Integrand size = 24, antiderivative size = 126 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx=\frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}+\frac {a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^4}+\frac {2 a c \cos ^3(e+f x)}{315 f \left (c^2-c^2 \sin (e+f x)\right )^3} \] Output:
1/9*a*c*cos(f*x+e)^3/f/(c-c*sin(f*x+e))^6+1/21*a*cos(f*x+e)^3/f/(c-c*sin(f *x+e))^5+2/105*a*cos(f*x+e)^3/c/f/(c-c*sin(f*x+e))^4+2/315*a*c*cos(f*x+e)^ 3/f/(c^2-c^2*sin(f*x+e))^3
Time = 5.88 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx=\frac {a \left (315 \cos \left (e+\frac {f x}{2}\right )-84 \cos \left (e+\frac {3 f x}{2}\right )+9 \cos \left (3 e+\frac {7 f x}{2}\right )+189 \sin \left (\frac {f x}{2}\right )+36 \sin \left (2 e+\frac {5 f x}{2}\right )-\sin \left (4 e+\frac {9 f x}{2}\right )\right )}{1260 c^5 f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \] Input:
Integrate[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^5,x]
Output:
(a*(315*Cos[e + (f*x)/2] - 84*Cos[e + (3*f*x)/2] + 9*Cos[3*e + (7*f*x)/2] + 189*Sin[(f*x)/2] + 36*Sin[2*e + (5*f*x)/2] - Sin[4*e + (9*f*x)/2]))/(126 0*c^5*f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9)
Time = 0.64 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3042, 3215, 3042, 3151, 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.
| \(\displaystyle \int \frac {a \sin (e+f x)+a}{(c-c \sin (e+f x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \sin (e+f x)+a}{(c-c \sin (e+f x))^5}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a c \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^6}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \int \frac {\cos (e+f x)^2}{(c-c \sin (e+f x))^6}dx\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a c \left (\frac {\int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^5}dx}{3 c}+\frac {\cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {\int \frac {\cos (e+f x)^2}{(c-c \sin (e+f x))^5}dx}{3 c}+\frac {\cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a c \left (\frac {\frac {2 \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^4}dx}{7 c}+\frac {\cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}}{3 c}+\frac {\cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {\frac {2 \int \frac {\cos (e+f x)^2}{(c-c \sin (e+f x))^4}dx}{7 c}+\frac {\cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}}{3 c}+\frac {\cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a c \left (\frac {\frac {2 \left (\frac {\int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^3}dx}{5 c}+\frac {\cos ^3(e+f x)}{5 f (c-c \sin (e+f x))^4}\right )}{7 c}+\frac {\cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}}{3 c}+\frac {\cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {\frac {2 \left (\frac {\int \frac {\cos (e+f x)^2}{(c-c \sin (e+f x))^3}dx}{5 c}+\frac {\cos ^3(e+f x)}{5 f (c-c \sin (e+f x))^4}\right )}{7 c}+\frac {\cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}}{3 c}+\frac {\cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}\right )\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle a c \left (\frac {\cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}+\frac {\frac {\cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}+\frac {2 \left (\frac {\cos ^3(e+f x)}{15 c f (c-c \sin (e+f x))^3}+\frac {\cos ^3(e+f x)}{5 f (c-c \sin (e+f x))^4}\right )}{7 c}}{3 c}\right )\) |
Input:
Int[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^5,x]
Output:
a*c*(Cos[e + f*x]^3/(9*f*(c - c*Sin[e + f*x])^6) + (Cos[e + f*x]^3/(7*f*(c - c*Sin[e + f*x])^5) + (2*(Cos[e + f*x]^3/(5*f*(c - c*Sin[e + f*x])^4) + Cos[e + f*x]^3/(15*c*f*(c - c*Sin[e + f*x])^3)))/(7*c))/(3*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 = 0.95 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(-\frac {4 i a \left (189 i {\mathrm e}^{4 i \left (f x +e \right )}+315 \,{\mathrm e}^{5 i \left (f x +e \right )}+36 i {\mathrm e}^{2 i \left (f x +e \right )}-84 \,{\mathrm e}^{3 i \left (f x +e \right )}-i+9 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{315 f \,c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9}}\) | \(85\) |
| parallelrisch | \(-\frac {2 a \left (315 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-945 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+2625 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-3465 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+3843 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-2247 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+1143 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-207 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+58\right )}{315 f \,c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(127\) |
| derivativedivides | \(\frac {2 a \left (-\frac {148}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {236}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {46}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {32}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {248}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\right )}{f \,c^{5}}\) | \(146\) |
| default | \(\frac {2 a \left (-\frac {148}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {236}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {46}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {32}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {248}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\right )}{f \,c^{5}}\) | \(146\) |
| norman | \(\frac {\frac {6 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{f c}-\frac {116 a}{315 c f}-\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{f c}+\frac {28 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{f c}-\frac {616 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{15 f c}-\frac {56 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{3 f c}+\frac {46 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35 f c}+\frac {544 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{15 f c}-\frac {1108 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{35 c f}+\frac {1636 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{105 f c}-\frac {2402 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{315 c f}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(241\) |
Input:
int((a+sin(f*x+e)*a)/(c-c*sin(f*x+e))^5,x,method=_RETURNVERBOSE)
Output:
-4/315*I*a*(189*I*exp(4*I*(f*x+e))+315*exp(5*I*(f*x+e))+36*I*exp(2*I*(f*x+ e))-84*exp(3*I*(f*x+e))-I+9*exp(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. 256 vs. \(2 (122) = 244\).
