\(\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx\) [49]

Optimal result

Integrand size = 36, antiderivative size = 118 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac {a^3 (2 A-9 B) c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac {a^3 (2 A-9 B) c \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7} \]

[Out]

1/11*a^3*(A+B)*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^9+1/99*a^3*(2*A-9*B)*c^2*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^8+
1/693*a^3*(2*A-9*B)*c*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^7

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3046, 2938, 2751, 2750} \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=\frac {a^3 c^3 (A+B) \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac {a^3 c^2 (2 A-9 B) \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac {a^3 c (2 A-9 B) \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7} \]

[In]

Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^6,x]

[Out]

(a^3*(A + B)*c^3*Cos[e + f*x]^7)/(11*f*(c - c*Sin[e + f*x])^9) + (a^3*(2*A - 9*B)*c^2*Cos[e + f*x]^7)/(99*f*(c
- c*Sin[e + f*x])^8) + (a^3*(2*A - 9*B)*c*Cos[e + f*x]^7)/(693*f*(c - c*Sin[e + f*x])^7)

Rule 2750

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[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 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[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[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 2938

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] + Dist[(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 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[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] && I
ntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))

Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^9} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac {1}{11} \left (a^3 (2 A-9 B) c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^8} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac {a^3 (2 A-9 B) c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac {1}{99} \left (a^3 (2 A-9 B) c\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{11 f (c-c \sin (e+f x))^9}+\frac {a^3 (2 A-9 B) c^2 \cos ^7(e+f x)}{99 f (c-c \sin (e+f x))^8}+\frac {a^3 (2 A-9 B) c \cos ^7(e+f x)}{693 f (c-c \sin (e+f x))^7} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(313\) vs. \(2(118)=236\).

Time = 12.46 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.65 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \left (462 (11 A+3 B) \cos \left (\frac {1}{2} (e+f x)\right )-594 (5 A+2 B) \cos \left (\frac {3}{2} (e+f x)\right )-924 A \cos \left (\frac {5}{2} (e+f x)\right )-693 B \cos \left (\frac {5}{2} (e+f x)\right )+110 A \cos \left (\frac {7}{2} (e+f x)\right )+198 B \cos \left (\frac {7}{2} (e+f x)\right )-2 A \cos \left (\frac {11}{2} (e+f x)\right )+9 B \cos \left (\frac {11}{2} (e+f x)\right )+4158 A \sin \left (\frac {1}{2} (e+f x)\right )+5544 B \sin \left (\frac {1}{2} (e+f x)\right )+2310 A \sin \left (\frac {3}{2} (e+f x)\right )+4158 B \sin \left (\frac {3}{2} (e+f x)\right )-594 A \sin \left (\frac {5}{2} (e+f x)\right )-2178 B \sin \left (\frac {5}{2} (e+f x)\right )-693 B \sin \left (\frac {7}{2} (e+f x)\right )-22 A \sin \left (\frac {9}{2} (e+f x)\right )+99 B \sin \left (\frac {9}{2} (e+f x)\right )\right )}{11088 c^6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (-1+\sin (e+f x))^6} \]

[In]

Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^6,x]

[Out]

(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(462*(11*A + 3*B)*Cos[(e + f*x)/2] - 594*(5*A
+ 2*B)*Cos[(3*(e + f*x))/2] - 924*A*Cos[(5*(e + f*x))/2] - 693*B*Cos[(5*(e + f*x))/2] + 110*A*Cos[(7*(e + f*x)
)/2] + 198*B*Cos[(7*(e + f*x))/2] - 2*A*Cos[(11*(e + f*x))/2] + 9*B*Cos[(11*(e + f*x))/2] + 4158*A*Sin[(e + f*
x)/2] + 5544*B*Sin[(e + f*x)/2] + 2310*A*Sin[(3*(e + f*x))/2] + 4158*B*Sin[(3*(e + f*x))/2] - 594*A*Sin[(5*(e
+ f*x))/2] - 2178*B*Sin[(5*(e + f*x))/2] - 693*B*Sin[(7*(e + f*x))/2] - 22*A*Sin[(9*(e + f*x))/2] + 99*B*Sin[(
9*(e + f*x))/2]))/(11088*c^6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-1 + Sin[e + f*x])^6)

