Integrand size = 38, antiderivative size = 241 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {4 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f \sqrt {c-c \sin (e+f x)}} \]
1/3*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c/f/(c-c*sin(f*x+e))^(7/2)-2/3*a*cos (f*x+e)*(a+a*sin(f*x+e))^(5/2)/c^2/f/(c-c*sin(f*x+e))^(5/2)+2*a^2*cos(f*x+ e)*(a+a*sin(f*x+e))^(3/2)/c^3/f/(c-c*sin(f*x+e))^(3/2)+8*a^4*cos(f*x+e)*ln (1-sin(f*x+e))/c^4/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)+4*a^3*c os(f*x+e)*(a+a*sin(f*x+e))^(1/2)/c^4/f/(c-c*sin(f*x+e))^(1/2)
Time = 13.80 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=-\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {a (1+\sin (e+f x))} \left (-563+3 \cos (4 (e+f x))-960 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+48 \cos (2 (e+f x)) \left (5+12 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )+690 \sin (e+f x)+1440 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)+18 \sin (3 (e+f x))-96 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))\right )}{24 c^4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x))^4 \sqrt {c-c \sin (e+f x)}} \]
-1/24*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x ])]*(-563 + 3*Cos[4*(e + f*x)] - 960*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/ 2]] + 48*Cos[2*(e + f*x)]*(5 + 12*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] ) + 690*Sin[e + f*x] + 1440*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + 18*Sin[3*(e + f*x)] - 96*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2] ]*Sin[3*(e + f*x)]))/(c^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Si n[e + f*x])^4*Sqrt[c - c*Sin[e + f*x]])
Time = 1.47 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.395, Rules used = {3042, 3320, 3042, 3218, 3042, 3218, 3042, 3218, 3042, 3219, 3042, 3216, 3042, 3146, 16}
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.
| \(\displaystyle \int \frac {\cos ^2(e+f x) (a \sin (e+f x)+a)^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (e+f x)^2 (a \sin (e+f x)+a)^{7/2}}{(c-c \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3320 |
| \(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{9/2}}{(c-c \sin (e+f x))^{7/2}}dx}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{9/2}}{(c-c \sin (e+f x))^{7/2}}dx}{a c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {4 a \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{5/2}}dx}{3 c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {4 a \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{5/2}}dx}{3 c}}{a c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {4 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{3/2}}dx}{2 c}\right )}{3 c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {4 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{3/2}}dx}{2 c}\right )}{3 c}}{a c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {4 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \int \frac {(\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{2 c}\right )}{3 c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {4 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \int \frac {(\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{2 c}\right )}{3 c}}{a c}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {4 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \left (2 a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{2 c}\right )}{3 c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {4 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \left (2 a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{2 c}\right )}{3 c}}{a c}\) |
| \(\Big \downarrow \) 3216 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {4 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \left (\frac {2 a^2 c \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{2 c}\right )}{3 c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {4 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \left (\frac {2 a^2 c \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{2 c}\right )}{3 c}}{a c}\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {4 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \left (-\frac {2 a^2 \cos (e+f x) \int \frac {1}{c-c \sin (e+f x)}d(-c \sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{2 c}\right )}{3 c}}{a c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {4 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {2 a \left (-\frac {2 a^2 \cos (e+f x) \log (c-c \sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{2 c}\right )}{3 c}}{a c}\) |
((a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(3*f*(c - c*Sin[e + f*x])^(7/ 2)) - (4*a*((a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(2*f*(c - c*Sin[e + f*x])^(5/2)) - (3*a*((a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(f*(c - c*Sin[e + f*x])^(3/2)) - (2*a*((-2*a^2*Cos[e + f*x]*Log[c - c*Sin[e + f*x ]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (a*Cos[e + f*x ]*Sqrt[a + a*Sin[e + f*x]])/(f*Sqrt[c - c*Sin[e + f*x]])))/c))/(2*c)))/(3* c))/(a*c)
3.1.39.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[a*c*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x ]]*Sqrt[c + d*Sin[e + f*x]])) Int[Cos[e + f*x]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0 ]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(2*n + 1))), x] - Simp[b*((2*m - 1)/(d*( 2*n + 1))) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b ^2, 0] && IGtQ[m - 1/2, 0] && LtQ[n, -1] && !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
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] + Simp[a*((2*m - 1)/(m + n )) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I GtQ[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])
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[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]
Time = 0.34 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.25
| method | result | size |
| default | \(-\frac {\left (48 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-24 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-3 \left (\cos ^{4}\left (f x +e \right )\right )-144 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+72 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-49 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-192 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \sin \left (f x +e \right )+96 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )+63 \left (\cos ^{2}\left (f x +e \right )\right )+192 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-96 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+76 \sin \left (f x +e \right )-60\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{3} \sec \left (f x +e \right )}{3 f \left (\cos ^{2}\left (f x +e \right )+2 \sin \left (f x +e \right )-2\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{4}}\) | \(301\) |
-1/3/f*(48*cos(f*x+e)^2*sin(f*x+e)*ln(csc(f*x+e)-cot(f*x+e)-1)-24*cos(f*x+ e)^2*sin(f*x+e)*ln(2/(1+cos(f*x+e)))-3*cos(f*x+e)^4-144*cos(f*x+e)^2*ln(cs c(f*x+e)-cot(f*x+e)-1)+72*cos(f*x+e)^2*ln(2/(1+cos(f*x+e)))-49*cos(f*x+e)^ 2*sin(f*x+e)-192*ln(csc(f*x+e)-cot(f*x+e)-1)*sin(f*x+e)+96*ln(2/(1+cos(f*x +e)))*sin(f*x+e)+63*cos(f*x+e)^2+192*ln(csc(f*x+e)-cot(f*x+e)-1)-96*ln(2/( 1+cos(f*x+e)))+76*sin(f*x+e)-60)*(a*(1+sin(f*x+e)))^(1/2)*a^3/(cos(f*x+e)^ 2+2*sin(f*x+e)-2)/(-c*(sin(f*x+e)-1))^(1/2)/c^4*sec(f*x+e)
\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]
integral(-(3*a^3*cos(f*x + e)^4 - 4*a^3*cos(f*x + e)^2 + (a^3*cos(f*x + e) ^4 - 4*a^3*cos(f*x + e)^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c* sin(f*x + e) + c)/(5*c^5*cos(f*x + e)^4 - 20*c^5*cos(f*x + e)^2 + 16*c^5 - (c^5*cos(f*x + e)^4 - 12*c^5*cos(f*x + e)^2 + 16*c^5)*sin(f*x + e)), x)
Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.71 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=-\frac {2 \, a^{\frac {7}{2}} \sqrt {c} {\left (\frac {3 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{c^{5} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {12 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{c^{5} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {18 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 30 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 13}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} c^{5} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{3 \, f} \]
-2/3*a^(7/2)*sqrt(c)*(3*cos(-1/4*pi + 1/2*f*x + 1/2*e)^2/(c^5*sgn(sin(-1/4 *pi + 1/2*f*x + 1/2*e))) + 12*log(-cos(-1/4*pi + 1/2*f*x + 1/2*e)^2 + 1)/( c^5*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) - (18*cos(-1/4*pi + 1/2*f*x + 1/2 *e)^4 - 30*cos(-1/4*pi + 1/2*f*x + 1/2*e)^2 + 13)/((cos(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 1)^3*c^5*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))))*sgn(cos(-1/4*p i + 1/2*f*x + 1/2*e))/f
Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \]