\(\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx\) [58]

Optimal result

Integrand size = 36, antiderivative size = 102 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=\frac {(A+B) \sec (e+f x)}{5 a c f (c-c \sin (e+f x))^2}+\frac {(3 A-2 B) \sec (e+f x)}{15 a f \left (c^3-c^3 \sin (e+f x)\right )}+\frac {2 (3 A-2 B) \tan (e+f x)}{15 a c^3 f} \] Output:

1/5*(A+B)*sec(f*x+e)/a/c/f/(c-c*sin(f*x+e))^2+1/15*(3*A-2*B)*sec(f*x+e)/a/ 
f/(c^3-c^3*sin(f*x+e))+2/15*(3*A-2*B)*tan(f*x+e)/a/c^3/f
 

Mathematica [A] (verified)

Time = 3.89 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.54 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=-\frac {\cos (e+f x) (80 B+5 (-9 A+B) \cos (e+f x)+32 (3 A-2 B) \cos (2 (e+f x))+9 A \cos (3 (e+f x))-B \cos (3 (e+f x))+120 A \sin (e+f x)-80 B \sin (e+f x)+36 A \sin (2 (e+f x))-4 B \sin (2 (e+f x))-24 A \sin (3 (e+f x))+16 B \sin (3 (e+f x)))}{240 a c^3 f (-1+\sin (e+f x))^3 (1+\sin (e+f x))} \] Input:

Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^ 
3),x]
 

Output:

-1/240*(Cos[e + f*x]*(80*B + 5*(-9*A + B)*Cos[e + f*x] + 32*(3*A - 2*B)*Co 
s[2*(e + f*x)] + 9*A*Cos[3*(e + f*x)] - B*Cos[3*(e + f*x)] + 120*A*Sin[e + 
 f*x] - 80*B*Sin[e + f*x] + 36*A*Sin[2*(e + f*x)] - 4*B*Sin[2*(e + f*x)] - 
 24*A*Sin[3*(e + f*x)] + 16*B*Sin[3*(e + f*x)]))/(a*c^3*f*(-1 + Sin[e + f* 
x])^3*(1 + Sin[e + f*x]))
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.92, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3446, 3042, 3338, 3042, 3151, 3042, 4254, 24}

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.

\(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a) (c-c \sin (e+f x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a) (c-c \sin (e+f x))^3}dx\)

\(\Big \downarrow \) 3446

\(\displaystyle \frac {\int \frac {\sec ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^2}dx}{a c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {A+B \sin (e+f x)}{\cos (e+f x)^2 (c-c \sin (e+f x))^2}dx}{a c}\)

\(\Big \downarrow \) 3338

\(\displaystyle \frac {\frac {(3 A-2 B) \int \frac {\sec ^2(e+f x)}{c-c \sin (e+f x)}dx}{5 c}+\frac {(A+B) \sec (e+f x)}{5 f (c-c \sin (e+f x))^2}}{a c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(3 A-2 B) \int \frac {1}{\cos (e+f x)^2 (c-c \sin (e+f x))}dx}{5 c}+\frac {(A+B) \sec (e+f x)}{5 f (c-c \sin (e+f x))^2}}{a c}\)

\(\Big \downarrow \) 3151

\(\displaystyle \frac {\frac {(3 A-2 B) \left (\frac {2 \int \sec ^2(e+f x)dx}{3 c}+\frac {\sec (e+f x)}{3 f (c-c \sin (e+f x))}\right )}{5 c}+\frac {(A+B) \sec (e+f x)}{5 f (c-c \sin (e+f x))^2}}{a c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(3 A-2 B) \left (\frac {2 \int \csc \left (e+f x+\frac {\pi }{2}\right )^2dx}{3 c}+\frac {\sec (e+f x)}{3 f (c-c \sin (e+f x))}\right )}{5 c}+\frac {(A+B) \sec (e+f x)}{5 f (c-c \sin (e+f x))^2}}{a c}\)

\(\Big \downarrow \) 4254

\(\displaystyle \frac {\frac {(3 A-2 B) \left (\frac {\sec (e+f x)}{3 f (c-c \sin (e+f x))}-\frac {2 \int 1d(-\tan (e+f x))}{3 c f}\right )}{5 c}+\frac {(A+B) \sec (e+f x)}{5 f (c-c \sin (e+f x))^2}}{a c}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {\frac {(A+B) \sec (e+f x)}{5 f (c-c \sin (e+f x))^2}+\frac {(3 A-2 B) \left (\frac {2 \tan (e+f x)}{3 c f}+\frac {\sec (e+f x)}{3 f (c-c \sin (e+f x))}\right )}{5 c}}{a c}\)

Input:

Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^3),x]
 

Output:

(((A + B)*Sec[e + f*x])/(5*f*(c - c*Sin[e + f*x])^2) + ((3*A - 2*B)*(Sec[e 
 + f*x]/(3*f*(c - c*Sin[e + f*x])) + (2*Tan[e + f*x])/(3*c*f)))/(5*c))/(a* 
c)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3151

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]
 

rule 3338

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - 
 a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1) 
)), x] + Simp[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1))   Int[(g*Cos[e 
+ f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, 
g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0 
]) && NeQ[2*m + p + 1, 0]
 

rule 3446

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + 
 (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si 
mp[a^m*c^m   Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin 
[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a* 
d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] 
&& GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
 

rule 4254

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, 
 d}, x] && IGtQ[n/2, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.09

method	result	size
risch	\(-\frac {4 \left (15 i A \,{\mathrm e}^{2 i \left (f x +e \right )}+12 A \,{\mathrm e}^{i \left (f x +e \right )}-10 i B \,{\mathrm e}^{2 i \left (f x +e \right )}+10 B \,{\mathrm e}^{3 i \left (f x +e \right )}+2 i B -8 B \,{\mathrm e}^{i \left (f x +e \right )}-3 i A \right )}{15 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) f a \,c^{3}}\)	\(111\)
parallelrisch	\(\frac {-30 A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (60 A -30 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (-60 A +40 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-40 B \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (18 A +8 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-12 A -2 B}{15 f a \,c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\)	\(128\)
derivativedivides	\(\frac {-\frac {2 \left (\frac {A}{8}-\frac {B}{8}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (2 A +2 B \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {4 A +4 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {\frac {5 A}{2}+\frac {3 B}{2}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {7 A}{8}+\frac {B}{8}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {2 \left (\frac {9 A}{2}+\frac {7 B}{2}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}}{f a \,c^{3}}\)	\(145\)
default	\(\frac {-\frac {2 \left (\frac {A}{8}-\frac {B}{8}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (2 A +2 B \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {4 A +4 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {\frac {5 A}{2}+\frac {3 B}{2}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {7 A}{8}+\frac {B}{8}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {2 \left (\frac {9 A}{2}+\frac {7 B}{2}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}}{f a \,c^{3}}\)	\(145\)
norman	\(\frac {-\frac {12 A +2 B}{15 a f c}+\frac {2 \left (6 A -7 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{3 a f c}-\frac {2 A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{a f c}-\frac {2 \left (-8 B +7 A \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{5 a f c}+\frac {2 \left (2 A -B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{a f c}-\frac {2 \left (9 A -4 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{3 a f c}-\frac {2 \left (2 A +7 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{5 a f c}+\frac {2 \left (9 A +4 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{15 a f c}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) c^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\)	\(260\)

Input:

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^3,x,method=_RETURNV 
ERBOSE)
 

Output:

-4/15*(15*I*A*exp(2*I*(f*x+e))+12*A*exp(I*(f*x+e))-10*I*B*exp(2*I*(f*x+e)) 
+10*B*exp(3*I*(f*x+e))+2*I*B-8*B*exp(I*(f*x+e))-3*I*A)/(exp(I*(f*x+e))-I)^ 
5/(exp(I*(f*x+e))+I)/f/a/c^3
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.05 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=-\frac {4 \, {\left (3 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{2} - {\left (2 \, {\left (3 \, A - 2 \, B\right )} \cos \left (f x + e\right )^{2} - 9 \, A + 6 \, B\right )} \sin \left (f x + e\right ) - 6 \, A + 9 \, B}{15 \, {\left (a c^{3} f \cos \left (f x + e\right )^{3} + 2 \, a c^{3} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a c^{3} f \cos \left (f x + e\right )\right )}} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^3,x, algorith 
m="fricas")
 

Output:

-1/15*(4*(3*A - 2*B)*cos(f*x + e)^2 - (2*(3*A - 2*B)*cos(f*x + e)^2 - 9*A 
+ 6*B)*sin(f*x + e) - 6*A + 9*B)/(a*c^3*f*cos(f*x + e)^3 + 2*a*c^3*f*cos(f 
*x + e)*sin(f*x + e) - 2*a*c^3*f*cos(f*x + e))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1236 vs. \(2 (85) = 170\).