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.79

method	result	size
parallelrisch	\(-\frac {2 \left (A \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (B -2 A \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {35 A}{3}+B \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 \left (-\frac {23 A}{3}+3 B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 \left (\frac {46 A}{3}+B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 \left (-11 A +4 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {12 \left (13 A +B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}+\frac {2 \left (-25 A +11 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}+\frac {\left (\frac {269 A}{9}+2 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}+\frac {\left (-\frac {16 A}{9}+B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7}+\frac {79 A}{693}-\frac {B}{77}\right ) a^{3}}{f \,c^{6} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\)	\(211\)
risch	\(-\frac {2 i a^{3} \left (2 i A +2970 i A \,{\mathrm e}^{4 i \left (f x +e \right )}+693 B \,{\mathrm e}^{9 i \left (f x +e \right )}+693 i B \,{\mathrm e}^{8 i \left (f x +e \right )}-2310 A \,{\mathrm e}^{7 i \left (f x +e \right )}-110 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-4158 B \,{\mathrm e}^{7 i \left (f x +e \right )}-9 i B +4158 A \,{\mathrm e}^{5 i \left (f x +e \right )}-1386 i B \,{\mathrm e}^{6 i \left (f x +e \right )}+5544 B \,{\mathrm e}^{5 i \left (f x +e \right )}-198 i B \,{\mathrm e}^{2 i \left (f x +e \right )}-594 A \,{\mathrm e}^{3 i \left (f x +e \right )}+924 i A \,{\mathrm e}^{8 i \left (f x +e \right )}-2178 B \,{\mathrm e}^{3 i \left (f x +e \right )}+1188 i B \,{\mathrm e}^{4 i \left (f x +e \right )}-22 A \,{\mathrm e}^{i \left (f x +e \right )}-5082 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+99 B \,{\mathrm e}^{i \left (f x +e \right )}\right )}{693 f \,c^{6} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{11}}\)	\(248\)
derivativedivides	\(\frac {2 a^{3} \left (-\frac {3008 A +2880 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {4352 A +3840 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {2960 A +1968 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {116 A +30 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {1460 A +780 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {256 A +256 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {504 A +200 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {16 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {4272 A +3344 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1280 A +1280 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}\right )}{f \,c^{6}}\)	\(249\)
default	\(\frac {2 a^{3} \left (-\frac {3008 A +2880 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {4352 A +3840 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {2960 A +1968 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {116 A +30 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {1460 A +780 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {256 A +256 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {504 A +200 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {16 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {4272 A +3344 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1280 A +1280 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}\right )}{f \,c^{6}}\)	\(249\)

[In]

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^6,x,method=_RETURNVERBOSE)

[Out]

-2*(A*tan(1/2*f*x+1/2*e)^10+(B-2*A)*tan(1/2*f*x+1/2*e)^9+(35/3*A+B)*tan(1/2*f*x+1/2*e)^8+2*(-23/3*A+3*B)*tan(1
/2*f*x+1/2*e)^7+2*(46/3*A+B)*tan(1/2*f*x+1/2*e)^6+2*(-11*A+4*B)*tan(1/2*f*x+1/2*e)^5+12/7*(13*A+B)*tan(1/2*f*x
+1/2*e)^4+2/7*(-25*A+11*B)*tan(1/2*f*x+1/2*e)^3+1/7*(269/9*A+2*B)*tan(1/2*f*x+1/2*e)^2+1/7*(-16/9*A+B)*tan(1/2
*f*x+1/2*e)+79/693*A-1/77*B)*a^3/f/c^6/(tan(1/2*f*x+1/2*e)-1)^11

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (115) = 230\).

