\(\int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx\) [51]

Optimal result

Integrand size = 36, antiderivative size = 197 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac {a^3 (4 A-11 B) c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {a^3 (4 A-11 B) c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac {2 a^3 (4 A-11 B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac {2 a^3 (4 A-11 B) \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7} \]

[Out]

1/15*a^3*(A+B)*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^11+1/195*a^3*(4*A-11*B)*c^2*cos(f*x+e)^7/f/(c-c*sin(f*x+e))
^10+1/715*a^3*(4*A-11*B)*c*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^9+2/6435*a^3*(4*A-11*B)*cos(f*x+e)^7/f/(c-c*sin(f*x
+e))^8+2/45045*a^3*(4*A-11*B)*cos(f*x+e)^7/c/f/(c-c*sin(f*x+e))^7

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3046, 2938, 2751, 2750} \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx=\frac {a^3 c^3 (A+B) \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac {a^3 c^2 (4 A-11 B) \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {2 a^3 (4 A-11 B) \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7}+\frac {2 a^3 (4 A-11 B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac {a^3 c (4 A-11 B) \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9} \]

[In]

Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^8,x]

[Out]

(a^3*(A + B)*c^3*Cos[e + f*x]^7)/(15*f*(c - c*Sin[e + f*x])^11) + (a^3*(4*A - 11*B)*c^2*Cos[e + f*x]^7)/(195*f
*(c - c*Sin[e + f*x])^10) + (a^3*(4*A - 11*B)*c*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x])^9) + (2*a^3*(4*A -
11*B)*Cos[e + f*x]^7)/(6435*f*(c - c*Sin[e + f*x])^8) + (2*a^3*(4*A - 11*B)*Cos[e + f*x]^7)/(45045*c*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))^{11}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac {1}{15} \left (a^3 (4 A-11 B) c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{10}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac {a^3 (4 A-11 B) c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {1}{65} \left (a^3 (4 A-11 B) c\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)}{15 f (c-c \sin (e+f x))^{11}}+\frac {a^3 (4 A-11 B) c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {a^3 (4 A-11 B) c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac {1}{715} \left (2 a^3 (4 A-11 B)\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)}{15 f (c-c \sin (e+f x))^{11}}+\frac {a^3 (4 A-11 B) c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {a^3 (4 A-11 B) c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac {2 a^3 (4 A-11 B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac {\left (2 a^3 (4 A-11 B)\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx}{6435 c} \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{15 f (c-c \sin (e+f x))^{11}}+\frac {a^3 (4 A-11 B) c^2 \cos ^7(e+f x)}{195 f (c-c \sin (e+f x))^{10}}+\frac {a^3 (4 A-11 B) c \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^9}+\frac {2 a^3 (4 A-11 B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^8}+\frac {2 a^3 (4 A-11 B) \cos ^7(e+f x)}{45045 c f (c-c \sin (e+f x))^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 14.87 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.85 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx=\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \left (6435 (72 A+47 B) \cos \left (\frac {1}{2} (e+f x)\right )-10010 (26 A+23 B) \cos \left (\frac {3}{2} (e+f x)\right )-72072 A \cos \left (\frac {5}{2} (e+f x)\right )-117117 B \cos \left (\frac {5}{2} (e+f x)\right )+5460 A \cos \left (\frac {7}{2} (e+f x)\right )+30030 B \cos \left (\frac {7}{2} (e+f x)\right )-420 A \cos \left (\frac {11}{2} (e+f x)\right )+1155 B \cos \left (\frac {11}{2} (e+f x)\right )+4 A \cos \left (\frac {15}{2} (e+f x)\right )-11 B \cos \left (\frac {15}{2} (e+f x)\right )+437580 A \sin \left (\frac {1}{2} (e+f x)\right )+373230 B \sin \left (\frac {1}{2} (e+f x)\right )+240240 A \sin \left (\frac {3}{2} (e+f x)\right )+285285 B \sin \left (\frac {3}{2} (e+f x)\right )-60060 A \sin \left (\frac {5}{2} (e+f x)\right )-150150 B \sin \left (\frac {5}{2} (e+f x)\right )-45045 B \sin \left (\frac {7}{2} (e+f x)\right )-1820 A \sin \left (\frac {9}{2} (e+f x)\right )+5005 B \sin \left (\frac {9}{2} (e+f x)\right )+60 A \sin \left (\frac {13}{2} (e+f x)\right )-165 B \sin \left (\frac {13}{2} (e+f x)\right )\right )}{1441440 c^8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (-1+\sin (e+f x))^8} \]

[In]

Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^8,x]

[Out]

