Integrand size = 26, antiderivative size = 69 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=\frac {a^3 c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}+\frac {a^3 c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^7} \]
1/9*a^3*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^8+1/63*a^3*c^2*cos(f*x+e)^7/f/ (c-c*sin(f*x+e))^7
Time = 2.41 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.96 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=\frac {a^3 \left (315 \cos \left (\frac {1}{2} (e+f x)\right )-189 \cos \left (\frac {3}{2} (e+f x)\right )-63 \cos \left (\frac {5}{2} (e+f x)\right )+9 \cos \left (\frac {7}{2} (e+f x)\right )+189 \sin \left (\frac {1}{2} (e+f x)\right )+105 \sin \left (\frac {3}{2} (e+f x)\right )-27 \sin \left (\frac {5}{2} (e+f x)\right )-\sin \left (\frac {9}{2} (e+f x)\right )\right )}{504 c^5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \]
(a^3*(315*Cos[(e + f*x)/2] - 189*Cos[(3*(e + f*x))/2] - 63*Cos[(5*(e + f*x ))/2] + 9*Cos[(7*(e + f*x))/2] + 189*Sin[(e + f*x)/2] + 105*Sin[(3*(e + f* x))/2] - 27*Sin[(5*(e + f*x))/2] - Sin[(9*(e + f*x))/2]))/(504*c^5*f*(Cos[ (e + f*x)/2] - Sin[(e + f*x)/2])^9)
Time = 0.40 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3042, 3215, 3042, 3151, 3042, 3150}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c-c \sin (e+f x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3}{(c-c \sin (e+f x))^5}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^8}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^8}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^3 c^3 \left (\frac {\int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7}dx}{9 c}+\frac {\cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {\int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^7}dx}{9 c}+\frac {\cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}\right )\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle a^3 c^3 \left (\frac {\cos ^7(e+f x)}{63 c f (c-c \sin (e+f x))^7}+\frac {\cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}\right )\) |
a^3*c^3*(Cos[e + f*x]^7/(9*f*(c - c*Sin[e + f*x])^8) + Cos[e + f*x]^7/(63* c*f*(c - c*Sin[e + f*x])^7))
3.3.57.3.1 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/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]
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/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[2*m + p + 1]) 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] && ILtQ[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 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]))
Result contains complex when optimal does not.
Time = 1.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.59
| method | result | size |
| risch | \(\frac {2 i a^{3} \left (105 i {\mathrm e}^{6 i \left (f x +e \right )}+63 \,{\mathrm e}^{7 i \left (f x +e \right )}-189 i {\mathrm e}^{4 i \left (f x +e \right )}-315 \,{\mathrm e}^{5 i \left (f x +e \right )}+27 i {\mathrm e}^{2 i \left (f x +e \right )}+189 \,{\mathrm e}^{3 i \left (f x +e \right )}+i-9 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{63 f \,c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9}}\) | \(110\) |
| parallelrisch | \(-\frac {2 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )-\left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {23 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-5 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+11 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-3 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {25 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7}+\frac {8}{63}\right ) a^{3}}{f \,c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(127\) |
| derivativedivides | \(\frac {2 a^{3} \left (-\frac {928}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {128}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {7}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {136}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {76}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {496}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {86}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{5}}\) | \(148\) |
| default | \(\frac {2 a^{3} \left (-\frac {928}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {128}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {7}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {136}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {76}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {496}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {86}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{5}}\) | \(148\) |
| norman | \(\frac {\frac {2 a^{3} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {16 a^{3} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {42 a^{3} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {16 a^{3}}{63 c f}-\frac {2 a^{3} \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {2 a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7 c f}+\frac {48 a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {64 a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {74 a^{3} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {166 a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f}+\frac {202 a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}+\frac {352 a^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {848 a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {928 a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21 c f}-\frac {6490 a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(351\) |
2/63*I*a^3*(105*I*exp(6*I*(f*x+e))+63*exp(7*I*(f*x+e))-189*I*exp(4*I*(f*x+ e))-315*exp(5*I*(f*x+e))+27*I*exp(2*I*(f*x+e))+189*exp(3*I*(f*x+e))+I-9*ex p(I*(f*x+e)))/f/c^5/(exp(I*(f*x+e))-I)^9
Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (67) = 134\).