Time = 0.27 (sec) , antiderivative size = 405, normalized size of antiderivative = 3.43 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=\frac {{\left (2 \, A - 9 \, B\right )} a^{3} \cos \left (f x + e\right )^{6} + 6 \, {\left (2 \, A - 9 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} - {\left (25 \, A + 234 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + 7 \, {\left (23 \, A + 45 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 28 \, {\left (16 \, A + 27 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 252 \, {\left (A + B\right )} a^{3} \cos \left (f x + e\right ) - 504 \, {\left (A + B\right )} a^{3} - {\left ({\left (2 \, A - 9 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} - 5 \, {\left (2 \, A - 9 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 7 \, {\left (5 \, A + 27 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 28 \, {\left (7 \, A + 18 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 252 \, {\left (A + B\right )} a^{3} \cos \left (f x + e\right ) + 504 \, {\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )}{693 \, {\left (c^{6} f \cos \left (f x + e\right )^{6} - 5 \, c^{6} f \cos \left (f x + e\right )^{5} - 18 \, c^{6} f \cos \left (f x + e\right )^{4} + 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 48 \, c^{6} f \cos \left (f x + e\right )^{2} - 16 \, c^{6} f \cos \left (f x + e\right ) - 32 \, c^{6} f + {\left (c^{6} f \cos \left (f x + e\right )^{5} + 6 \, c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} - 32 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f \cos \left (f x + e\right ) + 32 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^6,x, algorithm="fricas")

[Out]

1/693*((2*A - 9*B)*a^3*cos(f*x + e)^6 + 6*(2*A - 9*B)*a^3*cos(f*x + e)^5 - (25*A + 234*B)*a^3*cos(f*x + e)^4 +
7*(23*A + 45*B)*a^3*cos(f*x + e)^3 + 28*(16*A + 27*B)*a^3*cos(f*x + e)^2 - 252*(A + B)*a^3*cos(f*x + e) - 504
*(A + B)*a^3 - ((2*A - 9*B)*a^3*cos(f*x + e)^5 - 5*(2*A - 9*B)*a^3*cos(f*x + e)^4 - 7*(5*A + 27*B)*a^3*cos(f*x
+ e)^3 - 28*(7*A + 18*B)*a^3*cos(f*x + e)^2 + 252*(A + B)*a^3*cos(f*x + e) + 504*(A + B)*a^3)*sin(f*x + e))/(
c^6*f*cos(f*x + e)^6 - 5*c^6*f*cos(f*x + e)^5 - 18*c^6*f*cos(f*x + e)^4 + 20*c^6*f*cos(f*x + e)^3 + 48*c^6*f*c
os(f*x + e)^2 - 16*c^6*f*cos(f*x + e) - 32*c^6*f + (c^6*f*cos(f*x + e)^5 + 6*c^6*f*cos(f*x + e)^4 - 12*c^6*f*c
os(f*x + e)^3 - 32*c^6*f*cos(f*x + e)^2 + 16*c^6*f*cos(f*x + e) + 32*c^6*f)*sin(f*x + e))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4816 vs. \(2 (105) = 210\).

Time = 64.76 (sec) , antiderivative size = 4816, normalized size of antiderivative = 40.81 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**6,x)

[Out]

