\(\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx\) [50]

Optimal result

Integrand size = 36, antiderivative size = 156 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {a^3 (3 A-10 B) c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {2 a^3 (3 A-10 B) c \cos ^7(e+f x)}{1287 f (c-c \sin (e+f x))^8}+\frac {2 a^3 (3 A-10 B) \cos ^7(e+f x)}{9009 f (c-c \sin (e+f x))^7} \]

[Out]

1/13*a^3*(A+B)*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^10+1/143*a^3*(3*A-10*B)*c^2*cos(f*x+e)^7/f/(c-c*sin(f*x+e))
^9+2/1287*a^3*(3*A-10*B)*c*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^8+2/9009*a^3*(3*A-10*B)*cos(f*x+e)^7/f/(c-c*sin(f*x
+e))^7

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3046, 2938, 2751, 2750} \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\frac {a^3 c^3 (A+B) \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {a^3 c^2 (3 A-10 B) \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {2 a^3 (3 A-10 B) \cos ^7(e+f x)}{9009 f (c-c \sin (e+f x))^7}+\frac {2 a^3 c (3 A-10 B) \cos ^7(e+f x)}{1287 f (c-c \sin (e+f x))^8} \]

[In]

Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^7,x]

[Out]

(a^3*(A + B)*c^3*Cos[e + f*x]^7)/(13*f*(c - c*Sin[e + f*x])^10) + (a^3*(3*A - 10*B)*c^2*Cos[e + f*x]^7)/(143*f
*(c - c*Sin[e + f*x])^9) + (2*a^3*(3*A - 10*B)*c*Cos[e + f*x]^7)/(1287*f*(c - c*Sin[e + f*x])^8) + (2*a^3*(3*A
- 10*B)*Cos[e + f*x]^7)/(9009*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))^{10}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {1}{13} \left (a^3 (3 A-10 B) c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^9} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {a^3 (3 A-10 B) c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {1}{143} \left (2 a^3 (3 A-10 B) c\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)}{13 f (c-c \sin (e+f x))^{10}}+\frac {a^3 (3 A-10 B) c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {2 a^3 (3 A-10 B) c \cos ^7(e+f x)}{1287 f (c-c \sin (e+f x))^8}+\frac {\left (2 a^3 (3 A-10 B)\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx}{1287} \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{13 f (c-c \sin (e+f x))^{10}}+\frac {a^3 (3 A-10 B) c^2 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^9}+\frac {2 a^3 (3 A-10 B) c \cos ^7(e+f x)}{1287 f (c-c \sin (e+f x))^8}+\frac {2 a^3 (3 A-10 B) \cos ^7(e+f x)}{9009 f (c-c \sin (e+f x))^7} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(339\) vs. \(2(156)=312\).

Time = 13.83 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.17 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=-\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \left (6006 (9 A+5 B) \cos \left (\frac {1}{2} (e+f x)\right )-7722 (4 A+3 B) \cos \left (\frac {3}{2} (e+f x)\right )-9009 A \cos \left (\frac {5}{2} (e+f x)\right )-12012 B \cos \left (\frac {5}{2} (e+f x)\right )+858 A \cos \left (\frac {7}{2} (e+f x)\right )+3146 B \cos \left (\frac {7}{2} (e+f x)\right )-39 A \cos \left (\frac {11}{2} (e+f x)\right )+130 B \cos \left (\frac {11}{2} (e+f x)\right )+48906 A \sin \left (\frac {1}{2} (e+f x)\right )+47190 B \sin \left (\frac {1}{2} (e+f x)\right )+27027 A \sin \left (\frac {3}{2} (e+f x)\right )+36036 B \sin \left (\frac {3}{2} (e+f x)\right )-6864 A \sin \left (\frac {5}{2} (e+f x)\right )-19162 B \sin \left (\frac {5}{2} (e+f x)\right )-6006 B \sin \left (\frac {7}{2} (e+f x)\right )-234 A \sin \left (\frac {9}{2} (e+f x)\right )+780 B \sin \left (\frac {9}{2} (e+f x)\right )+3 A \sin \left (\frac {13}{2} (e+f x)\right )-10 B \sin \left (\frac {13}{2} (e+f x)\right )\right )}{144144 c^7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (-1+\sin (e+f x))^7} \]

