\(\int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx\) [36]

Optimal result

Integrand size = 36, antiderivative size = 156 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {a^2 (3 A-8 B) c \cos ^5(e+f x)}{99 f (c-c \sin (e+f x))^7}+\frac {2 a^2 (3 A-8 B) \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^6}+\frac {2 a^2 (3 A-8 B) \cos ^5(e+f x)}{3465 c f (c-c \sin (e+f x))^5} \]

[Out]

1/11*a^2*(A+B)*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^8+1/99*a^2*(3*A-8*B)*c*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^7+2/
693*a^2*(3*A-8*B)*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^6+2/3465*a^2*(3*A-8*B)*cos(f*x+e)^5/c/f/(c-c*sin(f*x+e))^5

Rubi [A] (verified)

Time = 0.23 (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} \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=\frac {a^2 c^2 (A+B) \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {2 a^2 (3 A-8 B) \cos ^5(e+f x)}{3465 c f (c-c \sin (e+f x))^5}+\frac {2 a^2 (3 A-8 B) \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^6}+\frac {a^2 c (3 A-8 B) \cos ^5(e+f x)}{99 f (c-c \sin (e+f x))^7} \]

[In]

Int[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^6,x]

[Out]

(a^2*(A + B)*c^2*Cos[e + f*x]^5)/(11*f*(c - c*Sin[e + f*x])^8) + (a^2*(3*A - 8*B)*c*Cos[e + f*x]^5)/(99*f*(c -
c*Sin[e + f*x])^7) + (2*a^2*(3*A - 8*B)*Cos[e + f*x]^5)/(693*f*(c - c*Sin[e + f*x])^6) + (2*a^2*(3*A - 8*B)*C
os[e + f*x]^5)/(3465*c*f*(c - c*Sin[e + f*x])^5)

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^2 c^2\right ) \int \frac {\cos ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8} \, dx \\ & = \frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {1}{11} \left (a^2 (3 A-8 B) c\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^7} \, dx \\ & = \frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {a^2 (3 A-8 B) c \cos ^5(e+f x)}{99 f (c-c \sin (e+f x))^7}+\frac {1}{99} \left (2 a^2 (3 A-8 B)\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^6} \, dx \\ & = \frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {a^2 (3 A-8 B) c \cos ^5(e+f x)}{99 f (c-c \sin (e+f x))^7}+\frac {2 a^2 (3 A-8 B) \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^6}+\frac {\left (2 a^2 (3 A-8 B)\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^5} \, dx}{693 c} \\ & = \frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {a^2 (3 A-8 B) c \cos ^5(e+f x)}{99 f (c-c \sin (e+f x))^7}+\frac {2 a^2 (3 A-8 B) \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^6}+\frac {2 a^2 (3 A-8 B) \cos ^5(e+f x)}{3465 c f (c-c \sin (e+f x))^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.59 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.83 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^2 \left (231 (27 A+28 B) \cos \left (\frac {1}{2} (e+f x)\right )-2475 (A+2 B) \cos \left (\frac {3}{2} (e+f x)\right )-2310 B \cos \left (\frac {5}{2} (e+f x)\right )-165 A \cos \left (\frac {7}{2} (e+f x)\right )+440 B \cos \left (\frac {7}{2} (e+f x)\right )+3 A \cos \left (\frac {11}{2} (e+f x)\right )-8 B \cos \left (\frac {11}{2} (e+f x)\right )+7623 A \sin \left (\frac {1}{2} (e+f x)\right )+2772 B \sin \left (\frac {1}{2} (e+f x)\right )+3465 A \sin \left (\frac {3}{2} (e+f x)\right )+2310 B \sin \left (\frac {3}{2} (e+f x)\right )-495 A \sin \left (\frac {5}{2} (e+f x)\right )-990 B \sin \left (\frac {5}{2} (e+f x)\right )+33 A \sin \left (\frac {9}{2} (e+f x)\right )-88 B \sin \left (\frac {9}{2} (e+f x)\right )\right )}{27720 c^6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (-1+\sin (e+f x))^6} \]

[In]

Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^6,x]

[Out]

(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(231*(27*A + 28*B)*Cos[(e + f*x)/2] - 2475*(A
+ 2*B)*Cos[(3*(e + f*x))/2] - 2310*B*Cos[(5*(e + f*x))/2] - 165*A*Cos[(7*(e + f*x))/2] + 440*B*Cos[(7*(e + f*x
))/2] + 3*A*Cos[(11*(e + f*x))/2] - 8*B*Cos[(11*(e + f*x))/2] + 7623*A*Sin[(e + f*x)/2] + 2772*B*Sin[(e + f*x)
/2] + 3465*A*Sin[(3*(e + f*x))/2] + 2310*B*Sin[(3*(e + f*x))/2] - 495*A*Sin[(5*(e + f*x))/2] - 990*B*Sin[(5*(e
+ f*x))/2] + 33*A*Sin[(9*(e + f*x))/2] - 88*B*Sin[(9*(e + f*x))/2]))/(27720*c^6*f*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2])^4*(-1 + Sin[e + f*x])^6)