Piecewise((-1386*A*a**3*tan(e/2 + f*x/2)**10/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**
10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7
- 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 +
114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6
*f) + 2772*A*a**3*tan(e/2 + f*x/2)**9/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38
115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 32016
6*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*
c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 1
6170*A*a**3*tan(e/2 + f*x/2)**8/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c*
*6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6
*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f
*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) + 21252*A
*a**3*tan(e/2 + f*x/2)**7/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*t
an(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan
(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e
/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 42504*A*a**3*
tan(e/2 + f*x/2)**6/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2
+ f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 +
f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f
*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) + 30492*A*a**3*tan(e/
2 + f*x/2)**5/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x
/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2
)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)*
*3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 30888*A*a**3*tan(e/2 + f*
x/2)**4/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9
- 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 +
320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 3
8115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) + 9900*A*a**3*tan(e/2 + f*x/2)**3
/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 1143
45*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166
*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c*
*6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 5918*A*a**3*tan(e/2 + f*x/2)**2/(693*c
**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6
*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f
*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*ta
n(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) + 352*A*a**3*tan(e/2 + f*x/2)/(693*c**6*f*tan(e
/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2
+ f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 +
f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x
/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 158*A*a**3/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*
f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*
tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*ta
n(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2
+ f*x/2) - 693*c**6*f) - 1386*B*a**3*tan(e/2 + f*x/2)**9/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e
/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2
+ f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 +
f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/
2) - 693*c**6*f) - 1386*B*a**3*tan(e/2 + f*x/2)**8/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*
x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/
2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)
**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 69
3*c**6*f) - 8316*B*a**3*tan(e/2 + f*x/2)**7/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**1
0 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 -
320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 1
14345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*
f) - 2772*B*a**3*tan(e/2 + f*x/2)**6/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 381
15*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166
*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c
**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 11
088*B*a**3*tan(e/2 + f*x/2)**5/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**
6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*
f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*
tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 2376*B*a
**3*tan(e/2 + f*x/2)**4/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan
(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e
/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2
+ f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 4356*B*a**3*tan
(e/2 + f*x/2)**3/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 +
f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*
x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/
2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 396*B*a**3*tan(e/2 + f
*x/2)**2/(693*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**
9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6
+ 320166*c**6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 -
38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) - 198*B*a**3*tan(e/2 + f*x/2)/(6
93*c**6*f*tan(e/2 + f*x/2)**11 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*
c**6*f*tan(e/2 + f*x/2)**8 + 228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c*
*6*f*tan(e/2 + f*x/2)**5 - 228690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*
f*tan(e/2 + f*x/2)**2 + 7623*c**6*f*tan(e/2 + f*x/2) - 693*c**6*f) + 18*B*a**3/(693*c**6*f*tan(e/2 + f*x/2)**1
1 - 7623*c**6*f*tan(e/2 + f*x/2)**10 + 38115*c**6*f*tan(e/2 + f*x/2)**9 - 114345*c**6*f*tan(e/2 + f*x/2)**8 +
228690*c**6*f*tan(e/2 + f*x/2)**7 - 320166*c**6*f*tan(e/2 + f*x/2)**6 + 320166*c**6*f*tan(e/2 + f*x/2)**5 - 22
8690*c**6*f*tan(e/2 + f*x/2)**4 + 114345*c**6*f*tan(e/2 + f*x/2)**3 - 38115*c**6*f*tan(e/2 + f*x/2)**2 + 7623*
c**6*f*tan(e/2 + f*x/2) - 693*c**6*f), Ne(f, 0)), (x*(A + B*sin(e))*(a*sin(e) + a)**3/(-c*sin(e) + c)**6, True
))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3390 vs. \(2 (115) = 230\).

Time = 0.31 (sec) , antiderivative size = 3390, normalized size of antiderivative = 28.73 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^6,x, algorithm="maxima")

[Out]