(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(6435*(72*A + 47*B)*Cos[(e + f*x)/2] - 10010*(
26*A + 23*B)*Cos[(3*(e + f*x))/2] - 72072*A*Cos[(5*(e + f*x))/2] - 117117*B*Cos[(5*(e + f*x))/2] + 5460*A*Cos[
(7*(e + f*x))/2] + 30030*B*Cos[(7*(e + f*x))/2] - 420*A*Cos[(11*(e + f*x))/2] + 1155*B*Cos[(11*(e + f*x))/2] +
4*A*Cos[(15*(e + f*x))/2] - 11*B*Cos[(15*(e + f*x))/2] + 437580*A*Sin[(e + f*x)/2] + 373230*B*Sin[(e + f*x)/2
] + 240240*A*Sin[(3*(e + f*x))/2] + 285285*B*Sin[(3*(e + f*x))/2] - 60060*A*Sin[(5*(e + f*x))/2] - 150150*B*Si
n[(5*(e + f*x))/2] - 45045*B*Sin[(7*(e + f*x))/2] - 1820*A*Sin[(9*(e + f*x))/2] + 5005*B*Sin[(9*(e + f*x))/2]
+ 60*A*Sin[(13*(e + f*x))/2] - 165*B*Sin[(13*(e + f*x))/2]))/(1441440*c^8*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/
2])^6*(-1 + Sin[e + f*x])^8)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.00 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.51

method	result	size
risch	\(\frac {4 i a^{3} \left (-4 i A -5460 i A \,{\mathrm e}^{4 i \left (f x +e \right )}+45045 B \,{\mathrm e}^{11 i \left (f x +e \right )}-302445 i B \,{\mathrm e}^{8 i \left (f x +e \right )}-240240 A \,{\mathrm e}^{9 i \left (f x +e \right )}+420 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-285285 B \,{\mathrm e}^{9 i \left (f x +e \right )}+11 i B +437580 A \,{\mathrm e}^{7 i \left (f x +e \right )}+72072 i A \,{\mathrm e}^{10 i \left (f x +e \right )}+373230 B \,{\mathrm e}^{7 i \left (f x +e \right )}+117117 i B \,{\mathrm e}^{10 i \left (f x +e \right )}-60060 A \,{\mathrm e}^{5 i \left (f x +e \right )}+230230 i B \,{\mathrm e}^{6 i \left (f x +e \right )}-150150 B \,{\mathrm e}^{5 i \left (f x +e \right )}-1155 i B \,{\mathrm e}^{2 i \left (f x +e \right )}-1820 A \,{\mathrm e}^{3 i \left (f x +e \right )}-463320 i A \,{\mathrm e}^{8 i \left (f x +e \right )}+5005 B \,{\mathrm e}^{3 i \left (f x +e \right )}-30030 i B \,{\mathrm e}^{4 i \left (f x +e \right )}+60 A \,{\mathrm e}^{i \left (f x +e \right )}+260260 i A \,{\mathrm e}^{6 i \left (f x +e \right )}-165 B \,{\mathrm e}^{i \left (f x +e \right )}\right )}{45045 f \,c^{8} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{15}}\)	\(298\)
parallelrisch	\(-\frac {2 \left (A \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-4 A +B \right ) \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {\left (71 A -B \right ) \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+10 \left (-6 A +B \right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {\left (741 A -34 B \right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {\left (-680 A +97 B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {\left (2195 A -123 B \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}+\frac {4 \left (-512 A +71 B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}+\frac {\left (2203 A -116 B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9}+\frac {\left (-404 A +61 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {\left (\frac {2263 A}{11}-7 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+\frac {2 \left (-\frac {950 A}{11}+17 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9}+\frac {\left (2 B +\frac {2527 A}{11}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{39}+\frac {\left (-\frac {1240 A}{11}+37 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{273}+\frac {4243 A}{45045}-\frac {37 B}{4095}\right ) a^{3}}{f \,c^{8} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{15}}\)	\(298\)
derivativedivides	\(\frac {2 a^{3} \left (-\frac {188 A +38 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {58816 A +40000 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {52736 A +49664 B}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {4536 A +1836 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {24320 A +23808 B}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {7168 A +7168 B}{14 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{14}}-\frac {94144 A +78144 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {84112 A +63856 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {13824 A +6936 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {81344 A +72512 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {1104 A +336 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {32288 A +19176 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {20 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {1024 A +1024 B}{15 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{15}}\right )}{f \,c^{8}}\)	\(337\)
default	\(\frac {2 a^{3} \left (-\frac {188 A +38 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {58816 A +40000 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {52736 A +49664 B}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {4536 A +1836 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {24320 A +23808 B}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {7168 A +7168 B}{14 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{14}}-\frac {94144 A +78144 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {84112 A +63856 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {13824 A +6936 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {81344 A +72512 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {1104 A +336 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {32288 A +19176 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {20 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {1024 A +1024 B}{15 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{15}}\right )}{f \,c^{8}}\)	\(337\)

