Integrand size = 23, antiderivative size = 102 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=-\frac {(c-d) \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {(2 c+3 d) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {(2 c+3 d) \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )} \] Output:
-1/5*(c-d)*cos(f*x+e)/f/(a+a*sin(f*x+e))^3-1/15*(2*c+3*d)*cos(f*x+e)/a/f/( a+a*sin(f*x+e))^2-1/15*(2*c+3*d)*cos(f*x+e)/f/(a^3+a^3*sin(f*x+e))
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=-\frac {\cos (e+f x) \left (7 c+3 d+(6 c+9 d) \sin (e+f x)+(2 c+3 d) \sin ^2(e+f x)\right )}{15 a^3 f (1+\sin (e+f x))^3} \] Input:
Integrate[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x])^3,x]
Output:
-1/15*(Cos[e + f*x]*(7*c + 3*d + (6*c + 9*d)*Sin[e + f*x] + (2*c + 3*d)*Si n[e + f*x]^2))/(a^3*f*(1 + Sin[e + f*x])^3)
Time = 0.39 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 3229, 3042, 3129, 3042, 3127}
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.
| \(\displaystyle \int \frac {c+d \sin (e+f x)}{(a \sin (e+f x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {c+d \sin (e+f x)}{(a \sin (e+f x)+a)^3}dx\) |
| \(\Big \downarrow \) 3229 |
| \(\displaystyle \frac {(2 c+3 d) \int \frac {1}{(\sin (e+f x) a+a)^2}dx}{5 a}-\frac {(c-d) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(2 c+3 d) \int \frac {1}{(\sin (e+f x) a+a)^2}dx}{5 a}-\frac {(c-d) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle \frac {(2 c+3 d) \left (\frac {\int \frac {1}{\sin (e+f x) a+a}dx}{3 a}-\frac {\cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}\right )}{5 a}-\frac {(c-d) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(2 c+3 d) \left (\frac {\int \frac {1}{\sin (e+f x) a+a}dx}{3 a}-\frac {\cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}\right )}{5 a}-\frac {(c-d) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle \frac {(2 c+3 d) \left (-\frac {\cos (e+f x)}{3 a f (a \sin (e+f x)+a)}-\frac {\cos (e+f x)}{3 f (a \sin (e+f x)+a)^2}\right )}{5 a}-\frac {(c-d) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3}\) |
Input:
Int[(c + d*Sin[e + f*x])/(a + a*Sin[e + f*x])^3,x]
Output:
-1/5*((c - d)*Cos[e + f*x])/(f*(a + a*Sin[e + f*x])^3) + ((2*c + 3*d)*(-1/ 3*Cos[e + f*x]/(f*(a + a*Sin[e + f*x])^2) - Cos[e + f*x]/(3*a*f*(a + a*Sin [e + f*x]))))/(5*a)
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b ^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((a + b*Sin[e + f* x])^m/(a*f*(2*m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.93
| method | result | size |
| risch | \(-\frac {2 i \left (20 i c \,{\mathrm e}^{2 i \left (f x +e \right )}+15 i d \,{\mathrm e}^{2 i \left (f x +e \right )}+15 d \,{\mathrm e}^{3 i \left (f x +e \right )}-2 i c -3 i d -10 c \,{\mathrm e}^{i \left (f x +e \right )}-15 d \,{\mathrm e}^{i \left (f x +e \right )}\right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(95\) |
| parallelrisch | \(\frac {-30 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (-60 c -30 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+\left (-80 c -30 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (-40 c -30 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-14 c -6 d}{15 f \,a^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(98\) |
| derivativedivides | \(\frac {-\frac {2 \left (8 c -6 d \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {-4 c +2 d}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (4 c -4 d \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 c}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-8 c +8 d}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}}{a^{3} f}\) | \(114\) |
| default | \(\frac {-\frac {2 \left (8 c -6 d \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {-4 c +2 d}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (4 c -4 d \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 c}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-8 c +8 d}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}}{a^{3} f}\) | \(114\) |
| norman | \(\frac {-\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{f a}+\frac {\left (-4 c -2 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{f a}+\frac {-14 c -6 d}{15 f a}+\frac {\left (-8 c -6 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 f a}+\frac {\left (-22 c -6 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{3 f a}+\frac {\left (-20 c -12 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 f a}+\frac {\left (-94 c -36 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{15 f a}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(196\) |
Input:
int((c+d*sin(f*x+e))/(a+sin(f*x+e)*a)^3,x,method=_RETURNVERBOSE)
Output:
-2/15*I*(20*I*c*exp(2*I*(f*x+e))+15*I*d*exp(2*I*(f*x+e))+15*d*exp(3*I*(f*x +e))-2*I*c-3*I*d-10*c*exp(I*(f*x+e))-15*d*exp(I*(f*x+e)))/f/a^3/(exp(I*(f* x+e))+I)^5
Time = 0.08 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.86 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=-\frac {{\left (2 \, c + 3 \, d\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (2 \, c + 3 \, d\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (3 \, c + 2 \, d\right )} \cos \left (f x + e\right ) - {\left ({\left (2 \, c + 3 \, d\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (2 \, c + 3 \, d\right )} \cos \left (f x + e\right ) - 3 \, c + 3 \, d\right )} \sin \left (f x + e\right ) - 3 \, c + 3 \, d}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^3,x, algorithm="fricas")
Output:
-1/15*((2*c + 3*d)*cos(f*x + e)^3 - 2*(2*c + 3*d)*cos(f*x + e)^2 - 3*(3*c + 2*d)*cos(f*x + e) - ((2*c + 3*d)*cos(f*x + e)^2 + 3*(2*c + 3*d)*cos(f*x + e) - 3*c + 3*d)*sin(f*x + e) - 3*c + 3*d)/(a^3*f*cos(f*x + e)^3 + 3*a^3* f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^3*f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 1015 vs. \(2 (87) = 174\).