-2/3465*(5*A*a^3*(913*sin(f*x + e)/(cos(f*x + e) + 1) - 4565*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 12540*sin(f
*x + e)^3/(cos(f*x + e) + 1)^3 - 25080*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33726*sin(f*x + e)^5/(cos(f*x + e
) + 1)^5 - 33726*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 23100*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 11550*sin(f
*x + e)^8/(cos(f*x + e) + 1)^8 + 3465*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 693*sin(f*x + e)^10/(cos(f*x + e)
+ 1)^10 - 146)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 16
5*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)
^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) +
1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(
f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) - 9*A*a^3*(671*sin(f*x + e)/(co
s(f*x + e) + 1) - 2200*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 6600*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 10890*
sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 15246*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 12936*sin(f*x + e)^6/(cos(f*
x + e) + 1)^6 + 9240*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3465*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 1155*sin
(f*x + e)^9/(cos(f*x + e) + 1)^9 - 61)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(
cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^
4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*
x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x +
e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) - 3*B*a
^3*(671*sin(f*x + e)/(cos(f*x + e) + 1) - 2200*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 6600*sin(f*x + e)^3/(cos(
f*x + e) + 1)^3 - 10890*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 15246*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 1293
6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 9240*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3465*sin(f*x + e)^8/(cos(f*
x + e) + 1)^8 + 1155*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 61)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) +
55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x +
e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e)
+ 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*s
in(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f
*x + e) + 1)^11) - 2*A*a^3*(341*sin(f*x + e)/(cos(f*x + e) + 1) - 1705*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 5
115*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 6765*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 9471*sin(f*x + e)^5/(cos(
f*x + e) + 1)^5 - 4851*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3465*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 31)/(c
^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e
)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e)
+ 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*s
in(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(
f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) - 6*B*a^3*(341*sin(f*x + e)/(cos(f*x + e) + 1) -
1705*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 5115*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 6765*sin(f*x + e)^4/(co
s(f*x + e) + 1)^4 + 9471*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 4851*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3465
*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 31)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)
^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) +
1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*si
n(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f
*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 1
2*A*a^3*(253*sin(f*x + e)/(cos(f*x + e) + 1) - 1265*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2640*sin(f*x + e)^3/
(cos(f*x + e) + 1)^3 - 5280*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5313*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 5
313*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2310*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 1155*sin(f*x + e)^8/(cos(
f*x + e) + 1)^8 - 23)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)
^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f
*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x
+ e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c
^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 12*B*a^3*(253*sin(f*x
+ e)/(cos(f*x + e) + 1) - 1265*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2640*sin(f*x + e)^3/(cos(f*x + e) + 1)^3
- 5280*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5313*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 5313*sin(f*x + e)^6/(c
os(f*x + e) + 1)^6 + 2310*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 1155*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 23)
/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x
+ e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x +
e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^
6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(c
os(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 48*B*a^3*(11*sin(f*x + e)/(cos(f*x + e) + 1
) - 55*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 165*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 330*sin(f*x + e)^4/(cos
(f*x + e) + 1)^4 + 231*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 231*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1)/(c^6
- 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^
3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) +
1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin
(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*
x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (115) = 230\).

Time = 0.76 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.99 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (693 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 1386 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 693 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 8085 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 693 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 10626 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 4158 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 21252 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 1386 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 15246 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 5544 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 15444 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1188 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 4950 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2178 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2959 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 198 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 176 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 99 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 79 \, A a^{3} - 9 \, B a^{3}\right )}}{693 \, c^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{11}} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^6,x, algorithm="giac")

[Out]

-2/693*(693*A*a^3*tan(1/2*f*x + 1/2*e)^10 - 1386*A*a^3*tan(1/2*f*x + 1/2*e)^9 + 693*B*a^3*tan(1/2*f*x + 1/2*e)
^9 + 8085*A*a^3*tan(1/2*f*x + 1/2*e)^8 + 693*B*a^3*tan(1/2*f*x + 1/2*e)^8 - 10626*A*a^3*tan(1/2*f*x + 1/2*e)^7
+ 4158*B*a^3*tan(1/2*f*x + 1/2*e)^7 + 21252*A*a^3*tan(1/2*f*x + 1/2*e)^6 + 1386*B*a^3*tan(1/2*f*x + 1/2*e)^6
- 15246*A*a^3*tan(1/2*f*x + 1/2*e)^5 + 5544*B*a^3*tan(1/2*f*x + 1/2*e)^5 + 15444*A*a^3*tan(1/2*f*x + 1/2*e)^4
+ 1188*B*a^3*tan(1/2*f*x + 1/2*e)^4 - 4950*A*a^3*tan(1/2*f*x + 1/2*e)^3 + 2178*B*a^3*tan(1/2*f*x + 1/2*e)^3 +
2959*A*a^3*tan(1/2*f*x + 1/2*e)^2 + 198*B*a^3*tan(1/2*f*x + 1/2*e)^2 - 176*A*a^3*tan(1/2*f*x + 1/2*e) + 99*B*a
^3*tan(1/2*f*x + 1/2*e) + 79*A*a^3 - 9*B*a^3)/(c^6*f*(tan(1/2*f*x + 1/2*e) - 1)^11)

Mupad [B] (verification not implemented)