Time = 0.26 (sec) , antiderivative size = 276, normalized size of antiderivative = 4.00 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=-\frac {a^{3} \cos \left (f x + e\right )^{5} - 4 \, a^{3} \cos \left (f x + e\right )^{4} + 19 \, a^{3} \cos \left (f x + e\right )^{3} + 52 \, a^{3} \cos \left (f x + e\right )^{2} - 28 \, a^{3} \cos \left (f x + e\right ) - 56 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{4} + 5 \, a^{3} \cos \left (f x + e\right )^{3} + 24 \, a^{3} \cos \left (f x + e\right )^{2} - 28 \, a^{3} \cos \left (f x + e\right ) - 56 \, a^{3}\right )} \sin \left (f x + e\right )}{63 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \]
-1/63*(a^3*cos(f*x + e)^5 - 4*a^3*cos(f*x + e)^4 + 19*a^3*cos(f*x + e)^3 + 52*a^3*cos(f*x + e)^2 - 28*a^3*cos(f*x + e) - 56*a^3 + (a^3*cos(f*x + e)^ 4 + 5*a^3*cos(f*x + e)^3 + 24*a^3*cos(f*x + e)^2 - 28*a^3*cos(f*x + e) - 5 6*a^3)*sin(f*x + e))/(c^5*f*cos(f*x + e)^5 + 5*c^5*f*cos(f*x + e)^4 - 8*c^ 5*f*cos(f*x + e)^3 - 20*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c ^5*f - (c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 1717 vs. \(2 (60) = 120\).
Time = 25.07 (sec) , antiderivative size = 1717, normalized size of antiderivative = 24.88 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \]
Piecewise((-126*a**3*tan(e/2 + f*x/2)**8/(63*c**5*f*tan(e/2 + f*x/2)**9 - 567*c**5*f*tan(e/2 + f*x/2)**8 + 2268*c**5*f*tan(e/2 + f*x/2)**7 - 5292*c* *5*f*tan(e/2 + f*x/2)**6 + 7938*c**5*f*tan(e/2 + f*x/2)**5 - 7938*c**5*f*t an(e/2 + f*x/2)**4 + 5292*c**5*f*tan(e/2 + f*x/2)**3 - 2268*c**5*f*tan(e/2 + f*x/2)**2 + 567*c**5*f*tan(e/2 + f*x/2) - 63*c**5*f) + 126*a**3*tan(e/2 + f*x/2)**7/(63*c**5*f*tan(e/2 + f*x/2)**9 - 567*c**5*f*tan(e/2 + f*x/2)* *8 + 2268*c**5*f*tan(e/2 + f*x/2)**7 - 5292*c**5*f*tan(e/2 + f*x/2)**6 + 7 938*c**5*f*tan(e/2 + f*x/2)**5 - 7938*c**5*f*tan(e/2 + f*x/2)**4 + 5292*c* *5*f*tan(e/2 + f*x/2)**3 - 2268*c**5*f*tan(e/2 + f*x/2)**2 + 567*c**5*f*ta n(e/2 + f*x/2) - 63*c**5*f) - 966*a**3*tan(e/2 + f*x/2)**6/(63*c**5*f*tan( e/2 + f*x/2)**9 - 567*c**5*f*tan(e/2 + f*x/2)**8 + 2268*c**5*f*tan(e/2 + f *x/2)**7 - 5292*c**5*f*tan(e/2 + f*x/2)**6 + 7938*c**5*f*tan(e/2 + f*x/2)* *5 - 7938*c**5*f*tan(e/2 + f*x/2)**4 + 5292*c**5*f*tan(e/2 + f*x/2)**3 - 2 268*c**5*f*tan(e/2 + f*x/2)**2 + 567*c**5*f*tan(e/2 + f*x/2) - 63*c**5*f) + 630*a**3*tan(e/2 + f*x/2)**5/(63*c**5*f*tan(e/2 + f*x/2)**9 - 567*c**5*f *tan(e/2 + f*x/2)**8 + 2268*c**5*f*tan(e/2 + f*x/2)**7 - 5292*c**5*f*tan(e /2 + f*x/2)**6 + 7938*c**5*f*tan(e/2 + f*x/2)**5 - 7938*c**5*f*tan(e/2 + f *x/2)**4 + 5292*c**5*f*tan(e/2 + f*x/2)**3 - 2268*c**5*f*tan(e/2 + f*x/2)* *2 + 567*c**5*f*tan(e/2 + f*x/2) - 63*c**5*f) - 1386*a**3*tan(e/2 + f*x/2) **4/(63*c**5*f*tan(e/2 + f*x/2)**9 - 567*c**5*f*tan(e/2 + f*x/2)**8 + 2...