Time = 4.82 (sec) , antiderivative size = 1236, normalized size of antiderivative = 12.12 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))**3,x)
 

Output:

Piecewise((-30*A*tan(e/2 + f*x/2)**5/(15*a*c**3*f*tan(e/2 + f*x/2)**6 - 60 
*a*c**3*f*tan(e/2 + f*x/2)**5 + 75*a*c**3*f*tan(e/2 + f*x/2)**4 - 75*a*c** 
3*f*tan(e/2 + f*x/2)**2 + 60*a*c**3*f*tan(e/2 + f*x/2) - 15*a*c**3*f) + 60 
*A*tan(e/2 + f*x/2)**4/(15*a*c**3*f*tan(e/2 + f*x/2)**6 - 60*a*c**3*f*tan( 
e/2 + f*x/2)**5 + 75*a*c**3*f*tan(e/2 + f*x/2)**4 - 75*a*c**3*f*tan(e/2 + 
f*x/2)**2 + 60*a*c**3*f*tan(e/2 + f*x/2) - 15*a*c**3*f) - 60*A*tan(e/2 + f 
*x/2)**3/(15*a*c**3*f*tan(e/2 + f*x/2)**6 - 60*a*c**3*f*tan(e/2 + f*x/2)** 
5 + 75*a*c**3*f*tan(e/2 + f*x/2)**4 - 75*a*c**3*f*tan(e/2 + f*x/2)**2 + 60 
*a*c**3*f*tan(e/2 + f*x/2) - 15*a*c**3*f) + 18*A*tan(e/2 + f*x/2)/(15*a*c* 
*3*f*tan(e/2 + f*x/2)**6 - 60*a*c**3*f*tan(e/2 + f*x/2)**5 + 75*a*c**3*f*t 
an(e/2 + f*x/2)**4 - 75*a*c**3*f*tan(e/2 + f*x/2)**2 + 60*a*c**3*f*tan(e/2 
 + f*x/2) - 15*a*c**3*f) - 12*A/(15*a*c**3*f*tan(e/2 + f*x/2)**6 - 60*a*c* 
*3*f*tan(e/2 + f*x/2)**5 + 75*a*c**3*f*tan(e/2 + f*x/2)**4 - 75*a*c**3*f*t 
an(e/2 + f*x/2)**2 + 60*a*c**3*f*tan(e/2 + f*x/2) - 15*a*c**3*f) - 30*B*ta 
n(e/2 + f*x/2)**4/(15*a*c**3*f*tan(e/2 + f*x/2)**6 - 60*a*c**3*f*tan(e/2 + 
 f*x/2)**5 + 75*a*c**3*f*tan(e/2 + f*x/2)**4 - 75*a*c**3*f*tan(e/2 + f*x/2 
)**2 + 60*a*c**3*f*tan(e/2 + f*x/2) - 15*a*c**3*f) + 40*B*tan(e/2 + f*x/2) 
**3/(15*a*c**3*f*tan(e/2 + f*x/2)**6 - 60*a*c**3*f*tan(e/2 + f*x/2)**5 + 7 
5*a*c**3*f*tan(e/2 + f*x/2)**4 - 75*a*c**3*f*tan(e/2 + f*x/2)**2 + 60*a*c* 
*3*f*tan(e/2 + f*x/2) - 15*a*c**3*f) - 40*B*tan(e/2 + f*x/2)**2/(15*a*c...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (98) = 196\).

Time = 0.05 (sec) , antiderivative size = 423, normalized size of antiderivative = 4.15 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=-\frac {2 \, {\left (\frac {B {\left (\frac {4 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {20 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {20 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 1\right )}}{a c^{3} - \frac {4 \, a c^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, a c^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {5 \, a c^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {4 \, a c^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {a c^{3} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}} + \frac {3 \, A {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {10 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {10 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {5 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - 2\right )}}{a c^{3} - \frac {4 \, a c^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, a c^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {5 \, a c^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {4 \, a c^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {a c^{3} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}}\right )}}{15 \, f} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^3,x, algorith 
m="maxima")
 

Output:

-2/15*(B*(4*sin(f*x + e)/(cos(f*x + e) + 1) - 20*sin(f*x + e)^2/(cos(f*x + 
 e) + 1)^2 + 20*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 15*sin(f*x + e)^4/(c 
os(f*x + e) + 1)^4 - 1)/(a*c^3 - 4*a*c^3*sin(f*x + e)/(cos(f*x + e) + 1) + 
 5*a*c^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 5*a*c^3*sin(f*x + e)^4/(cos 
(f*x + e) + 1)^4 + 4*a*c^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - a*c^3*sin 
(f*x + e)^6/(cos(f*x + e) + 1)^6) + 3*A*(3*sin(f*x + e)/(cos(f*x + e) + 1) 
 - 10*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 10*sin(f*x + e)^4/(cos(f*x + e 
) + 1)^4 - 5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 2)/(a*c^3 - 4*a*c^3*sin 
(f*x + e)/(cos(f*x + e) + 1) + 5*a*c^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 
 - 5*a*c^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 4*a*c^3*sin(f*x + e)^5/(c 
os(f*x + e) + 1)^5 - a*c^3*sin(f*x + e)^6/(cos(f*x + e) + 1)^6))/f
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.64 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=-\frac {\frac {15 \, {\left (A - B\right )}}{a c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {105 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 15 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 270 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 30 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 360 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 40 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 210 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 50 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 63 \, A - 7 \, B}{a c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{5}}}{60 \, f} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^3,x, algorith 
m="giac")
 