[In]

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^8,x,method=_RETURNVERBOSE)

[Out]

4/45045*I*a^3*(-4*I*A-5460*I*A*exp(4*I*(f*x+e))+45045*B*exp(11*I*(f*x+e))-302445*I*B*exp(8*I*(f*x+e))-240240*A
*exp(9*I*(f*x+e))+420*I*A*exp(2*I*(f*x+e))-285285*B*exp(9*I*(f*x+e))+11*I*B+437580*A*exp(7*I*(f*x+e))+72072*I*
A*exp(10*I*(f*x+e))+373230*B*exp(7*I*(f*x+e))+117117*I*B*exp(10*I*(f*x+e))-60060*A*exp(5*I*(f*x+e))+230230*I*B
*exp(6*I*(f*x+e))-150150*B*exp(5*I*(f*x+e))-1155*I*B*exp(2*I*(f*x+e))-1820*A*exp(3*I*(f*x+e))-463320*I*A*exp(8
*I*(f*x+e))+5005*B*exp(3*I*(f*x+e))-30030*I*B*exp(4*I*(f*x+e))+60*A*exp(I*(f*x+e))+260260*I*A*exp(6*I*(f*x+e))
-165*B*exp(I*(f*x+e)))/f/c^8/(exp(I*(f*x+e))-I)^15

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (192) = 384\).

Time = 0.29 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.75 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx=\frac {2 \, {\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{8} + 16 \, {\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{7} - 49 \, {\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{6} - 168 \, {\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} + 105 \, {\left (7 \, A + 88 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 231 \, {\left (31 \, A + 61 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 924 \, {\left (22 \, A + 37 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 12012 \, {\left (A + B\right )} a^{3} \cos \left (f x + e\right ) + 24024 \, {\left (A + B\right )} a^{3} - {\left (2 \, {\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{7} - 14 \, {\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{6} - 63 \, {\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} + 105 \, {\left (4 \, A - 11 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + 1155 \, {\left (A + 7 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 2772 \, {\left (3 \, A + 8 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 12012 \, {\left (A + B\right )} a^{3} \cos \left (f x + e\right ) - 24024 \, {\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )}{45045 \, {\left (c^{8} f \cos \left (f x + e\right )^{8} - 7 \, c^{8} f \cos \left (f x + e\right )^{7} - 32 \, c^{8} f \cos \left (f x + e\right )^{6} + 56 \, c^{8} f \cos \left (f x + e\right )^{5} + 160 \, c^{8} f \cos \left (f x + e\right )^{4} - 112 \, c^{8} f \cos \left (f x + e\right )^{3} - 256 \, c^{8} f \cos \left (f x + e\right )^{2} + 64 \, c^{8} f \cos \left (f x + e\right ) + 128 \, c^{8} f + {\left (c^{8} f \cos \left (f x + e\right )^{7} + 8 \, c^{8} f \cos \left (f x + e\right )^{6} - 24 \, c^{8} f \cos \left (f x + e\right )^{5} - 80 \, c^{8} f \cos \left (f x + e\right )^{4} + 80 \, c^{8} f \cos \left (f x + e\right )^{3} + 192 \, c^{8} f \cos \left (f x + e\right )^{2} - 64 \, c^{8} f \cos \left (f x + e\right ) - 128 \, c^{8} 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))^8,x, algorithm="fricas")

[Out]

1/45045*(2*(4*A - 11*B)*a^3*cos(f*x + e)^8 + 16*(4*A - 11*B)*a^3*cos(f*x + e)^7 - 49*(4*A - 11*B)*a^3*cos(f*x
+ e)^6 - 168*(4*A - 11*B)*a^3*cos(f*x + e)^5 + 105*(7*A + 88*B)*a^3*cos(f*x + e)^4 - 231*(31*A + 61*B)*a^3*cos
(f*x + e)^3 - 924*(22*A + 37*B)*a^3*cos(f*x + e)^2 + 12012*(A + B)*a^3*cos(f*x + e) + 24024*(A + B)*a^3 - (2*(
4*A - 11*B)*a^3*cos(f*x + e)^7 - 14*(4*A - 11*B)*a^3*cos(f*x + e)^6 - 63*(4*A - 11*B)*a^3*cos(f*x + e)^5 + 105
*(4*A - 11*B)*a^3*cos(f*x + e)^4 + 1155*(A + 7*B)*a^3*cos(f*x + e)^3 + 2772*(3*A + 8*B)*a^3*cos(f*x + e)^2 - 1
2012*(A + B)*a^3*cos(f*x + e) - 24024*(A + B)*a^3)*sin(f*x + e))/(c^8*f*cos(f*x + e)^8 - 7*c^8*f*cos(f*x + e)^
7 - 32*c^8*f*cos(f*x + e)^6 + 56*c^8*f*cos(f*x + e)^5 + 160*c^8*f*cos(f*x + e)^4 - 112*c^8*f*cos(f*x + e)^3 -
256*c^8*f*cos(f*x + e)^2 + 64*c^8*f*cos(f*x + e) + 128*c^8*f + (c^8*f*cos(f*x + e)^7 + 8*c^8*f*cos(f*x + e)^6
- 24*c^8*f*cos(f*x + e)^5 - 80*c^8*f*cos(f*x + e)^4 + 80*c^8*f*cos(f*x + e)^3 + 192*c^8*f*cos(f*x + e)^2 - 64*
c^8*f*cos(f*x + e) - 128*c^8*f)*sin(f*x + e))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8821 vs. \(2 (178) = 356\).