Leaf count of result is larger than twice the leaf count of optimal. 1389 vs. \(2 (67) = 134\).
Time = 0.24 (sec) , antiderivative size = 1389, normalized size of antiderivative = 20.13 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \]
-2/315*(a^3*(432*sin(f*x + e)/(cos(f*x + e) + 1) - 1728*sin(f*x + e)^2/(co s(f*x + e) + 1)^2 + 3612*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 5418*sin(f* x + e)^4/(cos(f*x + e) + 1)^4 + 5040*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 3360*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1260*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 315*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 83)/(c^5 - 9*c^5*si n(f*x + e)/(cos(f*x + e) + 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x + e)^4/(co s(f*x + e) + 1)^4 - 126*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*s in(f*x + e)^6/(cos(f*x + e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(f*x + e)^9/(co s(f*x + e) + 1)^9) - 15*a^3*(45*sin(f*x + e)/(cos(f*x + e) + 1) - 117*sin( f*x + e)^2/(cos(f*x + e) + 1)^2 + 273*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 315*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 315*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 147*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 63*sin(f*x + e)^7/(c os(f*x + e) + 1)^7 - 5)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 36* c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f *x + e) + 1)^8 - c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) - 10*a^3*(9*s...
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (67) = 134\).
Time = 0.34 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.22 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=-\frac {2 \, {\left (63 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 63 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 483 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 315 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 693 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 189 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 225 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 8 \, a^{3}\right )}}{63 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}} \]
-2/63*(63*a^3*tan(1/2*f*x + 1/2*e)^8 - 63*a^3*tan(1/2*f*x + 1/2*e)^7 + 483 *a^3*tan(1/2*f*x + 1/2*e)^6 - 315*a^3*tan(1/2*f*x + 1/2*e)^5 + 693*a^3*tan (1/2*f*x + 1/2*e)^4 - 189*a^3*tan(1/2*f*x + 1/2*e)^3 + 225*a^3*tan(1/2*f*x + 1/2*e)^2 - 9*a^3*tan(1/2*f*x + 1/2*e) + 8*a^3)/(c^5*f*(tan(1/2*f*x + 1/ 2*e) - 1)^9)
Time = 7.89 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.75 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^5} \, dx=\frac {\sqrt {2}\,a^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {37\,\cos \left (3\,e+3\,f\,x\right )}{8}-\frac {63\,\sin \left (e+f\,x\right )}{2}-\frac {113\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {257\,\cos \left (e+f\,x\right )}{8}+\frac {7\,\cos \left (4\,e+4\,f\,x\right )}{16}+\frac {63\,\sin \left (2\,e+2\,f\,x\right )}{8}+\frac {9\,\sin \left (3\,e+3\,f\,x\right )}{2}-\frac {9\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {1013}{16}\right )}{1008\,c^5\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^9} \]