\(\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx\) [69]

Optimal result

Integrand size = 36, antiderivative size = 175 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {(A+B) \sec ^3(e+f x)}{9 a^2 c^2 f (c-c \sin (e+f x))^3}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 c^3 f (c-c \sin (e+f x))^2}+\frac {(2 A-B) \sec ^3(e+f x)}{21 a^2 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {4 (2 A-B) \tan (e+f x)}{21 a^2 c^5 f}+\frac {4 (2 A-B) \tan ^3(e+f x)}{63 a^2 c^5 f} \] Output:

1/9*(A+B)*sec(f*x+e)^3/a^2/c^2/f/(c-c*sin(f*x+e))^3+1/21*(2*A-B)*sec(f*x+e 
)^3/a^2/c^3/f/(c-c*sin(f*x+e))^2+1/21*(2*A-B)*sec(f*x+e)^3/a^2/f/(c^5-c^5* 
sin(f*x+e))+4/21*(2*A-B)*tan(f*x+e)/a^2/c^5/f+4/63*(2*A-B)*tan(f*x+e)^3/a^ 
2/c^5/f
 

Mathematica [A] (verified)

Time = 7.68 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.88 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-10752 B+180 (31 A-5 B) \cos (e+f x)-6912 (2 A-B) \cos (2 (e+f x))+310 A \cos (3 (e+f x))-50 B \cos (3 (e+f x))-6144 A \cos (4 (e+f x))+3072 B \cos (4 (e+f x))-930 A \cos (5 (e+f x))+150 B \cos (5 (e+f x))+512 A \cos (6 (e+f x))-256 B \cos (6 (e+f x))-18432 A \sin (e+f x)+9216 B \sin (e+f x)-4185 A \sin (2 (e+f x))+675 B \sin (2 (e+f x))-1024 A \sin (3 (e+f x))+512 B \sin (3 (e+f x))-1860 A \sin (4 (e+f x))+300 B \sin (4 (e+f x))+3072 A \sin (5 (e+f x))-1536 B \sin (5 (e+f x))+155 A \sin (6 (e+f x))-25 B \sin (6 (e+f x)))}{64512 a^2 c^5 f (-1+\sin (e+f x))^5 (1+\sin (e+f x))^2} \] Input:

Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x] 
)^5),x]
 

Output:

((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2 
])*(-10752*B + 180*(31*A - 5*B)*Cos[e + f*x] - 6912*(2*A - B)*Cos[2*(e + f 
*x)] + 310*A*Cos[3*(e + f*x)] - 50*B*Cos[3*(e + f*x)] - 6144*A*Cos[4*(e + 
f*x)] + 3072*B*Cos[4*(e + f*x)] - 930*A*Cos[5*(e + f*x)] + 150*B*Cos[5*(e 
+ f*x)] + 512*A*Cos[6*(e + f*x)] - 256*B*Cos[6*(e + f*x)] - 18432*A*Sin[e 
+ f*x] + 9216*B*Sin[e + f*x] - 4185*A*Sin[2*(e + f*x)] + 675*B*Sin[2*(e + 
f*x)] - 1024*A*Sin[3*(e + f*x)] + 512*B*Sin[3*(e + f*x)] - 1860*A*Sin[4*(e 
 + f*x)] + 300*B*Sin[4*(e + f*x)] + 3072*A*Sin[5*(e + f*x)] - 1536*B*Sin[5 
*(e + f*x)] + 155*A*Sin[6*(e + f*x)] - 25*B*Sin[6*(e + f*x)]))/(64512*a^2* 
c^5*f*(-1 + Sin[e + f*x])^5*(1 + Sin[e + f*x])^2)
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.85, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {3042, 3446, 3042, 3338, 3042, 3151, 3042, 3151, 3042, 4254, 2009}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3446

\(\displaystyle \frac {\int \frac {\sec ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3}dx}{a^2 c^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {A+B \sin (e+f x)}{\cos (e+f x)^4 (c-c \sin (e+f x))^3}dx}{a^2 c^2}\)

\(\Big \downarrow \) 3338

\(\displaystyle \frac {\frac {(2 A-B) \int \frac {\sec ^4(e+f x)}{(c-c \sin (e+f x))^2}dx}{3 c}+\frac {(A+B) \sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(2 A-B) \int \frac {1}{\cos (e+f x)^4 (c-c \sin (e+f x))^2}dx}{3 c}+\frac {(A+B) \sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\)