Time = 162.97 (sec) , antiderivative size = 8821, normalized size of antiderivative = 44.78 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**8,x)

[Out]

Piecewise((-90090*A*a**3*tan(e/2 + f*x/2)**14/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x
/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2
+ f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*
f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 13527013
5*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 472
9725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) + 360360*A*a**3*tan(e/2 + f*x
/2)**13/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x
/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e
/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8
*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 6148642
5*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 6756
75*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 2132130*A*a**3*tan(e/2 + f*x/2)**12/(45045*c**8*f*tan(e/2 + f*x/2
)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f
*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*ta
n(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c*
*8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 204954
75*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c*
*8*f) + 5405400*A*a**3*tan(e/2 + f*x/2)**11/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2
)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 +
f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*
tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*
c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 47297
25*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 13351338*A*a**3*tan(e/2 + f*x
/2)**10/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x
/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e
/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8
*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 6148642
5*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 6756
75*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) + 20420400*A*a**3*tan(e/2 + f*x/2)**9/(45045*c**8*f*tan(e/2 + f*x/2
)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f
*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*ta
n(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c*
*8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 204954
75*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c*
*8*f) - 28249650*A*a**3*tan(e/2 + f*x/2)**8/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2
)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 +
f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*
tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*
c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 47297
25*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) + 26357760*A*a**3*tan(e/2 + f*x
/2)**7/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/
2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/
2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*
f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425
*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 67567
5*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 22052030*A*a**3*tan(e/2 + f*x/2)**6/(45045*c**8*f*tan(e/2 + f*x/2)
**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*
x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan
(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**
8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 2049547
5*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**
8*f) + 12132120*A*a**3*tan(e/2 + f*x/2)**5/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)
**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 +
f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*t
an(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c
**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 472972
5*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 6177990*A*a**3*tan(e/2 + f*x/2
)**4/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)
**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2
+ f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*
tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c
**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*
c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) + 1729000*A*a**3*tan(e/2 + f*x/2)**3/(45045*c**8*f*tan(e/2 + f*x/2)**1
5 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2
)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/
2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f
*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c
**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f
) - 530670*A*a**3*tan(e/2 + f*x/2)**2/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14
+ 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2
)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/
2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f
*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**
8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) + 37200*A*a**3*tan(e/2 + f*x/2)/(4504
5*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 204
95475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**
10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 +
f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(
e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan
(e/2 + f*x/2) - 45045*c**8*f) - 8486*A*a**3/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2
)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 +
f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*
tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*
c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 47297
25*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 90090*B*a**3*tan(e/2 + f*x/2)
**13/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)
**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2
+ f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*
tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c
**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*
c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) + 30030*B*a**3*tan(e/2 + f*x/2)**12/(45045*c**8*f*tan(e/2 + f*x/2)**15
- 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)
**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2
+ f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*
tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c*
*8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f)
- 900900*B*a**3*tan(e/2 + f*x/2)**11/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14
+ 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2
)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/
2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f
*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**
8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) + 612612*B*a**3*tan(e/2 + f*x/2)**10/
(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13
- 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x
/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e
/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f
*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*
f*tan(e/2 + f*x/2) - 45045*c**8*f) - 2912910*B*a**3*tan(e/2 + f*x/2)**9/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 6
75675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12
+ 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f
*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(
e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f
*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) + 1
583010*B*a**3*tan(e/2 + f*x/2)**8/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 47
29725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**1
1 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 +
f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan
(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*
tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 3655080*B*a**3*tan(e/2 + f*x/2)**7/(450
45*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20
495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)*
*10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 +
f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan
(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*ta
n(e/2 + f*x/2) - 45045*c**8*f) + 1161160*B*a**3*tan(e/2 + f*x/2)**6/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 67567
5*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 6
1486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2
)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2
+ f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan
(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 18318
30*B*a**3*tan(e/2 + f*x/2)**5/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 472972
5*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 -
135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/
2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2
+ f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(
e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) + 210210*B*a**3*tan(e/2 + f*x/2)**4/(45045*c*
*8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 2049547
5*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 +
225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/
2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2
+ f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2
+ f*x/2) - 45045*c**8*f) - 340340*B*a**3*tan(e/2 + f*x/2)**3/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8
*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 6148642
5*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 -
289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/
2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 +
f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 4620*B*a**3
*tan(e/2 + f*x/2)**2/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*
tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135
*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 2
89864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)
**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x
/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f) - 12210*B*a**3*tan(e/2 + f*x/2)/(45045*c**8*f*tan(e/2
+ f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c**8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(
e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 135270135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c*
*8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 22545
0225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 +
20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2 + f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 4
5045*c**8*f) + 814*B*a**3/(45045*c**8*f*tan(e/2 + f*x/2)**15 - 675675*c**8*f*tan(e/2 + f*x/2)**14 + 4729725*c*
*8*f*tan(e/2 + f*x/2)**13 - 20495475*c**8*f*tan(e/2 + f*x/2)**12 + 61486425*c**8*f*tan(e/2 + f*x/2)**11 - 1352
70135*c**8*f*tan(e/2 + f*x/2)**10 + 225450225*c**8*f*tan(e/2 + f*x/2)**9 - 289864575*c**8*f*tan(e/2 + f*x/2)**
8 + 289864575*c**8*f*tan(e/2 + f*x/2)**7 - 225450225*c**8*f*tan(e/2 + f*x/2)**6 + 135270135*c**8*f*tan(e/2 + f
*x/2)**5 - 61486425*c**8*f*tan(e/2 + f*x/2)**4 + 20495475*c**8*f*tan(e/2 + f*x/2)**3 - 4729725*c**8*f*tan(e/2
+ f*x/2)**2 + 675675*c**8*f*tan(e/2 + f*x/2) - 45045*c**8*f), Ne(f, 0)), (x*(A + B*sin(e))*(a*sin(e) + a)**3/(
-c*sin(e) + c)**8, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4765 vs. \(2 (192) = 384\).