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.40

method	result	size
parallelrisch	\(-\frac {2 a^{2} \left (A \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-3 A +B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (12 A -\frac {B}{3}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 \left (-10 A +\frac {7 B}{3}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {2 \left (81 A -\frac {13 B}{3}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {2 \left (-71 A +16 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {4 \left (41 A -2 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}+\frac {2 \left (-34 A +9 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}+\frac {\left (89 A +\frac {2 B}{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{21}+\frac {\left (-47 A +\frac {61 B}{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{105}+\frac {152 A}{1155}-\frac {61 B}{3465}\right )}{f \,c^{6} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\)	\(219\)
derivativedivides	\(\frac {2 a^{2} \left (-\frac {1752 A +1208 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {90 A +26 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {640 A +640 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {2304 A +2048 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {128 A +128 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {1536 A +1472 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {14 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {932 A +528 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2376 A +1896 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {352 A +152 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}\right )}{f \,c^{6}}\)	\(249\)
default	\(\frac {2 a^{2} \left (-\frac {1752 A +1208 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {90 A +26 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {640 A +640 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {2304 A +2048 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {128 A +128 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {1536 A +1472 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {14 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {932 A +528 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2376 A +1896 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {352 A +152 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}\right )}{f \,c^{6}}\)	\(249\)
risch	\(\frac {-\frac {4 A \,a^{2}}{1155}-\frac {16 i B \,a^{2} {\mathrm e}^{5 i \left (f x +e \right )}}{5}-\frac {44 i A \,a^{2} {\mathrm e}^{5 i \left (f x +e \right )}}{5}+\frac {32 B \,a^{2}}{3465}+\frac {32 i B \,a^{2} {\mathrm e}^{i \left (f x +e \right )}}{315}-\frac {4 i A \,a^{2} {\mathrm e}^{i \left (f x +e \right )}}{105}+\frac {8 i B \,a^{2} {\mathrm e}^{7 i \left (f x +e \right )}}{3}-\frac {32 B \,a^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{63}+4 i A \,a^{2} {\mathrm e}^{7 i \left (f x +e \right )}+\frac {8 i B \,a^{2} {\mathrm e}^{3 i \left (f x +e \right )}}{7}+\frac {4 i A \,a^{2} {\mathrm e}^{3 i \left (f x +e \right )}}{7}+\frac {8 B \,a^{2} {\mathrm e}^{8 i \left (f x +e \right )}}{3}-\frac {36 A \,a^{2} {\mathrm e}^{6 i \left (f x +e \right )}}{5}-\frac {112 B \,a^{2} {\mathrm e}^{6 i \left (f x +e \right )}}{15}+\frac {20 A \,a^{2} {\mathrm e}^{4 i \left (f x +e \right )}}{7}+\frac {40 B \,a^{2} {\mathrm e}^{4 i \left (f x +e \right )}}{7}+\frac {4 A \,a^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{21}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{11} f \,c^{6}}\)	\(269\)
norman	\(\frac {-\frac {912 A \,a^{2}-122 B \,a^{2}}{3465 c f}-\frac {2 A \,a^{2} \left (\tan ^{16}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {2 \left (3 A \,a^{2}-B \,a^{2}\right ) \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {2 \left (45 A \,a^{2}-B \,a^{2}\right ) \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {2 \left (87 A \,a^{2}-23 B \,a^{2}\right ) \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {\left (282 A \,a^{2}-122 B \,a^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{315 c f}+\frac {2 \left (487 A \,a^{2}-117 B \,a^{2}\right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}-\frac {2 \left (1071 A \,a^{2}-41 B \,a^{2}\right ) \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}+\frac {2 \left (1161 A \,a^{2}-331 B \,a^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}-\frac {2 \left (5517 A \,a^{2}-257 B \,a^{2}\right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 c f}+\frac {2 \left (5527 A \,a^{2}-1287 B \,a^{2}\right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 c f}+\frac {2 \left (6183 A \,a^{2}-1543 B \,a^{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}-\frac {2 \left (16053 A \,a^{2}-73 B \,a^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3465 c f}-\frac {2 \left (42201 A \,a^{2}-1271 B \,a^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{1155 c f}+\frac {2 \left (42459 A \,a^{2}-10009 B \,a^{2}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 c f}-\frac {2 \left (57873 A \,a^{2}-2813 B \,a^{2}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 c f}-\frac {2 \left (400317 A \,a^{2}-17617 B \,a^{2}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3465 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\)	\(556\)

[In]

int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^6,x,method=_RETURNVERBOSE)

[Out]

-2*a^2*(A*tan(1/2*f*x+1/2*e)^10+(-3*A+B)*tan(1/2*f*x+1/2*e)^9+(12*A-1/3*B)*tan(1/2*f*x+1/2*e)^8+2*(-10*A+7/3*B
)*tan(1/2*f*x+1/2*e)^7+2/5*(81*A-13/3*B)*tan(1/2*f*x+1/2*e)^6+2/5*(-71*A+16*B)*tan(1/2*f*x+1/2*e)^5+4/7*(41*A-
2*B)*tan(1/2*f*x+1/2*e)^4+2/7*(-34*A+9*B)*tan(1/2*f*x+1/2*e)^3+1/21*(89*A+2/3*B)*tan(1/2*f*x+1/2*e)^2+1/105*(-
47*A+61/3*B)*tan(1/2*f*x+1/2*e)+152/1155*A-61/3465*B)/f/c^6/(tan(1/2*f*x+1/2*e)-1)^11

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (152) = 304\).

