\(\int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx\) [37]

Optimal result

Integrand size = 36, antiderivative size = 197 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}+\frac {a^2 (4 A-9 B) c \cos ^5(e+f x)}{143 f (c-c \sin (e+f x))^8}+\frac {a^2 (4 A-9 B) \cos ^5(e+f x)}{429 f (c-c \sin (e+f x))^7}+\frac {2 a^2 (4 A-9 B) \cos ^5(e+f x)}{3003 c f (c-c \sin (e+f x))^6}+\frac {2 a^2 (4 A-9 B) \cos ^5(e+f x)}{15015 c^2 f (c-c \sin (e+f x))^5} \] Output:

1/13*a^2*(A+B)*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^9+1/143*a^2*(4*A-9*B)*c 
*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^8+1/429*a^2*(4*A-9*B)*cos(f*x+e)^5/f/(c-c 
*sin(f*x+e))^7+2/3003*a^2*(4*A-9*B)*cos(f*x+e)^5/c/f/(c-c*sin(f*x+e))^6+2/ 
15015*a^2*(4*A-9*B)*cos(f*x+e)^5/c^2/f/(c-c*sin(f*x+e))^5
 

Mathematica [A] (verified)

Time = 14.81 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.59 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=-\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^2 \left (6006 (8 A+7 B) \cos \left (\frac {1}{2} (e+f x)\right )-1716 (11 A+19 B) \cos \left (\frac {3}{2} (e+f x)\right )-15015 B \cos \left (\frac {5}{2} (e+f x)\right )-1144 A \cos \left (\frac {7}{2} (e+f x)\right )+2574 B \cos \left (\frac {7}{2} (e+f x)\right )+52 A \cos \left (\frac {11}{2} (e+f x)\right )-117 B \cos \left (\frac {11}{2} (e+f x)\right )+54912 A \sin \left (\frac {1}{2} (e+f x)\right )+26598 B \sin \left (\frac {1}{2} (e+f x)\right )+24024 A \sin \left (\frac {3}{2} (e+f x)\right )+21021 B \sin \left (\frac {3}{2} (e+f x)\right )-2860 A \sin \left (\frac {5}{2} (e+f x)\right )-8580 B \sin \left (\frac {5}{2} (e+f x)\right )+312 A \sin \left (\frac {9}{2} (e+f x)\right )-702 B \sin \left (\frac {9}{2} (e+f x)\right )-4 A \sin \left (\frac {13}{2} (e+f x)\right )+9 B \sin \left (\frac {13}{2} (e+f x)\right )\right )}{240240 c^7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (-1+\sin (e+f x))^7} \] Input:

Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x 
])^7,x]
 

Output:

-1/240240*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2* 
(6006*(8*A + 7*B)*Cos[(e + f*x)/2] - 1716*(11*A + 19*B)*Cos[(3*(e + f*x))/ 
2] - 15015*B*Cos[(5*(e + f*x))/2] - 1144*A*Cos[(7*(e + f*x))/2] + 2574*B*C 
os[(7*(e + f*x))/2] + 52*A*Cos[(11*(e + f*x))/2] - 117*B*Cos[(11*(e + f*x) 
)/2] + 54912*A*Sin[(e + f*x)/2] + 26598*B*Sin[(e + f*x)/2] + 24024*A*Sin[( 
3*(e + f*x))/2] + 21021*B*Sin[(3*(e + f*x))/2] - 2860*A*Sin[(5*(e + f*x))/ 
2] - 8580*B*Sin[(5*(e + f*x))/2] + 312*A*Sin[(9*(e + f*x))/2] - 702*B*Sin[ 
(9*(e + f*x))/2] - 4*A*Sin[(13*(e + f*x))/2] + 9*B*Sin[(13*(e + f*x))/2])) 
/(c^7*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-1 + Sin[e + f*x])^7)
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3446, 3042, 3338, 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.