Time = 0.41 (sec) , antiderivative size = 4765, normalized size of antiderivative = 24.19 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^8,x, algorithm="maxima")

[Out]

2/45045*(3*A*a^3*(17715*sin(f*x + e)/(cos(f*x + e) + 1) - 78960*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 342160*s
in(f*x + e)^3/(cos(f*x + e) + 1)^3 - 891345*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1960959*sin(f*x + e)^5/(cos(
f*x + e) + 1)^5 - 3043040*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3912480*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 -
3687255*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 2867865*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1585584*sin(f*x +
e)^10/(cos(f*x + e) + 1)^10 + 720720*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 195195*sin(f*x + e)^12/(cos(f*x +
e) + 1)^12 + 45045*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 1181)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1
) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f
*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f
*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 -
5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(
f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(c
os(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15
) + B*a^3*(17715*sin(f*x + e)/(cos(f*x + e) + 1) - 78960*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 342160*sin(f*x
+ e)^3/(cos(f*x + e) + 1)^3 - 891345*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1960959*sin(f*x + e)^5/(cos(f*x + e
) + 1)^5 - 3043040*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3912480*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3687255
*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 2867865*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1585584*sin(f*x + e)^10/(
cos(f*x + e) + 1)^10 + 720720*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 195195*sin(f*x + e)^12/(cos(f*x + e) + 1
)^12 + 45045*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 1181)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105
*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)
^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e)
+ 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c
^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e
)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(cos(f*x
+ e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15) - 7*A
*a^3*(7845*sin(f*x + e)/(cos(f*x + e) + 1) - 54915*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 222950*sin(f*x + e)^3
/(cos(f*x + e) + 1)^3 - 668850*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1444443*sin(f*x + e)^5/(cos(f*x + e) + 1)
^5 - 2407405*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3063060*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3063060*sin(f
*x + e)^8/(cos(f*x + e) + 1)^8 + 2357355*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1414413*sin(f*x + e)^10/(cos(f*
x + e) + 1)^10 + 630630*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 210210*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 +
45045*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 6435*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - 952)/(c^8 - 15*c^8
*sin(f*x + e)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f
*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 +
5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*
x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f
*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^
12 - 105*c^8*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*
x + e)^15/(cos(f*x + e) + 1)^15) - 12*A*a^3*(1740*sin(f*x + e)/(cos(f*x + e) + 1) - 12180*sin(f*x + e)^2/(cos(
f*x + e) + 1)^2 + 37765*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 113295*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 204
204*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 340340*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 373230*sin(f*x + e)^7/(
cos(f*x + e) + 1)^7 - 373230*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 240240*sin(f*x + e)^9/(cos(f*x + e) + 1)^9
- 144144*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 45045*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 15015*sin(f*x +
e)^12/(cos(f*x + e) + 1)^12 - 116)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(co
s(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4
- 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(
f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(
f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)
^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*s
in(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15) - 12*B*a^3*(1740*sin(f*x + e
)/(cos(f*x + e) + 1) - 12180*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 37765*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 -
113295*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 204204*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 340340*sin(f*x + e)
^6/(cos(f*x + e) + 1)^6 + 373230*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 373230*sin(f*x + e)^8/(cos(f*x + e) + 1
)^8 + 240240*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 144144*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 45045*sin(f*
x + e)^11/(cos(f*x + e) + 1)^11 - 15015*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 116)/(c^8 - 15*c^8*sin(f*x + e
)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*si
n(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(co
s(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)