[In]

Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^7,x]

[Out]

-1/144144*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(6006*(9*A + 5*B)*Cos[(e + f*x)/2] -
7722*(4*A + 3*B)*Cos[(3*(e + f*x))/2] - 9009*A*Cos[(5*(e + f*x))/2] - 12012*B*Cos[(5*(e + f*x))/2] + 858*A*Co
s[(7*(e + f*x))/2] + 3146*B*Cos[(7*(e + f*x))/2] - 39*A*Cos[(11*(e + f*x))/2] + 130*B*Cos[(11*(e + f*x))/2] +
48906*A*Sin[(e + f*x)/2] + 47190*B*Sin[(e + f*x)/2] + 27027*A*Sin[(3*(e + f*x))/2] + 36036*B*Sin[(3*(e + f*x))
/2] - 6864*A*Sin[(5*(e + f*x))/2] - 19162*B*Sin[(5*(e + f*x))/2] - 6006*B*Sin[(7*(e + f*x))/2] - 234*A*Sin[(9*
(e + f*x))/2] + 780*B*Sin[(9*(e + f*x))/2] + 3*A*Sin[(13*(e + f*x))/2] - 10*B*Sin[(13*(e + f*x))/2]))/(c^7*f*(
Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-1 + Sin[e + f*x])^7)

Maple [A] (verified)

Time = 1.69 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.60

method	result	size
parallelrisch	\(-\frac {2 \left (A \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-3 A +B \right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (17 A +\frac {B}{3}\right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-33 A +\frac {23 B}{3}\right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (72 A -B \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-82 A +\frac {50 B}{3}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {666 A}{7}-\frac {38 B}{21}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-\frac {426 A}{7}+\frac {90 B}{7}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {857 A}{21}-\frac {2 B}{63}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-\frac {263 A}{21}+\frac {215 B}{63}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {37 B}{231}+\frac {389 A}{77}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-\frac {79 A}{231}+\frac {97 B}{693}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {310 A}{3003}-\frac {97 B}{9009}\right ) a^{3}}{f \,c^{7} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}\)	\(250\)
derivativedivides	\(\frac {2 a^{3} \left (-\frac {6888 A +3928 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {18816 A +14464 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {18 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {3072 A +3072 B}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {8832 A +8576 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {512 A +512 B}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {768 A +264 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {150 A +34 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {2700 A +1240 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {20256 A +17248 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {13112 A +8840 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {16000 A +14720 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}\right )}{f \,c^{7}}\)	\(293\)
default	\(\frac {2 a^{3} \left (-\frac {6888 A +3928 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {18816 A +14464 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {18 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {3072 A +3072 B}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {8832 A +8576 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {512 A +512 B}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {768 A +264 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {150 A +34 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {2700 A +1240 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {20256 A +17248 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {13112 A +8840 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {16000 A +14720 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}\right )}{f \,c^{7}}\)	\(293\)
risch	\(-\frac {4 \left (-10 B \,a^{3}+3 A \,a^{3}-30030 i B \,a^{3} {\mathrm e}^{7 i \left (f x +e \right )}+39 i A \,a^{3} {\mathrm e}^{i \left (f x +e \right )}+12012 i B \,a^{3} {\mathrm e}^{9 i \left (f x +e \right )}-130 i B \,a^{3} {\mathrm e}^{i \left (f x +e \right )}-234 A \,a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-54054 i A \,a^{3} {\mathrm e}^{7 i \left (f x +e \right )}-3146 i B \,a^{3} {\mathrm e}^{3 i \left (f x +e \right )}-858 i A \,a^{3} {\mathrm e}^{3 i \left (f x +e \right )}+23166 i B \,a^{3} {\mathrm e}^{5 i \left (f x +e \right )}+9009 i A \,a^{3} {\mathrm e}^{9 i \left (f x +e \right )}+30888 i A \,a^{3} {\mathrm e}^{5 i \left (f x +e \right )}+48906 A \,a^{3} {\mathrm e}^{6 i \left (f x +e \right )}+47190 B \,a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-6864 A \,a^{3} {\mathrm e}^{4 i \left (f x +e \right )}-19162 B \,a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+780 B \,a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-27027 A \,a^{3} {\mathrm e}^{8 i \left (f x +e \right )}-36036 B \,a^{3} {\mathrm e}^{8 i \left (f x +e \right )}+6006 B \,a^{3} {\mathrm e}^{10 i \left (f x +e \right )}\right )}{9009 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{13} f \,c^{7}}\)	\(331\)