\(\Big \downarrow \) 3151

\(\displaystyle \frac {\frac {(2 A-B) \left (\frac {5 \int \frac {\sec ^4(e+f x)}{c-c \sin (e+f x)}dx}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}\right )}{3 c}+\frac {(A+B) \sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(2 A-B) \left (\frac {5 \int \frac {1}{\cos (e+f x)^4 (c-c \sin (e+f x))}dx}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}\right )}{3 c}+\frac {(A+B) \sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\)

\(\Big \downarrow \) 3151

\(\displaystyle \frac {\frac {(2 A-B) \left (\frac {5 \left (\frac {4 \int \sec ^4(e+f x)dx}{5 c}+\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}\right )}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}\right )}{3 c}+\frac {(A+B) \sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {(2 A-B) \left (\frac {5 \left (\frac {4 \int \csc \left (e+f x+\frac {\pi }{2}\right )^4dx}{5 c}+\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}\right )}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}\right )}{3 c}+\frac {(A+B) \sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\)

\(\Big \downarrow \) 4254

\(\displaystyle \frac {\frac {(2 A-B) \left (\frac {5 \left (\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}-\frac {4 \int \left (\tan ^2(e+f x)+1\right )d(-\tan (e+f x))}{5 c f}\right )}{7 c}+\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}\right )}{3 c}+\frac {(A+B) \sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}}{a^2 c^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {(A+B) \sec ^3(e+f x)}{9 f (c-c \sin (e+f x))^3}+\frac {(2 A-B) \left (\frac {\sec ^3(e+f x)}{7 f (c-c \sin (e+f x))^2}+\frac {5 \left (\frac {\sec ^3(e+f x)}{5 f (c-c \sin (e+f x))}-\frac {4 \left (-\frac {1}{3} \tan ^3(e+f x)-\tan (e+f x)\right )}{5 c f}\right )}{7 c}\right )}{3 c}}{a^2 c^2}\)

Input:

Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5),x 
]
 

Output:

(((A + B)*Sec[e + f*x]^3)/(9*f*(c - c*Sin[e + f*x])^3) + ((2*A - B)*(Sec[e 
 + f*x]^3/(7*f*(c - c*Sin[e + f*x])^2) + (5*(Sec[e + f*x]^3/(5*f*(c - c*Si 
n[e + f*x])) - (4*(-Tan[e + f*x] - Tan[e + f*x]^3/3))/(5*c*f)))/(7*c)))/(3 
*c))/(a^2*c^2)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 = 2.24 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.05