Time = 14.78 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.46 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=-\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (565\,A\,a^3\,\cos \left (2\,e+2\,f\,x\right )-\frac {837\,B\,a^3}{16}-922\,A\,a^3-\frac {3527\,A\,a^3\,\cos \left (3\,e+3\,f\,x\right )}{32}-29\,A\,a^3\,\cos \left (4\,e+4\,f\,x\right )+\frac {81\,A\,a^3\,\cos \left (5\,e+5\,f\,x\right )}{32}+\frac {225\,B\,a^3\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {207\,B\,a^3\,\cos \left (3\,e+3\,f\,x\right )}{16}+\frac {9\,B\,a^3\,\cos \left (4\,e+4\,f\,x\right )}{16}-\frac {9\,B\,a^3\,\cos \left (5\,e+5\,f\,x\right )}{16}-\frac {1617\,A\,a^3\,\sin \left (2\,e+2\,f\,x\right )}{8}-\frac {5049\,A\,a^3\,\sin \left (3\,e+3\,f\,x\right )}{32}+\frac {407\,A\,a^3\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {77\,A\,a^3\,\sin \left (5\,e+5\,f\,x\right )}{32}+\frac {693\,B\,a^3\,\sin \left (2\,e+2\,f\,x\right )}{8}+\frac {99\,B\,a^3\,\sin \left (3\,e+3\,f\,x\right )}{2}-\frac {99\,B\,a^3\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {6635\,A\,a^3\,\cos \left (e+f\,x\right )}{16}+18\,B\,a^3\,\cos \left (e+f\,x\right )+\frac {13629\,A\,a^3\,\sin \left (e+f\,x\right )}{16}-\frac {693\,B\,a^3\,\sin \left (e+f\,x\right )}{2}\right )}{693\,c^6\,f\,\left (\frac {231\,\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}{16}-\frac {165\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}-\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{16}-\frac {165\,\sqrt {2}\,\cos \left (\frac {5\,e}{2}+\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{32}+\frac {55\,\sqrt {2}\,\cos \left (\frac {7\,e}{2}-\frac {\pi }{4}+\frac {7\,f\,x}{2}\right )}{32}+\frac {11\,\sqrt {2}\,\cos \left (\frac {9\,e}{2}+\frac {\pi }{4}+\frac {9\,f\,x}{2}\right )}{32}-\frac {\sqrt {2}\,\cos \left (\frac {11\,e}{2}-\frac {\pi }{4}+\frac {11\,f\,x}{2}\right )}{32}\right )} \]

[In]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^6,x)

[Out]

-(2*cos(e/2 + (f*x)/2)*(565*A*a^3*cos(2*e + 2*f*x) - (837*B*a^3)/16 - 922*A*a^3 - (3527*A*a^3*cos(3*e + 3*f*x)
)/32 - 29*A*a^3*cos(4*e + 4*f*x) + (81*A*a^3*cos(5*e + 5*f*x))/32 + (225*B*a^3*cos(2*e + 2*f*x))/4 - (207*B*a^
3*cos(3*e + 3*f*x))/16 + (9*B*a^3*cos(4*e + 4*f*x))/16 - (9*B*a^3*cos(5*e + 5*f*x))/16 - (1617*A*a^3*sin(2*e +
2*f*x))/8 - (5049*A*a^3*sin(3*e + 3*f*x))/32 + (407*A*a^3*sin(4*e + 4*f*x))/16 + (77*A*a^3*sin(5*e + 5*f*x))/
32 + (693*B*a^3*sin(2*e + 2*f*x))/8 + (99*B*a^3*sin(3*e + 3*f*x))/2 - (99*B*a^3*sin(4*e + 4*f*x))/16 + (6635*A
*a^3*cos(e + f*x))/16 + 18*B*a^3*cos(e + f*x) + (13629*A*a^3*sin(e + f*x))/16 - (693*B*a^3*sin(e + f*x))/2))/(
693*c^6*f*((231*2^(1/2)*cos(e/2 + pi/4 + (f*x)/2))/16 - (165*2^(1/2)*cos((3*e)/2 - pi/4 + (3*f*x)/2))/16 - (16
5*2^(1/2)*cos((5*e)/2 + pi/4 + (5*f*x)/2))/32 + (55*2^(1/2)*cos((7*e)/2 - pi/4 + (7*f*x)/2))/32 + (11*2^(1/2)*
cos((9*e)/2 + pi/4 + (9*f*x)/2))/32 - (2^(1/2)*cos((11*e)/2 - pi/4 + (11*f*x)/2))/32))