[In]

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x,method=_RETURNVERBOSE)

[Out]

-2*(A*tan(1/2*f*x+1/2*e)^12+(-3*A+B)*tan(1/2*f*x+1/2*e)^11+(17*A+1/3*B)*tan(1/2*f*x+1/2*e)^10+(-33*A+23/3*B)*t
an(1/2*f*x+1/2*e)^9+(72*A-B)*tan(1/2*f*x+1/2*e)^8+(-82*A+50/3*B)*tan(1/2*f*x+1/2*e)^7+(666/7*A-38/21*B)*tan(1/
2*f*x+1/2*e)^6+(-426/7*A+90/7*B)*tan(1/2*f*x+1/2*e)^5+(857/21*A-2/63*B)*tan(1/2*f*x+1/2*e)^4+(-263/21*A+215/63
*B)*tan(1/2*f*x+1/2*e)^3+(37/231*B+389/77*A)*tan(1/2*f*x+1/2*e)^2+(-79/231*A+97/693*B)*tan(1/2*f*x+1/2*e)+310/
3003*A-97/9009*B)*a^3/f/c^7/(tan(1/2*f*x+1/2*e)-1)^13

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (152) = 304\).

Time = 0.27 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.04 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=-\frac {2 \, {\left (3 \, A - 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{7} - 12 \, {\left (3 \, A - 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{6} - 49 \, {\left (3 \, A - 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} + 7 \, {\left (30 \, A + 329 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 63 \, {\left (27 \, A + 53 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 252 \, {\left (19 \, A + 32 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 2772 \, {\left (A + B\right )} a^{3} \cos \left (f x + e\right ) + 5544 \, {\left (A + B\right )} a^{3} + {\left (2 \, {\left (3 \, A - 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{6} + 14 \, {\left (3 \, A - 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} - 35 \, {\left (3 \, A - 10 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 63 \, {\left (5 \, A + 31 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 252 \, {\left (8 \, A + 21 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 2772 \, {\left (A + B\right )} a^{3} \cos \left (f x + e\right ) + 5544 \, {\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )}{9009 \, {\left (c^{7} f \cos \left (f x + e\right )^{7} + 7 \, c^{7} f \cos \left (f x + e\right )^{6} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} - 56 \, c^{7} f \cos \left (f x + e\right )^{4} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} + 112 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f - {\left (c^{7} f \cos \left (f x + e\right )^{6} - 6 \, c^{7} f \cos \left (f x + e\right )^{5} - 24 \, c^{7} f \cos \left (f x + e\right )^{4} + 32 \, c^{7} f \cos \left (f x + e\right )^{3} + 80 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x, algorithm="fricas")

[Out]

-1/9009*(2*(3*A - 10*B)*a^3*cos(f*x + e)^7 - 12*(3*A - 10*B)*a^3*cos(f*x + e)^6 - 49*(3*A - 10*B)*a^3*cos(f*x
+ e)^5 + 7*(30*A + 329*B)*a^3*cos(f*x + e)^4 - 63*(27*A + 53*B)*a^3*cos(f*x + e)^3 - 252*(19*A + 32*B)*a^3*cos
(f*x + e)^2 + 2772*(A + B)*a^3*cos(f*x + e) + 5544*(A + B)*a^3 + (2*(3*A - 10*B)*a^3*cos(f*x + e)^6 + 14*(3*A
- 10*B)*a^3*cos(f*x + e)^5 - 35*(3*A - 10*B)*a^3*cos(f*x + e)^4 - 63*(5*A + 31*B)*a^3*cos(f*x + e)^3 - 252*(8*
A + 21*B)*a^3*cos(f*x + e)^2 + 2772*(A + B)*a^3*cos(f*x + e) + 5544*(A + B)*a^3)*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*cos(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. \(2 (143) = 286\).