method	result	size
risch	\(\frac {16 i \left (72 i A \,{\mathrm e}^{5 i \left (f x +e \right )}-36 i B \,{\mathrm e}^{5 i \left (f x +e \right )}+42 B \,{\mathrm e}^{6 i \left (f x +e \right )}+4 i A \,{\mathrm e}^{3 i \left (f x +e \right )}+54 A \,{\mathrm e}^{4 i \left (f x +e \right )}-2 i B \,{\mathrm e}^{3 i \left (f x +e \right )}-27 B \,{\mathrm e}^{4 i \left (f x +e \right )}-12 i A \,{\mathrm e}^{i \left (f x +e \right )}+24 A \,{\mathrm e}^{2 i \left (f x +e \right )}+6 i B \,{\mathrm e}^{i \left (f x +e \right )}-12 B \,{\mathrm e}^{2 i \left (f x +e \right )}-2 A +B \right )}{63 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3} f \,a^{2} c^{5}}\)	\(184\)
parallelrisch	\(\frac {-126 A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}+\left (378 A -126 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}+\left (-546 A +252 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+\left (-126 A -378 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+756 A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+\left (-588 A +84 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\left (-612 A -72 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (900 A -324 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (-470 A +256 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+\left (-78 A -150 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (102 A +12 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-38 A -2 B}{63 f \,a^{2} c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\)	\(242\)
derivativedivides	\(\frac {-\frac {2 \left (4 A +4 B \right )}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {16 A +16 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {2 \left (34 A +32 B \right )}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {46 A +40 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {\frac {9 A}{2}+\frac {13 B}{8}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {57 A}{64}+\frac {5 B}{64}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {\frac {59 A}{2}+\frac {39 B}{2}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {2 \left (\frac {57 A}{4}+\frac {59 B}{8}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {2 \left (\frac {175 A}{4}+\frac {135 B}{4}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {-\frac {A}{16}+\frac {B}{16}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (\frac {A}{16}-\frac {B}{16}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {7 A}{64}-\frac {5 B}{64}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{f \,a^{2} c^{5}}\)	\(277\)
default	\(\frac {-\frac {2 \left (4 A +4 B \right )}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {16 A +16 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {2 \left (34 A +32 B \right )}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {46 A +40 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {\frac {9 A}{2}+\frac {13 B}{8}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {57 A}{64}+\frac {5 B}{64}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {\frac {59 A}{2}+\frac {39 B}{2}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {2 \left (\frac {57 A}{4}+\frac {59 B}{8}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {2 \left (\frac {175 A}{4}+\frac {135 B}{4}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {-\frac {A}{16}+\frac {B}{16}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (\frac {A}{16}-\frac {B}{16}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (\frac {7 A}{64}-\frac {5 B}{64}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{f \,a^{2} c^{5}}\)	\(277\)
norman	\(\frac {\frac {\left (4 A -8 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{a f c}+\frac {\left (6 A -2 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{a f c}-\frac {38 A +2 B}{63 a f c}+\frac {8 \left (2 A -B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{7 a c f}-\frac {2 A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{a f c}-\frac {4 \left (92 A -67 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{63 a f c}-\frac {4 \left (8 A -3 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{3 a f c}+\frac {2 \left (5 A +6 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{3 a f c}+\frac {2 \left (17 A +2 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{21 a f c}-\frac {\left (34 A +14 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{3 a f c}+\frac {\left (274 A -158 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{21 a f c}+\frac {\left (104 A -80 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{21 a f c}-\frac {\left (116 A +152 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{63 a f c}-\frac {2 \left (541 A -92 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{63 a f c}}{a \left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\)	\(435\)

Input:

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x,method=_RETUR 
NVERBOSE)
 

Output:

16/63*I*(72*I*A*exp(5*I*(f*x+e))-36*I*B*exp(5*I*(f*x+e))+42*B*exp(6*I*(f*x 
+e))+4*I*A*exp(3*I*(f*x+e))+54*A*exp(4*I*(f*x+e))-2*I*B*exp(3*I*(f*x+e))-2 
7*B*exp(4*I*(f*x+e))-12*I*A*exp(I*(f*x+e))+24*A*exp(2*I*(f*x+e))+6*I*B*exp 
(I*(f*x+e))-12*B*exp(2*I*(f*x+e))-2*A+B)/(exp(I*(f*x+e))-I)^9/(exp(I*(f*x+ 
e))+I)^3/f/a^2/c^5
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.07 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {8 \, {\left (2 \, A - B\right )} \cos \left (f x + e\right )^{6} - 36 \, {\left (2 \, A - B\right )} \cos \left (f x + e\right )^{4} + 15 \, {\left (2 \, A - B\right )} \cos \left (f x + e\right )^{2} + {\left (24 \, {\left (2 \, A - B\right )} \cos \left (f x + e\right )^{4} - 20 \, {\left (2 \, A - B\right )} \cos \left (f x + e\right )^{2} - 14 \, A + 7 \, B\right )} \sin \left (f x + e\right ) + 7 \, A - 14 \, B}{63 \, {\left (3 \, a^{2} c^{5} f \cos \left (f x + e\right )^{5} - 4 \, a^{2} c^{5} f \cos \left (f x + e\right )^{3} - {\left (a^{2} c^{5} f \cos \left (f x + e\right )^{5} - 4 \, a^{2} c^{5} f \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )\right )}} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algori 
thm="fricas")
 

Output:

1/63*(8*(2*A - B)*cos(f*x + e)^6 - 36*(2*A - B)*cos(f*x + e)^4 + 15*(2*A - 
 B)*cos(f*x + e)^2 + (24*(2*A - B)*cos(f*x + e)^4 - 20*(2*A - B)*cos(f*x + 
 e)^2 - 14*A + 7*B)*sin(f*x + e) + 7*A - 14*B)/(3*a^2*c^5*f*cos(f*x + e)^5 
 - 4*a^2*c^5*f*cos(f*x + e)^3 - (a^2*c^5*f*cos(f*x + e)^5 - 4*a^2*c^5*f*co 
s(f*x + e)^3)*sin(f*x + e))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5868 vs. \(2 (160) = 320\).