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (e+f x)+a)^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7}dx\)

\(\Big \downarrow \) 3446

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3338

\(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^8}dx}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^8}dx}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\)

\(\Big \downarrow \) 3151

\(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \left (\frac {3 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^7}dx}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \left (\frac {3 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^7}dx}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\)

\(\Big \downarrow \) 3151

\(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \left (\frac {3 \left (\frac {2 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^6}dx}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \left (\frac {3 \left (\frac {2 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^6}dx}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\)

\(\Big \downarrow \) 3151

\(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^5}dx}{7 c}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^5}dx}{7 c}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\)

\(\Big \downarrow \) 3150

\(\displaystyle a^2 c^2 \left (\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}+\frac {(4 A-9 B) \left (\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {3 \left (\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac {2 \left (\frac {\cos ^5(e+f x)}{35 c f (c-c \sin (e+f x))^5}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}\right )}{11 c}\right )}{13 c}\right )\)

Input:

Int[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^7,x 
]
 

Output:

a^2*c^2*(((A + B)*Cos[e + f*x]^5)/(13*f*(c - c*Sin[e + f*x])^9) + ((4*A - 
9*B)*(Cos[e + f*x]^5/(11*f*(c - c*Sin[e + f*x])^8) + (3*(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)))/(11*c)))/(13 
*c))
 

Defintions of rubi rules used

rule 3042

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

rule 3150

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]
 

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]))
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.49 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.26

method	result	size
risch	\(-\frac {4 i a^{2} \left (-26598 i B \,{\mathrm e}^{6 i \left (f x +e \right )}+24024 i A \,{\mathrm e}^{8 i \left (f x +e \right )}+15015 B \,{\mathrm e}^{9 i \left (f x +e \right )}+8580 i B \,{\mathrm e}^{4 i \left (f x +e \right )}-48048 A \,{\mathrm e}^{7 i \left (f x +e \right )}+2860 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-42042 B \,{\mathrm e}^{7 i \left (f x +e \right )}-312 i A \,{\mathrm e}^{2 i \left (f x +e \right )}+18876 A \,{\mathrm e}^{5 i \left (f x +e \right )}+21021 i B \,{\mathrm e}^{8 i \left (f x +e \right )}+32604 B \,{\mathrm e}^{5 i \left (f x +e \right )}+702 i B \,{\mathrm e}^{2 i \left (f x +e \right )}+1144 A \,{\mathrm e}^{3 i \left (f x +e \right )}-9 i B -2574 B \,{\mathrm e}^{3 i \left (f x +e \right )}-54912 i A \,{\mathrm e}^{6 i \left (f x +e \right )}-52 A \,{\mathrm e}^{i \left (f x +e \right )}+4 i A +117 B \,{\mathrm e}^{i \left (f x +e \right )}\right )}{15015 f \,c^{7} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{13}}\)	\(248\)
parallelrisch	\(-\frac {2 a^{2} \left (A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}+\left (-4 A +B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}+\left (18 A -B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}+\left (-40 A +7 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+\frac {\left (391 A -31 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{5}+\frac {2 \left (-244 A +39 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{5}+\frac {18 \left (202 A -17 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{35}+\frac {2 \left (-1276 A +211 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{35}+\frac {\left (-22 B +\frac {923 A}{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{7}+\frac {\left (107 B -\frac {1636 A}{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{35}+\frac {\left (1986 A -41 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{385}+\frac {\left (71 B -\frac {608 A}{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{385}+\frac {1763 A}{15015}-\frac {71 B}{5005}\right )}{f \,c^{7} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}\)	\(258\)
derivativedivides	\(\frac {2 a^{2} \left (-\frac {4480 A +4352 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {7744 A +5368 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1536 A +1536 B}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {4320 A +2568 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {10560 A +8256 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {256 A +256 B}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {10896 A +9360 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {560 A +208 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {16 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {1816 A +884 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {8320 A +7680 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {120 A +30 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right )}{f \,c^{7}}\)	\(293\)
default	\(\frac {2 a^{2} \left (-\frac {4480 A +4352 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {7744 A +5368 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1536 A +1536 B}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {4320 A +2568 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {10560 A +8256 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {256 A +256 B}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {10896 A +9360 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {560 A +208 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {16 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {1816 A +884 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {8320 A +7680 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {120 A +30 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right )}{f \,c^{7}}\)	\(293\)

Input:

int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x,method=_RETUR 
NVERBOSE)
 

Output:

-4/15015*I*a^2*(-26598*I*B*exp(6*I*(f*x+e))+24024*I*A*exp(8*I*(f*x+e))+150 
15*B*exp(9*I*(f*x+e))+8580*I*B*exp(4*I*(f*x+e))-48048*A*exp(7*I*(f*x+e))+2 
860*I*A*exp(4*I*(f*x+e))-42042*B*exp(7*I*(f*x+e))-312*I*A*exp(2*I*(f*x+e)) 
+18876*A*exp(5*I*(f*x+e))+21021*I*B*exp(8*I*(f*x+e))+32604*B*exp(5*I*(f*x+ 
e))+702*I*B*exp(2*I*(f*x+e))+1144*A*exp(3*I*(f*x+e))-9*I*B-2574*B*exp(3*I* 
(f*x+e))-54912*I*A*exp(6*I*(f*x+e))-52*A*exp(I*(f*x+e))+4*I*A+117*B*exp(I* 
(f*x+e)))/f/c^7/(exp(I*(f*x+e))-I)^13
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (192) = 384\).