Time = 102.64 (sec) , antiderivative size = 6669, normalized size of antiderivative = 42.75 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**7,x)

[Out]

Piecewise((-18018*A*a**3*tan(e/2 + f*x/2)**12/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/
2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f
*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2
+ f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e
/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) + 54054*A*a
**3*tan(e/2 + f*x/2)**11/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*
f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c
**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 115945
83*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 7027
02*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) - 306306*A*a**3*tan(e/2 + f*x/2)
**10/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**
11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/
2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 +
f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 +
f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) + 594594*A*a**3*tan(e/2 + f*x/2)**9/(9009*c**7*f*tan
(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*t
an(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7
*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c
**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c*
*7*f*tan(e/2 + f*x/2) - 9009*c**7*f) - 1297296*A*a**3*tan(e/2 + f*x/2)**8/(9009*c**7*f*tan(e/2 + f*x/2)**13 -
117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10
+ 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)*
*7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/
2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2
) - 9009*c**7*f) + 1477476*A*a**3*tan(e/2 + f*x/2)**7/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/
2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan
(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f
*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7
*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) - 1
714284*A*a**3*tan(e/2 + f*x/2)**6/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702
702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 1
1594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6
+ 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)*
*3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) + 1096524*A*a**3*tan(e/
2 + f*x/2)**5/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 +
f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e
/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*t
an(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*t
an(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) - 735306*A*a**3*tan(e/2 + f*x/2)**4/(9009*c
**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574
*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 1545
9444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 -
6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 +
117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) + 225654*A*a**3*tan(e/2 + f*x/2)**3/(9009*c**7*f*tan(e/2 + f*x/2
)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x
/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 +
f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/
2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2
+ f*x/2) - 9009*c**7*f) - 91026*A*a**3*tan(e/2 + f*x/2)**2/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*
tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7
*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*
c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 257657
4*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*
f) + 6162*A*a**3*tan(e/2 + f*x/2)/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702
702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 1
1594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6
+ 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)*
*3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) - 1860*A*a**3/(9009*c**
7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c
**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 154594
44*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 64
41435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 11
7117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) - 18018*B*a**3*tan(e/2 + f*x/2)**11/(9009*c**7*f*tan(e/2 + f*x/2)*
*13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2
)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f
*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2
+ f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 +
f*x/2) - 9009*c**7*f) - 6006*B*a**3*tan(e/2 + f*x/2)**10/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*ta
n(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f
*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c*
*7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*
c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f)
- 138138*B*a**3*tan(e/2 + f*x/2)**9/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 +
702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9
- 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)
**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/
2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) + 18018*B*a**3*tan(e
/2 + f*x/2)**8/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2
+ f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(
e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*
tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*
tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) - 300300*B*a**3*tan(e/2 + f*x/2)**7/(9009*
c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 257657
4*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 154
59444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 -
6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 +
117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) + 32604*B*a**3*tan(e/2 + f*x/2)**6/(9009*c**7*f*tan(e/2 + f*x/2
)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x
/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 +
f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/
2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2
+ f*x/2) - 9009*c**7*f) - 231660*B*a**3*tan(e/2 + f*x/2)**5/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f
*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**
7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444
*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 25765
74*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7
*f) + 572*B*a**3*tan(e/2 + f*x/2)**4/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 +
702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9
- 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)
**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/
2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) - 61490*B*a**3*tan(e
/2 + f*x/2)**3/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2
+ f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(
e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*
tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*
tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) - 2886*B*a**3*tan(e/2 + f*x/2)**2/(9009*c*
*7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*
c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459
444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6
441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 1
17117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f) - 2522*B*a**3*tan(e/2 + f*x/2)/(9009*c**7*f*tan(e/2 + f*x/2)**13
- 117117*c**7*f*tan(e/2 + f*x/2)**12 + 702702*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**1
0 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 11594583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2
)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 + 11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*
x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3 - 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x
/2) - 9009*c**7*f) + 194*B*a**3/(9009*c**7*f*tan(e/2 + f*x/2)**13 - 117117*c**7*f*tan(e/2 + f*x/2)**12 + 70270
2*c**7*f*tan(e/2 + f*x/2)**11 - 2576574*c**7*f*tan(e/2 + f*x/2)**10 + 6441435*c**7*f*tan(e/2 + f*x/2)**9 - 115
94583*c**7*f*tan(e/2 + f*x/2)**8 + 15459444*c**7*f*tan(e/2 + f*x/2)**7 - 15459444*c**7*f*tan(e/2 + f*x/2)**6 +
11594583*c**7*f*tan(e/2 + f*x/2)**5 - 6441435*c**7*f*tan(e/2 + f*x/2)**4 + 2576574*c**7*f*tan(e/2 + f*x/2)**3
- 702702*c**7*f*tan(e/2 + f*x/2)**2 + 117117*c**7*f*tan(e/2 + f*x/2) - 9009*c**7*f), Ne(f, 0)), (x*(A + B*sin
(e))*(a*sin(e) + a)**3/(-c*sin(e) + c)**7, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4078 vs. \(2 (152) = 304\).