Time = 34.90 (sec) , antiderivative size = 5868, normalized size of antiderivative = 33.53 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))**2/(c-c*sin(f*x+e))**5,x)
 

Output:

Piecewise((-126*A*tan(e/2 + f*x/2)**11/(63*a**2*c**5*f*tan(e/2 + f*x/2)**1 
2 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2 
)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f 
*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 
 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan( 
e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*ta 
n(e/2 + f*x/2) - 63*a**2*c**5*f) + 378*A*tan(e/2 + f*x/2)**10/(63*a**2*c** 
5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + f*x/2)**11 + 756*a**2 
*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e/2 + f*x/2)**9 - 1701* 
a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*tan(e/2 + f*x/2)**7 - 2 
268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5*f*tan(e/2 + f*x/2)**4 
 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c**5*f*tan(e/2 + f*x/2)* 
*2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5*f) - 546*A*tan(e/2 + 
f*x/2)**9/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - 378*a**2*c**5*f*tan(e/2 + 
 f*x/2)**11 + 756*a**2*c**5*f*tan(e/2 + f*x/2)**10 - 126*a**2*c**5*f*tan(e 
/2 + f*x/2)**9 - 1701*a**2*c**5*f*tan(e/2 + f*x/2)**8 + 2268*a**2*c**5*f*t 
an(e/2 + f*x/2)**7 - 2268*a**2*c**5*f*tan(e/2 + f*x/2)**5 + 1701*a**2*c**5 
*f*tan(e/2 + f*x/2)**4 + 126*a**2*c**5*f*tan(e/2 + f*x/2)**3 - 756*a**2*c* 
*5*f*tan(e/2 + f*x/2)**2 + 378*a**2*c**5*f*tan(e/2 + f*x/2) - 63*a**2*c**5 
*f) - 126*A*tan(e/2 + f*x/2)**8/(63*a**2*c**5*f*tan(e/2 + f*x/2)**12 - ...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (168) = 336\).

Time = 0.07 (sec) , antiderivative size = 998, normalized size of antiderivative = 5.70 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algori 
thm="maxima")
 

Output:

-2/63*(A*(51*sin(f*x + e)/(cos(f*x + e) + 1) - 39*sin(f*x + e)^2/(cos(f*x 
+ e) + 1)^2 - 235*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 450*sin(f*x + e)^4 
/(cos(f*x + e) + 1)^4 - 306*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 294*sin( 
f*x + e)^6/(cos(f*x + e) + 1)^6 + 378*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 
- 63*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 273*sin(f*x + e)^9/(cos(f*x + e 
) + 1)^9 + 189*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 63*sin(f*x + e)^11/ 
(cos(f*x + e) + 1)^11 - 19)/(a^2*c^5 - 6*a^2*c^5*sin(f*x + e)/(cos(f*x + e 
) + 1) + 12*a^2*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 2*a^2*c^5*sin(f* 
x + e)^3/(cos(f*x + e) + 1)^3 - 27*a^2*c^5*sin(f*x + e)^4/(cos(f*x + e) + 
1)^4 + 36*a^2*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 36*a^2*c^5*sin(f*x 
 + e)^7/(cos(f*x + e) + 1)^7 + 27*a^2*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1 
)^8 + 2*a^2*c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 12*a^2*c^5*sin(f*x + 
 e)^10/(cos(f*x + e) + 1)^10 + 6*a^2*c^5*sin(f*x + e)^11/(cos(f*x + e) + 1 
)^11 - a^2*c^5*sin(f*x + e)^12/(cos(f*x + e) + 1)^12) + B*(6*sin(f*x + e)/ 
(cos(f*x + e) + 1) - 75*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 128*sin(f*x 
+ e)^3/(cos(f*x + e) + 1)^3 - 162*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 36 
*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 42*sin(f*x + e)^6/(cos(f*x + e) + 1 
)^6 - 189*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 126*sin(f*x + e)^9/(cos(f* 
x + e) + 1)^9 - 63*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1)/(a^2*c^5 - 6 
*a^2*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 12*a^2*c^5*sin(f*x + e)^2/(c...
 