^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8
*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(c
os(f*x + e) + 1)^15) + 6*A*a^3*(675*sin(f*x + e)/(cos(f*x + e) + 1) - 4725*sin(f*x + e)^2/(cos(f*x + e) + 1)^2
+ 20475*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 46410*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 102102*sin(f*x + e)
^5/(cos(f*x + e) + 1)^5 - 130130*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 167310*sin(f*x + e)^7/(cos(f*x + e) + 1
)^7 - 122265*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 95095*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 33033*sin(f*x +
e)^10/(cos(f*x + e) + 1)^10 + 15015*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 45)/(c^8 - 15*c^8*sin(f*x + e)/(c
os(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 +
1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*
x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*
x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10
- 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin
(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f
*x + e) + 1)^15) + 18*B*a^3*(675*sin(f*x + e)/(cos(f*x + e) + 1) - 4725*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 +
20475*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 46410*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 102102*sin(f*x + e)^5/
(cos(f*x + e) + 1)^5 - 130130*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 167310*sin(f*x + e)^7/(cos(f*x + e) + 1)^7
- 122265*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 95095*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 33033*sin(f*x + e)
^10/(cos(f*x + e) + 1)^10 + 15015*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 45)/(c^8 - 15*c^8*sin(f*x + e)/(cos(
f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 13
65*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x +
e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x +
e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1
365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*
x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x
+ e) + 1)^15) - 48*B*a^3*(60*sin(f*x + e)/(cos(f*x + e) + 1) - 420*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1820*
sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 5460*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 9009*sin(f*x + e)^5/(cos(f*x
+ e) + 1)^5 - 15015*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 12870*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 12870*si
n(f*x + e)^8/(cos(f*x + e) + 1)^8 + 5005*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 3003*sin(f*x + e)^10/(cos(f*x +
e) + 1)^10 - 4)/(c^8 - 15*c^8*sin(f*x + e)/(cos(f*x + e) + 1) + 105*c^8*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 -
455*c^8*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 1365*c^8*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 3003*c^8*sin(f*x
+ e)^5/(cos(f*x + e) + 1)^5 + 5005*c^8*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 6435*c^8*sin(f*x + e)^7/(cos(f*x
+ e) + 1)^7 + 6435*c^8*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 5005*c^8*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 3
003*c^8*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 1365*c^8*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 455*c^8*sin(f
*x + e)^12/(cos(f*x + e) + 1)^12 - 105*c^8*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 15*c^8*sin(f*x + e)^14/(cos
(f*x + e) + 1)^14 - c^8*sin(f*x + e)^15/(cos(f*x + e) + 1)^15))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (192) = 384\).

Time = 0.45 (sec) , antiderivative size = 489, normalized size of antiderivative = 2.48 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx=-\frac {2 \, {\left (45045 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{14} - 180180 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} + 45045 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{13} + 1066065 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 15015 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 2702700 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 450450 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 6675669 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 306306 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 10210200 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 1456455 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 14124825 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 791505 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 13178880 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 1827540 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 11026015 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 580580 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 6066060 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 915915 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3088995 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 105105 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 864500 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 170170 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 265335 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2310 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 18600 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6105 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4243 \, A a^{3} - 407 \, B a^{3}\right )}}{45045 \, c^{8} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{15}} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^8,x, algorithm="giac")

[Out]