Time = 3.30 (sec) , antiderivative size = 1015, normalized size of antiderivative = 9.95 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))**3,x)
Output:
Piecewise((-30*c*tan(e/2 + f*x/2)**4/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a **3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*ta n(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 60*c*tan(e/2 + f*x/2)**3/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)** 4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a **3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 80*c*tan(e/2 + f*x/2)**2/(15*a**3*f* tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 40*c*tan(e/2 + f*x/2)/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a* *3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan (e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 14*c/(15*a**3 *f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/ 2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2 ) + 15*a**3*f) - 30*d*tan(e/2 + f*x/2)**3/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3 *f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 30*d*ta n(e/2 + f*x/2)**2/(15*a**3*f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x /2)**4 + 150*a**3*f*tan(e/2 + f*x/2)**3 + 150*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 30*d*tan(e/2 + f*x/2)/(15*a**3* f*tan(e/2 + f*x/2)**5 + 75*a**3*f*tan(e/2 + f*x/2)**4 + 150*a**3*f*tan(...
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (96) = 192\).
Time = 0.04 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.79 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (\frac {c {\left (\frac {20 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {40 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 7\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, d {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}\right )}}{15 \, f} \] Input:
integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^3,x, algorithm="maxima")
Output:
-2/15*(c*(20*sin(f*x + e)/(cos(f*x + e) + 1) + 40*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 30*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15*sin(f*x + e)^4/( cos(f*x + e) + 1)^4 + 7)/(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) + 3*d*(5*sin(f*x + e)/(cos(f*x + e) + 1) + 5*sin( f*x + e)^2/(cos(f*x + e) + 1)^2 + 5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(c os(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*si n(f*x + e)^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^ 5))/f
Time = 0.14 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.20 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (15 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, c + 3 \, d\right )}}{15 \, a^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} \] Input:
integrate((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^3,x, algorithm="giac")
Output:
-2/15*(15*c*tan(1/2*f*x + 1/2*e)^4 + 30*c*tan(1/2*f*x + 1/2*e)^3 + 15*d*ta n(1/2*f*x + 1/2*e)^3 + 40*c*tan(1/2*f*x + 1/2*e)^2 + 15*d*tan(1/2*f*x + 1/ 2*e)^2 + 20*c*tan(1/2*f*x + 1/2*e) + 15*d*tan(1/2*f*x + 1/2*e) + 7*c + 3*d )/(a^3*f*(tan(1/2*f*x + 1/2*e) + 1)^5)
Time = 16.68 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.47 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {53\,c}{4}+3\,d-4\,c\,\cos \left (e+f\,x\right )+\frac {3\,d\,\cos \left (e+f\,x\right )}{2}+\frac {25\,c\,\sin \left (e+f\,x\right )}{2}+\frac {15\,d\,\sin \left (e+f\,x\right )}{2}-\frac {9\,c\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {3\,d\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {5\,c\,\sin \left (2\,e+2\,f\,x\right )}{4}\right )}{15\,a^3\,f\,\left (\frac {5\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}+\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{4}-\frac {5\,\sqrt {2}\,\cos \left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\right )}{2}+\frac {\sqrt {2}\,\cos \left (\frac {5\,e}{2}-\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{4}\right )} \] Input:
int((c + d*sin(e + f*x))/(a + a*sin(e + f*x))^3,x)
Output:
(2*cos(e/2 + (f*x)/2)*((53*c)/4 + 3*d - 4*c*cos(e + f*x) + (3*d*cos(e + f* x))/2 + (25*c*sin(e + f*x))/2 + (15*d*sin(e + f*x))/2 - (9*c*cos(2*e + 2*f *x))/4 - (3*d*cos(2*e + 2*f*x))/2 - (5*c*sin(2*e + 2*f*x))/4))/(15*a^3*f*( (5*2^(1/2)*cos((3*e)/2 + pi/4 + (3*f*x)/2))/4 - (5*2^(1/2)*cos(e/2 - pi/4 + (f*x)/2))/2 + (2^(1/2)*cos((5*e)/2 - pi/4 + (5*f*x)/2))/4))
Time = 0.17 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.57 \[ \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^3} \, dx=\frac {\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} c}{5}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} d -\frac {4 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) c}{3}-2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) d -\frac {8 c}{15}-\frac {2 d}{5}}{a^{3} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+10 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )} \] Input:
int((c+d*sin(f*x+e))/(a+a*sin(f*x+e))^3,x)
Output:
(2*(3*tan((e + f*x)/2)**5*c - 15*tan((e + f*x)/2)**3*d - 10*tan((e + f*x)/ 2)**2*c - 15*tan((e + f*x)/2)**2*d - 5*tan((e + f*x)/2)*c - 15*tan((e + f* x)/2)*d - 4*c - 3*d))/(15*a**3*f*(tan((e + f*x)/2)**5 + 5*tan((e + f*x)/2) **4 + 10*tan((e + f*x)/2)**3 + 10*tan((e + f*x)/2)**2 + 5*tan((e + f*x)/2) + 1))