Time = 0.39 (sec) , antiderivative size = 4078, normalized size of antiderivative = 26.14 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x, algorithm="maxima")

[Out]

-2/45045*(6*A*a^3*(4771*sin(f*x + e)/(cos(f*x + e) + 1) - 28626*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 74932*si
n(f*x + e)^3/(cos(f*x + e) + 1)^3 - 187330*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 - 2072
07*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/(co
s(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) + 6*B*a^3*(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) + 15*A*a^3*(3796*sin(f*x + e)/(cos(f*x + e) + 1) - 22776*sin(f*x + e
)^2/(cos(f*x + e) + 1)^2 + 77506*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 193765*sin(f*x + e)^4/(cos(f*x + e) + 1
)^4 + 339768*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 453024*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 444444*sin(f*x
+ e)^7/(cos(f*x + e) + 1)^7 - 333333*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 180180*sin(f*x + e)^9/(cos(f*x + e
) + 1)^9 - 72072*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 18018*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 3003*si
n(f*x + e)^12/(cos(f*x + e) + 1)^12 - 523)/(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) - 105*A*a^3*(
611*sin(f*x + e)/(cos(f*x + e) + 1) - 2379*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 8723*sin(f*x + e)^3/(cos(f*x
+ e) + 1)^3 - 18590*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33462*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 40326*si
n(f*x + e)^6/(cos(f*x + e) + 1)^6 + 40326*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 27027*sin(f*x + e)^8/(cos(f*x
+ e) + 1)^8 + 15015*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 4719*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 1287*si
n(f*x + e)^11/(cos(f*x + e) + 1)^11 - 47)/(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/(c
os(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) - 35*B*a^3*(61
1*sin(f*x + e)/(cos(f*x + e) + 1) - 2379*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 8723*sin(f*x + e)^3/(cos(f*x +
e) + 1)^3 - 18590*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33462*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 40326*sin(
f*x + e)^6/(cos(f*x + e) + 1)^6 + 40326*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 27027*sin(f*x + e)^8/(cos(f*x +
e) + 1)^8 + 15015*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 4719*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 1287*sin(
f*x + e)^11/(cos(f*x + e) + 1)^11 - 47)/(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*si
n(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)^1
1 + 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) + 8*B*a^3*(559*s
in(f*x + e)/(cos(f*x + e) + 1) - 3354*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 12298*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3 - 30745*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 37323*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 49764*sin(f*
x + e)^6/(cos(f*x + e) + 1)^6 + 24024*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 18018*sin(f*x + e)^8/(cos(f*x + e)
+ 1)^8 - 43)/(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) - 462*A*a^3*(13*sin(f*x + e)/(cos(f*x + e)
+ 1) - 78*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 286*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 520*sin(f*x + e)^4/
(cos(f*x + e) + 1)^4 + 936*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 858*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 858
*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 351*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 195*sin(f*x + e)^9/(cos(f*x +
e) + 1)^9 - 1)/(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 - 2
86*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) - 1386*B*a^3*(13*sin(f*x + e)/(cos(f*x +
e) + 1) - 78*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 286*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 520*sin(f*x + e)
^4/(cos(f*x + e) + 1)^4 + 936*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 858*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 +
858*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 351*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 195*sin(f*x + e)^9/(cos(f*
x + e) + 1)^9 - 1)/(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 + 28
6*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))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (152) = 304\).