-2/45045*(45045*A*a^3*tan(1/2*f*x + 1/2*e)^14 - 180180*A*a^3*tan(1/2*f*x + 1/2*e)^13 + 45045*B*a^3*tan(1/2*f*x
+ 1/2*e)^13 + 1066065*A*a^3*tan(1/2*f*x + 1/2*e)^12 - 15015*B*a^3*tan(1/2*f*x + 1/2*e)^12 - 2702700*A*a^3*tan
(1/2*f*x + 1/2*e)^11 + 450450*B*a^3*tan(1/2*f*x + 1/2*e)^11 + 6675669*A*a^3*tan(1/2*f*x + 1/2*e)^10 - 306306*B
*a^3*tan(1/2*f*x + 1/2*e)^10 - 10210200*A*a^3*tan(1/2*f*x + 1/2*e)^9 + 1456455*B*a^3*tan(1/2*f*x + 1/2*e)^9 +
14124825*A*a^3*tan(1/2*f*x + 1/2*e)^8 - 791505*B*a^3*tan(1/2*f*x + 1/2*e)^8 - 13178880*A*a^3*tan(1/2*f*x + 1/2
*e)^7 + 1827540*B*a^3*tan(1/2*f*x + 1/2*e)^7 + 11026015*A*a^3*tan(1/2*f*x + 1/2*e)^6 - 580580*B*a^3*tan(1/2*f*
x + 1/2*e)^6 - 6066060*A*a^3*tan(1/2*f*x + 1/2*e)^5 + 915915*B*a^3*tan(1/2*f*x + 1/2*e)^5 + 3088995*A*a^3*tan(
1/2*f*x + 1/2*e)^4 - 105105*B*a^3*tan(1/2*f*x + 1/2*e)^4 - 864500*A*a^3*tan(1/2*f*x + 1/2*e)^3 + 170170*B*a^3*
tan(1/2*f*x + 1/2*e)^3 + 265335*A*a^3*tan(1/2*f*x + 1/2*e)^2 + 2310*B*a^3*tan(1/2*f*x + 1/2*e)^2 - 18600*A*a^3
*tan(1/2*f*x + 1/2*e) + 6105*B*a^3*tan(1/2*f*x + 1/2*e) + 4243*A*a^3 - 407*B*a^3)/(c^8*f*(tan(1/2*f*x + 1/2*e)
- 1)^15)

Mupad [B] (verification not implemented)

Time = 14.90 (sec) , antiderivative size = 577, normalized size of antiderivative = 2.93 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx=\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {544369\,A\,a^3}{4}-\frac {21791\,B\,a^3}{4}-\frac {257861\,A\,a^3\,\cos \left (2\,e+2\,f\,x\right )}{2}+\frac {3497111\,A\,a^3\,\cos \left (3\,e+3\,f\,x\right )}{128}+\frac {72047\,A\,a^3\,\cos \left (4\,e+4\,f\,x\right )}{4}-\frac {378579\,A\,a^3\,\cos \left (5\,e+5\,f\,x\right )}{128}-\frac {1059\,A\,a^3\,\cos \left (6\,e+6\,f\,x\right )}{2}+\frac {4251\,A\,a^3\,\cos \left (7\,e+7\,f\,x\right )}{128}+\frac {219769\,B\,a^3\,\cos \left (2\,e+2\,f\,x\right )}{32}-\frac {191389\,B\,a^3\,\cos \left (3\,e+3\,f\,x\right )}{128}-1672\,B\,a^3\,\cos \left (4\,e+4\,f\,x\right )+\frac {38841\,B\,a^3\,\cos \left (5\,e+5\,f\,x\right )}{128}+\frac {1551\,B\,a^3\,\cos \left (6\,e+6\,f\,x\right )}{32}-\frac {429\,B\,a^3\,\cos \left (7\,e+7\,f\,x\right )}{128}+\frac {2633345\,A\,a^3\,\sin \left (2\,e+2\,f\,x\right )}{64}+\frac {7210775\,A\,a^3\,\sin \left (3\,e+3\,f\,x\right )}{128}-\frac {89375\,A\,a^3\,\sin \left (4\,e+4\,f\,x\right )}{8}-\frac {504205\,A\,a^3\,\sin \left (5\,e+5\,f\,x\right )}{128}+\frac {29765\,A\,a^3\,\sin \left (6\,e+6\,f\,x\right )}{64}+\frac {4235\,A\,a^3\,\sin \left (7\,e+7\,f\,x\right )}{128}-\frac {451165\,B\,a^3\,\sin \left (2\,e+2\,f\,x\right )}{64}-\frac {854425\,B\,a^3\,\sin \left (3\,e+3\,f\,x\right )}{128}+\frac {9295\,B\,a^3\,\sin \left (4\,e+4\,f\,x\right )}{8}+\frac {46475\,B\,a^3\,\sin \left (5\,e+5\,f\,x\right )}{128}-\frac {3025\,B\,a^3\,\sin \left (6\,e+6\,f\,x\right )}{64}-\frac {385\,B\,a^3\,\sin \left (7\,e+7\,f\,x\right )}{128}-\frac {5734111\,A\,a^3\,\cos \left (e+f\,x\right )}{128}+\frac {126929\,B\,a^3\,\cos \left (e+f\,x\right )}{128}-\frac {25501905\,A\,a^3\,\sin \left (e+f\,x\right )}{128}+\frac {3970395\,B\,a^3\,\sin \left (e+f\,x\right )}{128}\right )}{45045\,c^8\,f\,\left (\frac {6435\,\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}{128}-\frac {5005\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}-\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{128}-\frac {3003\,\sqrt {2}\,\cos \left (\frac {5\,e}{2}+\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{128}+\frac {1365\,\sqrt {2}\,\cos \left (\frac {7\,e}{2}-\frac {\pi }{4}+\frac {7\,f\,x}{2}\right )}{128}+\frac {455\,\sqrt {2}\,\cos \left (\frac {9\,e}{2}+\frac {\pi }{4}+\frac {9\,f\,x}{2}\right )}{128}-\frac {105\,\sqrt {2}\,\cos \left (\frac {11\,e}{2}-\frac {\pi }{4}+\frac {11\,f\,x}{2}\right )}{128}-\frac {15\,\sqrt {2}\,\cos \left (\frac {13\,e}{2}+\frac {\pi }{4}+\frac {13\,f\,x}{2}\right )}{128}+\frac {\sqrt {2}\,\cos \left (\frac {15\,e}{2}-\frac {\pi }{4}+\frac {15\,f\,x}{2}\right )}{128}\right )} \]