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.90 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=-\frac {\frac {21 \, {\left (21 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 24 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 19 \, A - 13 \, B\right )}}{a^{2} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {3591 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 315 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 19656 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 756 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 56196 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 4200 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 95760 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 11340 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 107730 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 14994 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 79464 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 13356 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 38484 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 6768 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10944 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2196 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1615 \, A - 209 \, B}{a^{2} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}}}{2016 \, f} \] Input:

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algori 
thm="giac")
 

Output:

-1/2016*(21*(21*A*tan(1/2*f*x + 1/2*e)^2 - 15*B*tan(1/2*f*x + 1/2*e)^2 + 3 
6*A*tan(1/2*f*x + 1/2*e) - 24*B*tan(1/2*f*x + 1/2*e) + 19*A - 13*B)/(a^2*c 
^5*(tan(1/2*f*x + 1/2*e) + 1)^3) + (3591*A*tan(1/2*f*x + 1/2*e)^8 + 315*B* 
tan(1/2*f*x + 1/2*e)^8 - 19656*A*tan(1/2*f*x + 1/2*e)^7 + 756*B*tan(1/2*f* 
x + 1/2*e)^7 + 56196*A*tan(1/2*f*x + 1/2*e)^6 - 4200*B*tan(1/2*f*x + 1/2*e 
)^6 - 95760*A*tan(1/2*f*x + 1/2*e)^5 + 11340*B*tan(1/2*f*x + 1/2*e)^5 + 10 
7730*A*tan(1/2*f*x + 1/2*e)^4 - 14994*B*tan(1/2*f*x + 1/2*e)^4 - 79464*A*t 
an(1/2*f*x + 1/2*e)^3 + 13356*B*tan(1/2*f*x + 1/2*e)^3 + 38484*A*tan(1/2*f 
*x + 1/2*e)^2 - 6768*B*tan(1/2*f*x + 1/2*e)^2 - 10944*A*tan(1/2*f*x + 1/2* 
e) + 2196*B*tan(1/2*f*x + 1/2*e) + 1615*A - 209*B)/(a^2*c^5*(tan(1/2*f*x + 
 1/2*e) - 1)^9))/f
 

Mupad [B] (verification not implemented)

Time = 36.80 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.93 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {2\,\left (7\,A-14\,B-14\,A\,\sin \left (e+f\,x\right )+7\,B\,\sin \left (e+f\,x\right )+30\,A\,{\cos \left (e+f\,x\right )}^2-76\,A\,{\cos \left (e+f\,x\right )}^3-72\,A\,{\cos \left (e+f\,x\right )}^4+57\,A\,{\cos \left (e+f\,x\right )}^5+16\,A\,{\cos \left (e+f\,x\right )}^6-15\,B\,{\cos \left (e+f\,x\right )}^2-4\,B\,{\cos \left (e+f\,x\right )}^3+36\,B\,{\cos \left (e+f\,x\right )}^4+3\,B\,{\cos \left (e+f\,x\right )}^5-8\,B\,{\cos \left (e+f\,x\right )}^6-40\,A\,{\cos \left (e+f\,x\right )}^2\,\sin \left (e+f\,x\right )+76\,A\,{\cos \left (e+f\,x\right )}^3\,\sin \left (e+f\,x\right )+48\,A\,{\cos \left (e+f\,x\right )}^4\,\sin \left (e+f\,x\right )-19\,A\,{\cos \left (e+f\,x\right )}^5\,\sin \left (e+f\,x\right )+20\,B\,{\cos \left (e+f\,x\right )}^2\,\sin \left (e+f\,x\right )+4\,B\,{\cos \left (e+f\,x\right )}^3\,\sin \left (e+f\,x\right )-24\,B\,{\cos \left (e+f\,x\right )}^4\,\sin \left (e+f\,x\right )-B\,{\cos \left (e+f\,x\right )}^5\,\sin \left (e+f\,x\right )\right )}{63\,a^2\,c^5\,f\,\left (8\,{\cos \left (e+f\,x\right )}^3\,\sin \left (e+f\,x\right )-2\,{\cos \left (e+f\,x\right )}^5\,\sin \left (e+f\,x\right )-8\,{\cos \left (e+f\,x\right )}^3+6\,{\cos \left (e+f\,x\right )}^5\right )} \] Input:

int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^2*(c - c*sin(e + f*x))^5),x 
)
 