Time = 0.43 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.70 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=-\frac {2 \, {\left (9009 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 27027 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 9009 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 153153 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} + 3003 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 297297 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 69069 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 648648 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 9009 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 738738 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 150150 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 857142 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 16302 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 548262 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 115830 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 367653 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 286 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 112827 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 30745 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 45513 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1443 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3081 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1261 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 930 \, A a^{3} - 97 \, B a^{3}\right )}}{9009 \, c^{7} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{13}} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^7,x, algorithm="giac")

[Out]

-2/9009*(9009*A*a^3*tan(1/2*f*x + 1/2*e)^12 - 27027*A*a^3*tan(1/2*f*x + 1/2*e)^11 + 9009*B*a^3*tan(1/2*f*x + 1
/2*e)^11 + 153153*A*a^3*tan(1/2*f*x + 1/2*e)^10 + 3003*B*a^3*tan(1/2*f*x + 1/2*e)^10 - 297297*A*a^3*tan(1/2*f*
x + 1/2*e)^9 + 69069*B*a^3*tan(1/2*f*x + 1/2*e)^9 + 648648*A*a^3*tan(1/2*f*x + 1/2*e)^8 - 9009*B*a^3*tan(1/2*f
*x + 1/2*e)^8 - 738738*A*a^3*tan(1/2*f*x + 1/2*e)^7 + 150150*B*a^3*tan(1/2*f*x + 1/2*e)^7 + 857142*A*a^3*tan(1
/2*f*x + 1/2*e)^6 - 16302*B*a^3*tan(1/2*f*x + 1/2*e)^6 - 548262*A*a^3*tan(1/2*f*x + 1/2*e)^5 + 115830*B*a^3*ta
n(1/2*f*x + 1/2*e)^5 + 367653*A*a^3*tan(1/2*f*x + 1/2*e)^4 - 286*B*a^3*tan(1/2*f*x + 1/2*e)^4 - 112827*A*a^3*t
an(1/2*f*x + 1/2*e)^3 + 30745*B*a^3*tan(1/2*f*x + 1/2*e)^3 + 45513*A*a^3*tan(1/2*f*x + 1/2*e)^2 + 1443*B*a^3*t
an(1/2*f*x + 1/2*e)^2 - 3081*A*a^3*tan(1/2*f*x + 1/2*e) + 1261*B*a^3*tan(1/2*f*x + 1/2*e) + 930*A*a^3 - 97*B*a
^3)/(c^7*f*(tan(1/2*f*x + 1/2*e) - 1)^13)

Mupad [B] (verification not implemented)

