*c*((-2*c*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(9/2))/f - 8*c*((8*c^2*Sec[e + f*x]^5*(c - c*Sin[e + f*x ])^(5/2))/(15*f) - (2*c*Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(7/2))/(3*f))) ))/3)/(a^3*c^3)
3.4.32.3.1 Defintions of rubi rules used
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 - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]
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] + Simp[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] && IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[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] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 179.74 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.57
| method | result | size |
| default | \(-\frac {2 c^{5} \left (\sin \left (f x +e \right )-1\right ) \left (5 \left (\sin ^{4}\left (f x +e \right )\right )-100 \left (\sin ^{3}\left (f x +e \right )\right )-690 \left (\sin ^{2}\left (f x +e \right )\right )-900 \sin \left (f x +e \right )-363\right )}{15 a^{3} \left (\sin \left (f x +e \right )+1\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(91\) |
-2/15*c^5/a^3*(sin(f*x+e)-1)/(sin(f*x+e)+1)^2*(5*sin(f*x+e)^4-100*sin(f*x+ e)^3-690*sin(f*x+e)^2-900*sin(f*x+e)-363)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2 )/f
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.74 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (5 \, c^{4} \cos \left (f x + e\right )^{4} + 680 \, c^{4} \cos \left (f x + e\right )^{2} - 1048 \, c^{4} + 100 \, {\left (c^{4} \cos \left (f x + e\right )^{2} - 10 \, c^{4}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} - 2 \, a^{3} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} f \cos \left (f x + e\right )\right )}} \]
-2/15*(5*c^4*cos(f*x + e)^4 + 680*c^4*cos(f*x + e)^2 - 1048*c^4 + 100*(c^4 *cos(f*x + e)^2 - 10*c^4)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(a^3*f*c os(f*x + e)^3 - 2*a^3*f*cos(f*x + e)*sin(f*x + e) - 2*a^3*f*cos(f*x + e))
Timed out. \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (156) = 312\).
Time = 0.30 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.93 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=\frac {2 \, {\left (363 \, c^{\frac {9}{2}} + \frac {1800 \, c^{\frac {9}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5301 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {11600 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {21343 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {30200 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {40065 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {40800 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {40065 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {30200 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {21343 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {11600 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} + \frac {5301 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} + \frac {1800 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{13}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{13}} + \frac {363 \, c^{\frac {9}{2}} \sin \left (f x + e\right )^{14}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{14}}\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )} f {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {9}{2}}} \]
2/15*(363*c^(9/2) + 1800*c^(9/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 5301*c^ (9/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 11600*c^(9/2)*sin(f*x + e)^3/( cos(f*x + e) + 1)^3 + 21343*c^(9/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 30200*c^(9/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 40065*c^(9/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 40800*c^(9/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 40065*c^(9/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 30200*c^(9/2)* sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 21343*c^(9/2)*sin(f*x + e)^10/(cos(f *x + e) + 1)^10 + 11600*c^(9/2)*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 53 01*c^(9/2)*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 + 1800*c^(9/2)*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 363*c^(9/2)*sin(f*x + e)^14/(cos(f*x + e) + 1)^14)/((a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e) ^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a ^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5)*f*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(9/2))
Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (156) = 312\).
Time = 0.39 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.65 \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=-\frac {8 \, \sqrt {2} \sqrt {c} {\left (\frac {5 \, {\left (11 \, c^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - \frac {24 \, c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {9 \, c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}\right )}}{a^{3} {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - 1\right )}^{3}} - \frac {73 \, c^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + \frac {320 \, c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {490 \, c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} + \frac {240 \, c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {45 \, c^{4} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}}}{a^{3} {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}^{5}}\right )}}{15 \, f} \]
-8/15*sqrt(2)*sqrt(c)*(5*(11*c^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 24* c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e ))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 9*c^4*(cos(-1/4*pi + 1/2*f*x + 1 /2*e) - 1)^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2)/(a^3*((cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1 /2*f*x + 1/2*e) + 1) - 1)^3) - (73*c^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 320*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 490*c^4*(cos(-1/4*pi + 1/ 2*f*x + 1/2*e) - 1)^2*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1 /2*f*x + 1/2*e) + 1)^2 + 240*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^3*sg n(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 + 45*c^4*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^4)/(a^3*((cos(-1/4*pi + 1/2* f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 1)^5))/f
Timed out. \[ \int \frac {(c-c \sin (e+f x))^{9/2}}{(3+3 \sin (e+f x))^3} \, dx=\int \frac {{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]