Time = 0.08 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.03 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx=-\frac {2 \, a \cos \left (f x + e\right )^{5} - 8 \, a \cos \left (f x + e\right )^{4} - 25 \, a \cos \left (f x + e\right )^{3} + 20 \, a \cos \left (f x + e\right )^{2} - 35 \, a \cos \left (f x + e\right ) + {\left (2 \, a \cos \left (f x + e\right )^{4} + 10 \, a \cos \left (f x + e\right )^{3} - 15 \, a \cos \left (f x + e\right )^{2} - 35 \, a \cos \left (f x + e\right ) - 70 \, a\right )} \sin \left (f x + e\right ) - 70 \, a}{315 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="fricas")
Output:
-1/315*(2*a*cos(f*x + e)^5 - 8*a*cos(f*x + e)^4 - 25*a*cos(f*x + e)^3 + 20 *a*cos(f*x + e)^2 - 35*a*cos(f*x + e) + (2*a*cos(f*x + e)^4 + 10*a*cos(f*x + e)^3 - 15*a*cos(f*x + e)^2 - 35*a*cos(f*x + e) - 70*a)*sin(f*x + e) - 7 0*a)/(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*co s(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. 1700 vs. \(2 (112) = 224\).
Time = 10.23 (sec) , antiderivative size = 1700, normalized size of antiderivative = 13.49 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))**5,x)
Output:
Piecewise((-630*a*tan(e/2 + f*x/2)**8/(315*c**5*f*tan(e/2 + f*x/2)**9 - 28 35*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*ta n(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) + 1890*a*ta n(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) - 5250*a*tan(e/2 + f*x/2)**6/(3 15*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) + 6930*a*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 - 1 1340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5* f) - 7686*a*tan(e/2 + f*x/2)**4/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*...
Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (122) = 244\).
Time = 0.05 (sec) , antiderivative size = 733, normalized size of antiderivative = 5.82 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="maxima")
Output:
-2/315*(a*(432*sin(f*x + e)/(cos(f*x + e) + 1) - 1728*sin(f*x + e)^2/(cos( 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 - 3 360*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*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) - 5*a*(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/(cos(f* x + e) + 1)^7 - 5)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 36*c^5*s in(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))/f
Time = 0.16 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.07 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx=-\frac {2 \, {\left (315 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 945 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 2625 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 3465 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3843 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2247 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1143 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 207 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 58 \, a\right )}}{315 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}} \] Input:
integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="giac")
Output:
-2/315*(315*a*tan(1/2*f*x + 1/2*e)^8 - 945*a*tan(1/2*f*x + 1/2*e)^7 + 2625 *a*tan(1/2*f*x + 1/2*e)^6 - 3465*a*tan(1/2*f*x + 1/2*e)^5 + 3843*a*tan(1/2 *f*x + 1/2*e)^4 - 2247*a*tan(1/2*f*x + 1/2*e)^3 + 1143*a*tan(1/2*f*x + 1/2 *e)^2 - 207*a*tan(1/2*f*x + 1/2*e) + 58*a)/(c^5*f*(tan(1/2*f*x + 1/2*e) - 1)^9)
Time = 19.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.94 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx=\frac {\sqrt {2}\,a\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {121\,\cos \left (3\,e+3\,f\,x\right )}{4}-\frac {1575\,\sin \left (e+f\,x\right )}{4}-\frac {625\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {635\,\cos \left (e+f\,x\right )}{4}+\frac {7\,\cos \left (4\,e+4\,f\,x\right )}{2}+\frac {399\,\sin \left (2\,e+2\,f\,x\right )}{4}+\frac {141\,\sin \left (3\,e+3\,f\,x\right )}{4}-\frac {15\,\sin \left (4\,e+4\,f\,x\right )}{4}+\frac {1357}{4}\right )}{5040\,c^5\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^9} \] Input:
int((a + a*sin(e + f*x))/(c - c*sin(e + f*x))^5,x)
Output:
(2^(1/2)*a*cos(e/2 + (f*x)/2)*((121*cos(3*e + 3*f*x))/4 - (1575*sin(e + f* x))/4 - (625*cos(2*e + 2*f*x))/4 - (635*cos(e + f*x))/4 + (7*cos(4*e + 4*f *x))/2 + (399*sin(2*e + 2*f*x))/4 + (141*sin(3*e + 3*f*x))/4 - (15*sin(4*e + 4*f*x))/4 + 1357/4))/(5040*c^5*f*cos(e/2 + pi/4 + (f*x)/2)^9)
Time = 0.18 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.95 \[ \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx=\frac {a \left (-10 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}+38 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}-51 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+23 \cos \left (f x +e \right ) \sin \left (f x +e \right )-70 \cos \left (f x +e \right )-14 \sin \left (f x +e \right )^{5}+68 \sin \left (f x +e \right )^{4}-131 \sin \left (f x +e \right )^{3}+124 \sin \left (f x +e \right )^{2}+23 \sin \left (f x +e \right )+70\right )}{315 c^{5} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}-4 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}+6 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}-4 \cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )+\sin \left (f x +e \right )^{5}-5 \sin \left (f x +e \right )^{4}+10 \sin \left (f x +e \right )^{3}-10 \sin \left (f x +e \right )^{2}+5 \sin \left (f x +e \right )-1\right )} \] Input:
int((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x)
Output:
(a*( - 10*cos(e + f*x)*sin(e + f*x)**4 + 38*cos(e + f*x)*sin(e + f*x)**3 - 51*cos(e + f*x)*sin(e + f*x)**2 + 23*cos(e + f*x)*sin(e + f*x) - 70*cos(e + f*x) - 14*sin(e + f*x)**5 + 68*sin(e + f*x)**4 - 131*sin(e + f*x)**3 + 124*sin(e + f*x)**2 + 23*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*si n(e + f*x)**4 + 10*sin(e + f*x)**3 - 10*sin(e + f*x)**2 + 5*sin(e + f*x) - 1))