Output:

(2*(7*A - 14*B - 14*A*sin(e + f*x) + 7*B*sin(e + f*x) + 30*A*cos(e + f*x)^ 
2 - 76*A*cos(e + f*x)^3 - 72*A*cos(e + f*x)^4 + 57*A*cos(e + f*x)^5 + 16*A 
*cos(e + f*x)^6 - 15*B*cos(e + f*x)^2 - 4*B*cos(e + f*x)^3 + 36*B*cos(e + 
f*x)^4 + 3*B*cos(e + f*x)^5 - 8*B*cos(e + f*x)^6 - 40*A*cos(e + f*x)^2*sin 
(e + f*x) + 76*A*cos(e + f*x)^3*sin(e + f*x) + 48*A*cos(e + f*x)^4*sin(e + 
 f*x) - 19*A*cos(e + f*x)^5*sin(e + f*x) + 20*B*cos(e + f*x)^2*sin(e + f*x 
) + 4*B*cos(e + f*x)^3*sin(e + f*x) - 24*B*cos(e + f*x)^4*sin(e + f*x) - B 
*cos(e + f*x)^5*sin(e + f*x)))/(63*a^2*c^5*f*(8*cos(e + f*x)^3*sin(e + f*x 
) - 2*cos(e + f*x)^5*sin(e + f*x) - 8*cos(e + f*x)^3 + 6*cos(e + f*x)^5))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.19 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^5} \, dx=\frac {19 a +b +16 \sin \left (f x +e \right )^{6} a +\cos \left (f x +e \right ) b -8 \sin \left (f x +e \right )^{6} b -48 \sin \left (f x +e \right )^{5} a -2 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} a -3 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b +\cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} b +6 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a +2 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b +6 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a +6 a \sin \left (f x +e \right )+2 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b -4 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a -3 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b -4 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a -12 \sin \left (f x +e \right )^{4} b +56 \sin \left (f x +e \right )^{3} a +24 \sin \left (f x +e \right )^{5} b +24 \sin \left (f x +e \right )^{4} a -2 \cos \left (f x +e \right ) a -28 \sin \left (f x +e \right )^{3} b -66 \sin \left (f x +e \right )^{2} a +33 \sin \left (f x +e \right )^{2} b -3 \sin \left (f x +e \right ) b}{63 \cos \left (f x +e \right ) a^{2} c^{5} f \left (\sin \left (f x +e \right )^{5}-3 \sin \left (f x +e \right )^{4}+2 \sin \left (f x +e \right )^{3}+2 \sin \left (f x +e \right )^{2}-3 \sin \left (f x +e \right )+1\right )} \] Input:

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x)
 

Output:

( - 2*cos(e + f*x)*sin(e + f*x)**5*a + cos(e + f*x)*sin(e + f*x)**5*b + 6* 
cos(e + f*x)*sin(e + f*x)**4*a - 3*cos(e + f*x)*sin(e + f*x)**4*b - 4*cos( 
e + f*x)*sin(e + f*x)**3*a + 2*cos(e + f*x)*sin(e + f*x)**3*b - 4*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 - 3*cos(e + f*x)*sin(e + f*x)*b - 2*cos(e + f*x)*a + cos(e 
 + f*x)*b + 16*sin(e + f*x)**6*a - 8*sin(e + f*x)**6*b - 48*sin(e + f*x)** 
5*a + 24*sin(e + f*x)**5*b + 24*sin(e + f*x)**4*a - 12*sin(e + f*x)**4*b + 
 56*sin(e + f*x)**3*a - 28*sin(e + f*x)**3*b - 66*sin(e + f*x)**2*a + 33*s 
in(e + f*x)**2*b + 6*sin(e + f*x)*a - 3*sin(e + f*x)*b + 19*a + b)/(63*cos 
(e + f*x)*a**2*c**5*f*(sin(e + f*x)**5 - 3*sin(e + f*x)**4 + 2*sin(e + f*x 
)**3 + 2*sin(e + f*x)**2 - 3*sin(e + f*x) + 1))