Integrand size = 26, antiderivative size = 98 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx=\frac {a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac {2 a^2 c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6}+\frac {2 a^2 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5} \] Output:
1/9*a^2*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^7+2/63*a^2*c*cos(f*x+e)^5/f/(c -c*sin(f*x+e))^6+2/315*a^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^5
Time = 4.58 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.23 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx=\frac {a^2 \left (315 \cos \left (\frac {1}{2} (e+f x)\right )-126 \cos \left (\frac {3}{2} (e+f x)\right )-9 \cos \left (\frac {7}{2} (e+f x)\right )+441 \sin \left (\frac {1}{2} (e+f x)\right )+210 \sin \left (\frac {3}{2} (e+f x)\right )-36 \sin \left (\frac {5}{2} (e+f x)\right )+\sin \left (\frac {9}{2} (e+f x)\right )\right )}{1260 c^5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \] Input:
Integrate[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^5,x]
Output:
(a^2*(315*Cos[(e + f*x)/2] - 126*Cos[(3*(e + f*x))/2] - 9*Cos[(7*(e + f*x) )/2] + 441*Sin[(e + f*x)/2] + 210*Sin[(3*(e + f*x))/2] - 36*Sin[(5*(e + f* x))/2] + Sin[(9*(e + f*x))/2]))/(1260*c^5*f*(Cos[(e + f*x)/2] - Sin[(e + f *x)/2])^9)
Time = 0.55 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 3215, 3042, 3151, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2}{(c-c \sin (e+f x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2}{(c-c \sin (e+f x))^5}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^2 c^2 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^7}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^7}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^2 c^2 \left (\frac {2 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^6}dx}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {2 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^6}dx}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^2 c^2 \left (\frac {2 \left (\frac {\int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^5}dx}{7 c}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {2 \left (\frac {\int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^5}dx}{7 c}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle a^2 c^2 \left (\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac {2 \left (\frac {\cos ^5(e+f x)}{35 c f (c-c \sin (e+f x))^5}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}\right )\) |
Input:
Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^5,x]
Output:
a^2*c^2*(Cos[e + f*x]^5/(9*f*(c - c*Sin[e + f*x])^7) + (2*(Cos[e + f*x]^5/ (7*f*(c - c*Sin[e + f*x])^6) + Cos[e + f*x]^5/(35*c*f*(c - c*Sin[e + f*x]) ^5)))/(9*c))
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 = 1.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(-\frac {4 \left (-126 i a^{2} {\mathrm e}^{3 i \left (f x +e \right )}-441 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}-9 i a^{2} {\mathrm e}^{i \left (f x +e \right )}+36 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}-a^{2}+315 i a^{2} {\mathrm e}^{5 i \left (f x +e \right )}+210 a^{2} {\mathrm e}^{6 i \left (f x +e \right )}\right )}{315 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9} f \,c^{5}}\) | \(116\) |
| parallelrisch | \(-\frac {2 a^{2} \left (315 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-630 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+2310 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-2520 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+3402 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-1638 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+1062 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-108 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+47\right )}{315 f \,c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(129\) |
| derivativedivides | \(\frac {2 a^{2} \left (-\frac {404}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {6}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {64}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {50}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {480}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {272}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}\right )}{f \,c^{5}}\) | \(148\) |
| default | \(\frac {2 a^{2} \left (-\frac {404}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {6}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {64}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {50}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {480}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {272}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}\right )}{f \,c^{5}}\) | \(148\) |
| norman | \(\frac {\frac {24 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{f c}-\frac {94 a^{2}}{315 c f}-\frac {2 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{f c}+\frac {4 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{f c}+\frac {24 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35 c f}-\frac {56 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{3 f c}+\frac {232 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{5 f c}+\frac {412 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{35 c f}-\frac {794 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{15 f c}-\frac {2312 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{315 c f}-\frac {6784 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{105 f c}+\frac {1312 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{35 c f}-\frac {11146 a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{315 c f}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{2} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(307\) |
Input:
int((a+sin(f*x+e)*a)^2/(c-c*sin(f*x+e))^5,x,method=_RETURNVERBOSE)
Output:
-4/315*(-126*I*a^2*exp(3*I*(f*x+e))-441*a^2*exp(4*I*(f*x+e))-9*I*a^2*exp(I *(f*x+e))+36*a^2*exp(2*I*(f*x+e))-a^2+315*I*a^2*exp(5*I*(f*x+e))+210*a^2*e xp(6*I*(f*x+e)))/(exp(I*(f*x+e))-I)^9/f/c^5
Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (95) = 190\).
Time = 0.09 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.84 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx=\frac {2 \, a^{2} \cos \left (f x + e\right )^{5} - 8 \, a^{2} \cos \left (f x + e\right )^{4} - 25 \, a^{2} \cos \left (f x + e\right )^{3} - 85 \, a^{2} \cos \left (f x + e\right )^{2} + 70 \, a^{2} \cos \left (f x + e\right ) + 140 \, a^{2} + {\left (2 \, a^{2} \cos \left (f x + e\right )^{4} + 10 \, a^{2} \cos \left (f x + e\right )^{3} - 15 \, a^{2} \cos \left (f x + e\right )^{2} + 70 \, a^{2} \cos \left (f x + e\right ) + 140 \, a^{2}\right )} \sin \left (f x + e\right )}{315 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm="fricas")
Output:
1/315*(2*a^2*cos(f*x + e)^5 - 8*a^2*cos(f*x + e)^4 - 25*a^2*cos(f*x + e)^3 - 85*a^2*cos(f*x + e)^2 + 70*a^2*cos(f*x + e) + 140*a^2 + (2*a^2*cos(f*x + e)^4 + 10*a^2*cos(f*x + e)^3 - 15*a^2*cos(f*x + e)^2 + 70*a^2*cos(f*x + e) + 140*a^2)*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*co s(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. \(2 (88) = 176\).