Time = 0.10 (sec) , antiderivative size = 475, normalized size of antiderivative = 2.41 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\frac {2 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{7} - 12 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{6} - 49 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} + 70 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} + 105 \, {\left (7 \, A + 20 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} + 105 \, {\left (25 \, A + 51 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2310 \, {\left (A + B\right )} a^{2} \cos \left (f x + e\right ) - 4620 \, {\left (A + B\right )} a^{2} + {\left (2 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{6} + 14 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} - 35 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} - 105 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} + 105 \, {\left (3 \, A + 29 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2310 \, {\left (A + B\right )} a^{2} \cos \left (f x + e\right ) - 4620 \, {\left (A + B\right )} a^{2}\right )} \sin \left (f x + e\right )}{15015 \, {\left (c^{7} f \cos \left (f x + e\right )^{7} + 7 \, c^{7} f \cos \left (f x + e\right )^{6} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} - 56 \, c^{7} f \cos \left (f x + e\right )^{4} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} + 112 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f - {\left (c^{7} f \cos \left (f x + e\right )^{6} - 6 \, c^{7} f \cos \left (f x + e\right )^{5} - 24 \, c^{7} f \cos \left (f x + e\right )^{4} + 32 \, c^{7} f \cos \left (f x + e\right )^{3} + 80 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f\right )} \sin \left (f x + e\right )\right )}} \] Input:

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x, algori 
thm="fricas")
 

Output:

1/15015*(2*(4*A - 9*B)*a^2*cos(f*x + e)^7 - 12*(4*A - 9*B)*a^2*cos(f*x + e 
)^6 - 49*(4*A - 9*B)*a^2*cos(f*x + e)^5 + 70*(4*A - 9*B)*a^2*cos(f*x + e)^ 
4 + 105*(7*A + 20*B)*a^2*cos(f*x + e)^3 + 105*(25*A + 51*B)*a^2*cos(f*x + 
e)^2 - 2310*(A + B)*a^2*cos(f*x + e) - 4620*(A + B)*a^2 + (2*(4*A - 9*B)*a 
^2*cos(f*x + e)^6 + 14*(4*A - 9*B)*a^2*cos(f*x + e)^5 - 35*(4*A - 9*B)*a^2 
*cos(f*x + e)^4 - 105*(4*A - 9*B)*a^2*cos(f*x + e)^3 + 105*(3*A + 29*B)*a^ 
2*cos(f*x + e)^2 - 2310*(A + B)*a^2*cos(f*x + e) - 4620*(A + B)*a^2)*sin(f 
*x + e))/(c^7*f*cos(f*x + e)^7 + 7*c^7*f*cos(f*x + e)^6 - 18*c^7*f*cos(f*x 
 + e)^5 - 56*c^7*f*cos(f*x + e)^4 + 48*c^7*f*cos(f*x + e)^3 + 112*c^7*f*co 
s(f*x + e)^2 - 32*c^7*f*cos(f*x + e) - 64*c^7*f - (c^7*f*cos(f*x + e)^6 - 
6*c^7*f*cos(f*x + e)^5 - 24*c^7*f*cos(f*x + e)^4 + 32*c^7*f*cos(f*x + e)^3 
 + 80*c^7*f*cos(f*x + e)^2 - 32*c^7*f*cos(f*x + e) - 64*c^7*f)*sin(f*x + e 
))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6669 vs. \(2 (178) = 356\).