[In]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^8,x)

[Out]

(2*cos(e/2 + (f*x)/2)*((544369*A*a^3)/4 - (21791*B*a^3)/4 - (257861*A*a^3*cos(2*e + 2*f*x))/2 + (3497111*A*a^3
*cos(3*e + 3*f*x))/128 + (72047*A*a^3*cos(4*e + 4*f*x))/4 - (378579*A*a^3*cos(5*e + 5*f*x))/128 - (1059*A*a^3*
cos(6*e + 6*f*x))/2 + (4251*A*a^3*cos(7*e + 7*f*x))/128 + (219769*B*a^3*cos(2*e + 2*f*x))/32 - (191389*B*a^3*c
os(3*e + 3*f*x))/128 - 1672*B*a^3*cos(4*e + 4*f*x) + (38841*B*a^3*cos(5*e + 5*f*x))/128 + (1551*B*a^3*cos(6*e
+ 6*f*x))/32 - (429*B*a^3*cos(7*e + 7*f*x))/128 + (2633345*A*a^3*sin(2*e + 2*f*x))/64 + (7210775*A*a^3*sin(3*e
+ 3*f*x))/128 - (89375*A*a^3*sin(4*e + 4*f*x))/8 - (504205*A*a^3*sin(5*e + 5*f*x))/128 + (29765*A*a^3*sin(6*e
+ 6*f*x))/64 + (4235*A*a^3*sin(7*e + 7*f*x))/128 - (451165*B*a^3*sin(2*e + 2*f*x))/64 - (854425*B*a^3*sin(3*e
+ 3*f*x))/128 + (9295*B*a^3*sin(4*e + 4*f*x))/8 + (46475*B*a^3*sin(5*e + 5*f*x))/128 - (3025*B*a^3*sin(6*e +
6*f*x))/64 - (385*B*a^3*sin(7*e + 7*f*x))/128 - (5734111*A*a^3*cos(e + f*x))/128 + (126929*B*a^3*cos(e + f*x))
/128 - (25501905*A*a^3*sin(e + f*x))/128 + (3970395*B*a^3*sin(e + f*x))/128))/(45045*c^8*f*((6435*2^(1/2)*cos(
e/2 + pi/4 + (f*x)/2))/128 - (5005*2^(1/2)*cos((3*e)/2 - pi/4 + (3*f*x)/2))/128 - (3003*2^(1/2)*cos((5*e)/2 +
pi/4 + (5*f*x)/2))/128 + (1365*2^(1/2)*cos((7*e)/2 - pi/4 + (7*f*x)/2))/128 + (455*2^(1/2)*cos((9*e)/2 + pi/4
+ (9*f*x)/2))/128 - (105*2^(1/2)*cos((11*e)/2 - pi/4 + (11*f*x)/2))/128 - (15*2^(1/2)*cos((13*e)/2 + pi/4 + (1
3*f*x)/2))/128 + (2^(1/2)*cos((15*e)/2 - pi/4 + (15*f*x)/2))/128))