Time = 16.65 (sec) , antiderivative size = 1717, normalized size of antiderivative = 17.52 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))**2/(c-c*sin(f*x+e))**5,x)
Output:
Piecewise((-630*a**2*tan(e/2 + f*x/2)**8/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 2646 0*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c* *5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f *tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) + 1260*a **2*tan(e/2 + f*x/2)**7/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan( e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f* x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2) **2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 4620*a**2*tan(e/2 + f*x /2)**6/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 396 90*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c **5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f *tan(e/2 + f*x/2) - 315*c**5*f) + 5040*a**2*tan(e/2 + f*x/2)**5/(315*c**5* f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan (e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f *x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 6804*a**2*tan(e/2 + f*x/2)**4/(315*c**5*f*tan(e/2 + f*...
Leaf count of result is larger than twice the leaf count of optimal. 1073 vs. \(2 (95) = 190\).
Time = 0.07 (sec) , antiderivative size = 1073, normalized size of antiderivative = 10.95 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm="maxima")
Output:
-2/315*(a^2*(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) - 10*a^2*(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) + 14*a^2*(9*s...
Time = 0.17 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.56 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx=-\frac {2 \, {\left (315 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 630 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 2310 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3402 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1638 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1062 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 108 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 47 \, a^{2}\right )}}{315 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}} \] Input:
integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm="giac")
Output:
-2/315*(315*a^2*tan(1/2*f*x + 1/2*e)^8 - 630*a^2*tan(1/2*f*x + 1/2*e)^7 + 2310*a^2*tan(1/2*f*x + 1/2*e)^6 - 2520*a^2*tan(1/2*f*x + 1/2*e)^5 + 3402*a ^2*tan(1/2*f*x + 1/2*e)^4 - 1638*a^2*tan(1/2*f*x + 1/2*e)^3 + 1062*a^2*tan (1/2*f*x + 1/2*e)^2 - 108*a^2*tan(1/2*f*x + 1/2*e) + 47*a^2)/(c^5*f*(tan(1 /2*f*x + 1/2*e) - 1)^9)
Time = 18.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.23 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx=\frac {\sqrt {2}\,a^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {89\,\cos \left (3\,e+3\,f\,x\right )}{4}-\frac {2205\,\sin \left (e+f\,x\right )}{8}-\frac {265\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {625\,\cos \left (e+f\,x\right )}{4}+\frac {49\,\cos \left (4\,e+4\,f\,x\right )}{16}+\frac {567\,\sin \left (2\,e+2\,f\,x\right )}{8}+\frac {243\,\sin \left (3\,e+3\,f\,x\right )}{8}-\frac {45\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {4967}{16}\right )}{5040\,c^5\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^9} \] Input:
int((a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^5,x)
Output:
(2^(1/2)*a^2*cos(e/2 + (f*x)/2)*((89*cos(3*e + 3*f*x))/4 - (2205*sin(e + f *x))/8 - (265*cos(2*e + 2*f*x))/2 - (625*cos(e + f*x))/4 + (49*cos(4*e + 4 *f*x))/16 + (567*sin(2*e + 2*f*x))/8 + (243*sin(3*e + 3*f*x))/8 - (45*sin( 4*e + 4*f*x))/16 + 4967/16))/(5040*c^5*f*cos(e/2 + pi/4 + (f*x)/2)^9)
Time = 0.17 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.53 \[ \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx=\frac {a^{2} \left (-25 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}+102 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}-159 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+12 \cos \left (f x +e \right ) \sin \left (f x +e \right )-70 \cos \left (f x +e \right )-21 \sin \left (f x +e \right )^{5}+107 \sin \left (f x +e \right )^{4}-219 \sin \left (f x +e \right )^{3}+331 \sin \left (f x +e \right )^{2}+12 \sin \left (f x +e \right )+70\right )}{315 c^{5} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}-4 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}+6 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}-4 \cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )+\sin \left (f x +e \right )^{5}-5 \sin \left (f x +e \right )^{4}+10 \sin \left (f x +e \right )^{3}-10 \sin \left (f x +e \right )^{2}+5 \sin \left (f x +e \right )-1\right )} \] Input:
int((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x)
Output:
(a**2*( - 25*cos(e + f*x)*sin(e + f*x)**4 + 102*cos(e + f*x)*sin(e + f*x)* *3 - 159*cos(e + f*x)*sin(e + f*x)**2 + 12*cos(e + f*x)*sin(e + f*x) - 70* cos(e + f*x) - 21*sin(e + f*x)**5 + 107*sin(e + f*x)**4 - 219*sin(e + f*x) **3 + 331*sin(e + f*x)**2 + 12*sin(e + f*x) + 70))/(315*c**5*f*(cos(e + f* x)*sin(e + f*x)**4 - 4*cos(e + f*x)*sin(e + f*x)**3 + 6*cos(e + f*x)*sin(e + f*x)**2 - 4*cos(e + f*x)*sin(e + f*x) + cos(e + f*x) + sin(e + f*x)**5 - 5*sin(e + f*x)**4 + 10*sin(e + f*x)**3 - 10*sin(e + f*x)**2 + 5*sin(e + f*x) - 1))