Time = 77.72 (sec) , antiderivative size = 6669, normalized size of antiderivative = 33.85 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))**2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**7,x)
 

Output:

Piecewise((-30030*A*a**2*tan(e/2 + f*x/2)**12/(15015*c**7*f*tan(e/2 + f*x/ 
2)**13 - 195195*c**7*f*tan(e/2 + f*x/2)**12 + 1171170*c**7*f*tan(e/2 + f*x 
/2)**11 - 4294290*c**7*f*tan(e/2 + f*x/2)**10 + 10735725*c**7*f*tan(e/2 + 
f*x/2)**9 - 19324305*c**7*f*tan(e/2 + f*x/2)**8 + 25765740*c**7*f*tan(e/2 
+ f*x/2)**7 - 25765740*c**7*f*tan(e/2 + f*x/2)**6 + 19324305*c**7*f*tan(e/ 
2 + f*x/2)**5 - 10735725*c**7*f*tan(e/2 + f*x/2)**4 + 4294290*c**7*f*tan(e 
/2 + f*x/2)**3 - 1171170*c**7*f*tan(e/2 + f*x/2)**2 + 195195*c**7*f*tan(e/ 
2 + f*x/2) - 15015*c**7*f) + 120120*A*a**2*tan(e/2 + f*x/2)**11/(15015*c** 
7*f*tan(e/2 + f*x/2)**13 - 195195*c**7*f*tan(e/2 + f*x/2)**12 + 1171170*c* 
*7*f*tan(e/2 + f*x/2)**11 - 4294290*c**7*f*tan(e/2 + f*x/2)**10 + 10735725 
*c**7*f*tan(e/2 + f*x/2)**9 - 19324305*c**7*f*tan(e/2 + f*x/2)**8 + 257657 
40*c**7*f*tan(e/2 + f*x/2)**7 - 25765740*c**7*f*tan(e/2 + f*x/2)**6 + 1932 
4305*c**7*f*tan(e/2 + f*x/2)**5 - 10735725*c**7*f*tan(e/2 + f*x/2)**4 + 42 
94290*c**7*f*tan(e/2 + f*x/2)**3 - 1171170*c**7*f*tan(e/2 + f*x/2)**2 + 19 
5195*c**7*f*tan(e/2 + f*x/2) - 15015*c**7*f) - 540540*A*a**2*tan(e/2 + f*x 
/2)**10/(15015*c**7*f*tan(e/2 + f*x/2)**13 - 195195*c**7*f*tan(e/2 + f*x/2 
)**12 + 1171170*c**7*f*tan(e/2 + f*x/2)**11 - 4294290*c**7*f*tan(e/2 + f*x 
/2)**10 + 10735725*c**7*f*tan(e/2 + f*x/2)**9 - 19324305*c**7*f*tan(e/2 + 
f*x/2)**8 + 25765740*c**7*f*tan(e/2 + f*x/2)**7 - 25765740*c**7*f*tan(e/2 
+ f*x/2)**6 + 19324305*c**7*f*tan(e/2 + f*x/2)**5 - 10735725*c**7*f*tan...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3120 vs. \(2 (192) = 384\).

Time = 0.15 (sec) , antiderivative size = 3120, normalized size of antiderivative = 15.84 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\text {Too large to display} \] Input:

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x, algori 
thm="maxima")
 

Output:

-2/45045*(2*A*a^2*(4771*sin(f*x + e)/(cos(f*x + e) + 1) - 28626*sin(f*x + 
e)^2/(cos(f*x + e) + 1)^2 + 74932*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 18 
7330*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 265122*sin(f*x + e)^5/(cos(f*x 
+ e) + 1)^5 - 353496*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 276276*sin(f*x 
+ e)^7/(cos(f*x + e) + 1)^7 - 207207*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 
 75075*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 30030*sin(f*x + e)^10/(cos(f* 
x + e) + 1)^10 - 367)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c 
^7*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + 
 e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*sin(f* 
x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1) 
^6 - 1716*c^7*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^ 
8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 286 
*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f 
*x + e) + 1)^11 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f 
*x + e)^13/(cos(f*x + e) + 1)^13) + 4*B*a^2*(4771*sin(f*x + e)/(cos(f*x + 
e) + 1) - 28626*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 74932*sin(f*x + e)^3 
/(cos(f*x + e) + 1)^3 - 187330*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 26512 
2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 353496*sin(f*x + e)^6/(cos(f*x + e 
) + 1)^6 + 276276*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 207207*sin(f*x + e 
)^8/(cos(f*x + e) + 1)^8 + 75075*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (192) = 384\).