Time = 14.36 (sec) , antiderivative size = 500, normalized size of antiderivative = 3.21 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {2363\,B\,a^3}{32}-\frac {279183\,A\,a^3}{16}+\frac {220269\,A\,a^3\,\cos \left (2\,e+2\,f\,x\right )}{16}-\frac {46095\,A\,a^3\,\cos \left (3\,e+3\,f\,x\right )}{16}-\frac {20829\,A\,a^3\,\cos \left (4\,e+4\,f\,x\right )}{16}+\frac {2811\,A\,a^3\,\cos \left (5\,e+5\,f\,x\right )}{16}+\frac {231\,A\,a^3\,\cos \left (6\,e+6\,f\,x\right )}{16}-\frac {8995\,B\,a^3\,\cos \left (2\,e+2\,f\,x\right )}{64}+\frac {497\,B\,a^3\,\cos \left (3\,e+3\,f\,x\right )}{16}+\frac {3725\,B\,a^3\,\cos \left (4\,e+4\,f\,x\right )}{32}-\frac {361\,B\,a^3\,\cos \left (5\,e+5\,f\,x\right )}{16}-\frac {77\,B\,a^3\,\cos \left (6\,e+6\,f\,x\right )}{64}-\frac {19305\,A\,a^3\,\sin \left (2\,e+2\,f\,x\right )}{4}-\frac {81081\,A\,a^3\,\sin \left (3\,e+3\,f\,x\right )}{16}+\frac {15015\,A\,a^3\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {3237\,A\,a^3\,\sin \left (5\,e+5\,f\,x\right )}{16}-\frac {117\,A\,a^3\,\sin \left (6\,e+6\,f\,x\right )}{8}+\frac {77649\,B\,a^3\,\sin \left (2\,e+2\,f\,x\right )}{64}+\frac {27027\,B\,a^3\,\sin \left (3\,e+3\,f\,x\right )}{32}-\frac {1001\,B\,a^3\,\sin \left (4\,e+4\,f\,x\right )}{8}-\frac {559\,B\,a^3\,\sin \left (5\,e+5\,f\,x\right )}{32}+\frac {117\,B\,a^3\,\sin \left (6\,e+6\,f\,x\right )}{64}+\frac {26979\,A\,a^3\,\cos \left (e+f\,x\right )}{4}+40\,B\,a^3\,\cos \left (e+f\,x\right )+\frac {173745\,A\,a^3\,\sin \left (e+f\,x\right )}{8}-\frac {80223\,B\,a^3\,\sin \left (e+f\,x\right )}{16}\right )}{9009\,c^7\,f\,\left (\frac {1287\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}-\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{64}-\frac {429\,\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}{16}+\frac {715\,\sqrt {2}\,\cos \left (\frac {5\,e}{2}+\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{64}-\frac {143\,\sqrt {2}\,\cos \left (\frac {7\,e}{2}-\frac {\pi }{4}+\frac {7\,f\,x}{2}\right )}{32}-\frac {39\,\sqrt {2}\,\cos \left (\frac {9\,e}{2}+\frac {\pi }{4}+\frac {9\,f\,x}{2}\right )}{32}+\frac {13\,\sqrt {2}\,\cos \left (\frac {11\,e}{2}-\frac {\pi }{4}+\frac {11\,f\,x}{2}\right )}{64}+\frac {\sqrt {2}\,\cos \left (\frac {13\,e}{2}+\frac {\pi }{4}+\frac {13\,f\,x}{2}\right )}{64}\right )} \]

[In]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^7,x)

[Out]

(2*cos(e/2 + (f*x)/2)*((2363*B*a^3)/32 - (279183*A*a^3)/16 + (220269*A*a^3*cos(2*e + 2*f*x))/16 - (46095*A*a^3
*cos(3*e + 3*f*x))/16 - (20829*A*a^3*cos(4*e + 4*f*x))/16 + (2811*A*a^3*cos(5*e + 5*f*x))/16 + (231*A*a^3*cos(
6*e + 6*f*x))/16 - (8995*B*a^3*cos(2*e + 2*f*x))/64 + (497*B*a^3*cos(3*e + 3*f*x))/16 + (3725*B*a^3*cos(4*e +
4*f*x))/32 - (361*B*a^3*cos(5*e + 5*f*x))/16 - (77*B*a^3*cos(6*e + 6*f*x))/64 - (19305*A*a^3*sin(2*e + 2*f*x))
/4 - (81081*A*a^3*sin(3*e + 3*f*x))/16 + (15015*A*a^3*sin(4*e + 4*f*x))/16 + (3237*A*a^3*sin(5*e + 5*f*x))/16
- (117*A*a^3*sin(6*e + 6*f*x))/8 + (77649*B*a^3*sin(2*e + 2*f*x))/64 + (27027*B*a^3*sin(3*e + 3*f*x))/32 - (10
01*B*a^3*sin(4*e + 4*f*x))/8 - (559*B*a^3*sin(5*e + 5*f*x))/32 + (117*B*a^3*sin(6*e + 6*f*x))/64 + (26979*A*a^
3*cos(e + f*x))/4 + 40*B*a^3*cos(e + f*x) + (173745*A*a^3*sin(e + f*x))/8 - (80223*B*a^3*sin(e + f*x))/16))/(9
009*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 + (7
15*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 - (39*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))