Output:

-1/60*(15*(A - B)/(a*c^3*(tan(1/2*f*x + 1/2*e) + 1)) + (105*A*tan(1/2*f*x 
+ 1/2*e)^4 + 15*B*tan(1/2*f*x + 1/2*e)^4 - 270*A*tan(1/2*f*x + 1/2*e)^3 + 
30*B*tan(1/2*f*x + 1/2*e)^3 + 360*A*tan(1/2*f*x + 1/2*e)^2 - 40*B*tan(1/2* 
f*x + 1/2*e)^2 - 210*A*tan(1/2*f*x + 1/2*e) + 50*B*tan(1/2*f*x + 1/2*e) + 
63*A - 7*B)/(a*c^3*(tan(1/2*f*x + 1/2*e) - 1)^5))/f
 

Mupad [B] (verification not implemented)

Time = 37.30 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.75 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=\frac {2\,\left (\frac {5\,B\,\sin \left (e+f\,x\right )}{2}-\frac {15\,A\,\cos \left (e+f\,x\right )}{4}-\frac {5\,B\,\cos \left (e+f\,x\right )}{8}-\frac {15\,A\,\sin \left (e+f\,x\right )}{4}-\frac {5\,B}{2}-3\,A\,\cos \left (2\,e+2\,f\,x\right )+\frac {3\,A\,\cos \left (3\,e+3\,f\,x\right )}{4}+2\,B\,\cos \left (2\,e+2\,f\,x\right )+\frac {B\,\cos \left (3\,e+3\,f\,x\right )}{8}+3\,A\,\sin \left (2\,e+2\,f\,x\right )+\frac {3\,A\,\sin \left (3\,e+3\,f\,x\right )}{4}+\frac {B\,\sin \left (2\,e+2\,f\,x\right )}{2}-\frac {B\,\sin \left (3\,e+3\,f\,x\right )}{2}\right )}{15\,a\,c^3\,f\,\left (\frac {\cos \left (3\,e+3\,f\,x\right )}{4}-\frac {5\,\cos \left (e+f\,x\right )}{4}+\sin \left (2\,e+2\,f\,x\right )\right )} \] Input:

int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))*(c - c*sin(e + f*x))^3),x)
 

Output:

(2*((5*B*sin(e + f*x))/2 - (15*A*cos(e + f*x))/4 - (5*B*cos(e + f*x))/8 - 
(15*A*sin(e + f*x))/4 - (5*B)/2 - 3*A*cos(2*e + 2*f*x) + (3*A*cos(3*e + 3* 
f*x))/4 + 2*B*cos(2*e + 2*f*x) + (B*cos(3*e + 3*f*x))/8 + 3*A*sin(2*e + 2* 
f*x) + (3*A*sin(3*e + 3*f*x))/4 + (B*sin(2*e + 2*f*x))/2 - (B*sin(3*e + 3* 
f*x))/2))/(15*a*c^3*f*(cos(3*e + 3*f*x)/4 - (5*cos(e + f*x))/4 + sin(2*e + 
 2*f*x)))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.86 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^3} \, dx=\frac {-3 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a +2 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b +6 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a -4 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b -3 \cos \left (f x +e \right ) a +2 \cos \left (f x +e \right ) b +12 \sin \left (f x +e \right )^{3} a -8 \sin \left (f x +e \right )^{3} b -24 \sin \left (f x +e \right )^{2} a +16 \sin \left (f x +e \right )^{2} b +6 a \sin \left (f x +e \right )-4 \sin \left (f x +e \right ) b +12 a +2 b}{30 \cos \left (f x +e \right ) a \,c^{3} f \left (\sin \left (f x +e \right )^{2}-2 \sin \left (f x +e \right )+1\right )} \] Input:

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^3,x)
 

Output:

( - 3*cos(e + f*x)*sin(e + f*x)**2*a + 2*cos(e + f*x)*sin(e + f*x)**2*b + 
6*cos(e + f*x)*sin(e + f*x)*a - 4*cos(e + f*x)*sin(e + f*x)*b - 3*cos(e + 
f*x)*a + 2*cos(e + f*x)*b + 12*sin(e + f*x)**3*a - 8*sin(e + f*x)**3*b - 2 
4*sin(e + f*x)**2*a + 16*sin(e + f*x)**2*b + 6*sin(e + f*x)*a - 4*sin(e + 
f*x)*b + 12*a + 2*b)/(30*cos(e + f*x)*a*c**3*f*(sin(e + f*x)**2 - 2*sin(e 
+ f*x) + 1))