Time = 0.29 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.14 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=-\frac {2 \, {\left (15015 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 60060 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 15015 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 270270 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 15015 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 600600 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 105105 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 1174173 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 93093 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 1465464 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 234234 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 1559844 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 131274 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1094808 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 181038 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 659945 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 47190 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 233948 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 45903 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 77454 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1599 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7904 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2769 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1763 \, A a^{2} - 213 \, B a^{2}\right )}}{15015 \, c^{7} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{13}} \] Input:

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x, algori 
thm="giac")
 

Output:

-2/15015*(15015*A*a^2*tan(1/2*f*x + 1/2*e)^12 - 60060*A*a^2*tan(1/2*f*x + 
1/2*e)^11 + 15015*B*a^2*tan(1/2*f*x + 1/2*e)^11 + 270270*A*a^2*tan(1/2*f*x 
 + 1/2*e)^10 - 15015*B*a^2*tan(1/2*f*x + 1/2*e)^10 - 600600*A*a^2*tan(1/2* 
f*x + 1/2*e)^9 + 105105*B*a^2*tan(1/2*f*x + 1/2*e)^9 + 1174173*A*a^2*tan(1 
/2*f*x + 1/2*e)^8 - 93093*B*a^2*tan(1/2*f*x + 1/2*e)^8 - 1465464*A*a^2*tan 
(1/2*f*x + 1/2*e)^7 + 234234*B*a^2*tan(1/2*f*x + 1/2*e)^7 + 1559844*A*a^2* 
tan(1/2*f*x + 1/2*e)^6 - 131274*B*a^2*tan(1/2*f*x + 1/2*e)^6 - 1094808*A*a 
^2*tan(1/2*f*x + 1/2*e)^5 + 181038*B*a^2*tan(1/2*f*x + 1/2*e)^5 + 659945*A 
*a^2*tan(1/2*f*x + 1/2*e)^4 - 47190*B*a^2*tan(1/2*f*x + 1/2*e)^4 - 233948* 
A*a^2*tan(1/2*f*x + 1/2*e)^3 + 45903*B*a^2*tan(1/2*f*x + 1/2*e)^3 + 77454* 
A*a^2*tan(1/2*f*x + 1/2*e)^2 - 1599*B*a^2*tan(1/2*f*x + 1/2*e)^2 - 7904*A* 
a^2*tan(1/2*f*x + 1/2*e) + 2769*B*a^2*tan(1/2*f*x + 1/2*e) + 1763*A*a^2 - 
213*B*a^2)/(c^7*f*(tan(1/2*f*x + 1/2*e) - 1)^13)
 

Mupad [B] (verification not implemented)

Time = 37.50 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.54 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx =\text {Too large to display} \] Input:

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^2)/(c - c*sin(e + f*x))^7,x 
)
 

Output:

-(2*cos(e/2 + (f*x)/2)*((994249*A*a^2)/32 - (63639*B*a^2)/32 - (1609013*A* 
a^2*cos(2*e + 2*f*x))/64 + (85687*A*a^2*cos(3*e + 3*f*x))/16 + (79591*A*a^ 
2*cos(4*e + 4*f*x))/32 - (5261*A*a^2*cos(5*e + 5*f*x))/16 - (1771*A*a^2*co 
s(6*e + 6*f*x))/64 + (140553*B*a^2*cos(2*e + 2*f*x))/64 - (4431*B*a^2*cos( 
3*e + 3*f*x))/8 - (10161*B*a^2*cos(4*e + 4*f*x))/32 + 36*B*a^2*cos(5*e + 5 
*f*x) + (231*B*a^2*cos(6*e + 6*f*x))/64 + (636207*A*a^2*sin(2*e + 2*f*x))/ 
64 + (309309*A*a^2*sin(3*e + 3*f*x))/32 - (7007*A*a^2*sin(4*e + 4*f*x))/4 
- (12389*A*a^2*sin(5*e + 5*f*x))/32 + (1755*A*a^2*sin(6*e + 6*f*x))/64 - ( 
121407*B*a^2*sin(2*e + 2*f*x))/64 - (39039*B*a^2*sin(3*e + 3*f*x))/32 + (3 
003*B*a^2*sin(4*e + 4*f*x))/16 + (1599*B*a^2*sin(5*e + 5*f*x))/32 - (195*B 
*a^2*sin(6*e + 6*f*x))/64 - (93221*A*a^2*cos(e + f*x))/8 + (3291*B*a^2*cos 
(e + f*x))/8 - (704847*A*a^2*sin(e + f*x))/16 + (125697*B*a^2*sin(e + f*x) 
)/16))/(15015*c^7*f*((1287*2^(1/2)*cos((3*e)/2 - pi/4 + (3*f*x)/2))/64 - ( 
429*2^(1/2)*cos(e/2 + pi/4 + (f*x)/2))/16 + (715*2^(1/2)*cos((5*e)/2 + pi/ 
4 + (5*f*x)/2))/64 - (143*2^(1/2)*cos((7*e)/2 - pi/4 + (7*f*x)/2))/32 - (3 
9*2^(1/2)*cos((9*e)/2 + pi/4 + (9*f*x)/2))/32 + (13*2^(1/2)*cos((11*e)/2 - 
 pi/4 + (11*f*x)/2))/64 + (2^(1/2)*cos((13*e)/2 + pi/4 + (13*f*x)/2))/64))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.76 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\frac {a^{2} \left (2310 a -231 \sin \left (f x +e \right )^{7} b +3781 \sin \left (f x +e \right )^{6} a +1599 \sin \left (f x +e \right )^{6} b -11371 \sin \left (f x +e \right )^{5} a -555 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} a -195 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} b +3338 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} a -213 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b +1152 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} b -8377 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a -539 \sin \left (f x +e \right )^{7} a -6123 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b +608 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a +608 a \sin \left (f x +e \right )+3567 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b -8572 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a -2808 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b +11248 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a +7761 \sin \left (f x +e \right )^{4} b -19086 \sin \left (f x +e \right )^{3} a -4734 \sin \left (f x +e \right )^{5} b +19009 \sin \left (f x +e \right )^{4} a -2310 \cos \left (f x +e \right ) a -3834 \sin \left (f x +e \right )^{3} b +14528 \sin \left (f x +e \right )^{2} a +8892 \sin \left (f x +e \right )^{2} b -213 \sin \left (f x +e \right ) b \right )}{15015 c^{7} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )^{6}-6 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5}+15 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}-20 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}+15 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}-6 \cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )+\sin \left (f x +e \right )^{7}-7 \sin \left (f x +e \right )^{6}+21 \sin \left (f x +e \right )^{5}-35 \sin \left (f x +e \right )^{4}+35 \sin \left (f x +e \right )^{3}-21 \sin \left (f x +e \right )^{2}+7 \sin \left (f x +e \right )-1\right )} \] Input:

int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x)
 

Output:

(a**2*( - 555*cos(e + f*x)*sin(e + f*x)**6*a - 195*cos(e + f*x)*sin(e + f* 
x)**6*b + 3338*cos(e + f*x)*sin(e + f*x)**5*a + 1152*cos(e + f*x)*sin(e + 
f*x)**5*b - 8377*cos(e + f*x)*sin(e + f*x)**4*a - 2808*cos(e + f*x)*sin(e 
+ f*x)**4*b + 11248*cos(e + f*x)*sin(e + f*x)**3*a + 3567*cos(e + f*x)*sin 
(e + f*x)**3*b - 8572*cos(e + f*x)*sin(e + f*x)**2*a - 6123*cos(e + f*x)*s 
in(e + f*x)**2*b + 608*cos(e + f*x)*sin(e + f*x)*a - 213*cos(e + f*x)*sin( 
e + f*x)*b - 2310*cos(e + f*x)*a - 539*sin(e + f*x)**7*a - 231*sin(e + f*x 
)**7*b + 3781*sin(e + f*x)**6*a + 1599*sin(e + f*x)**6*b - 11371*sin(e + f 
*x)**5*a - 4734*sin(e + f*x)**5*b + 19009*sin(e + f*x)**4*a + 7761*sin(e + 
 f*x)**4*b - 19086*sin(e + f*x)**3*a - 3834*sin(e + f*x)**3*b + 14528*sin( 
e + f*x)**2*a + 8892*sin(e + f*x)**2*b + 608*sin(e + f*x)*a - 213*sin(e + 
f*x)*b + 2310*a))/(15015*c**7*f*(cos(e + f*x)*sin(e + f*x)**6 - 6*cos(e + 
f*x)*sin(e + f*x)**5 + 15*cos(e + f*x)*sin(e + f*x)**4 - 20*cos(e + f*x)*s 
in(e + f*x)**3 + 15*cos(e + f*x)*sin(e + f*x)**2 - 6*cos(e + f*x)*sin(e + 
f*x) + cos(e + f*x) + sin(e + f*x)**7 - 7*sin(e + f*x)**6 + 21*sin(e + f*x 
)**5 - 35*sin(e + f*x)**4 + 35*sin(e + f*x)**3 - 21*sin(e + f*x)**2 + 7*si 
n(e + f*x) - 1))