Time = 0.25 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.61 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (3 \, A - 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{6} + 12 \, {\left (3 \, A - 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} - 25 \, {\left (3 \, A - 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} - 35 \, {\left (6 \, A + 17 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} - 35 \, {\left (21 \, A + 43 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} + 630 \, {\left (A + B\right )} a^{2} \cos \left (f x + e\right ) + 1260 \, {\left (A + B\right )} a^{2} - {\left (2 \, {\left (3 \, A - 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} - 10 \, {\left (3 \, A - 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} - 35 \, {\left (3 \, A - 8 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} + 35 \, {\left (3 \, A + 25 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 630 \, {\left (A + B\right )} a^{2} \cos \left (f x + e\right ) - 1260 \, {\left (A + B\right )} a^{2}\right )} \sin \left (f x + e\right )}{3465 \, {\left (c^{6} f \cos \left (f x + e\right )^{6} - 5 \, c^{6} f \cos \left (f x + e\right )^{5} - 18 \, c^{6} f \cos \left (f x + e\right )^{4} + 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 48 \, c^{6} f \cos \left (f x + e\right )^{2} - 16 \, c^{6} f \cos \left (f x + e\right ) - 32 \, c^{6} f + {\left (c^{6} f \cos \left (f x + e\right )^{5} + 6 \, c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} - 32 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f \cos \left (f x + e\right ) + 32 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^6,x, algorithm="fricas")

[Out]

-1/3465*(2*(3*A - 8*B)*a^2*cos(f*x + e)^6 + 12*(3*A - 8*B)*a^2*cos(f*x + e)^5 - 25*(3*A - 8*B)*a^2*cos(f*x + e
)^4 - 35*(6*A + 17*B)*a^2*cos(f*x + e)^3 - 35*(21*A + 43*B)*a^2*cos(f*x + e)^2 + 630*(A + B)*a^2*cos(f*x + e)
+ 1260*(A + B)*a^2 - (2*(3*A - 8*B)*a^2*cos(f*x + e)^5 - 10*(3*A - 8*B)*a^2*cos(f*x + e)^4 - 35*(3*A - 8*B)*a^
2*cos(f*x + e)^3 + 35*(3*A + 25*B)*a^2*cos(f*x + e)^2 - 630*(A + B)*a^2*cos(f*x + e) - 1260*(A + B)*a^2)*sin(f
*x + e))/(c^6*f*cos(f*x + e)^6 - 5*c^6*f*cos(f*x + e)^5 - 18*c^6*f*cos(f*x + e)^4 + 20*c^6*f*cos(f*x + e)^3 +
48*c^6*f*cos(f*x + e)^2 - 16*c^6*f*cos(f*x + e) - 32*c^6*f + (c^6*f*cos(f*x + e)^5 + 6*c^6*f*cos(f*x + e)^4 -
12*c^6*f*cos(f*x + e)^3 - 32*c^6*f*cos(f*x + e)^2 + 16*c^6*f*cos(f*x + e) + 32*c^6*f)*sin(f*x + e))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4816 vs. \(2 (141) = 282\).

Time = 44.16 (sec) , antiderivative size = 4816, normalized size of antiderivative = 30.87 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))**2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**6,x)

[Out]

Piecewise((-6930*A*a**2*tan(e/2 + f*x/2)**10/(3465*c**6*f*tan(e/2 + f*x/2)**11 - 38115*c**6*f*tan(e/2 + f*x/2)
**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 + 1143450*c**6*f*tan(e/2 + f*x/2)
**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5 - 1143450*c**6*f*tan(e/2 + f*x/2
)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 + 38115*c**6*f*tan(e/2 + f*x/2) -
3465*c**6*f) + 20790*A*a**2*tan(e/2 + f*x/2)**9/(3465*c**6*f*tan(e/2 + f*x/2)**11 - 38115*c**6*f*tan(e/2 + f*
x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 + 1143450*c**6*f*tan(e/2 + f*
x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5 - 1143450*c**6*f*tan(e/2 + f
*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 + 38115*c**6*f*tan(e/2 + f*x/
2) - 3465*c**6*f) - 83160*A*a**2*tan(e/2 + f*x/2)**8/(3465*c**6*f*tan(e/2 + f*x/2)**11 - 38115*c**6*f*tan(e/2
+ f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 + 1143450*c**6*f*tan(e/2
+ f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5 - 1143450*c**6*f*tan(e/2
+ f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 + 38115*c**6*f*tan(e/2 +
f*x/2) - 3465*c**6*f) + 138600*A*a**2*tan(e/2 + f*x/2)**7/(3465*c**6*f*tan(e/2 + f*x/2)**11 - 38115*c**6*f*tan
(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 + 1143450*c**6*f*tan
(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5 - 1143450*c**6*f*ta
n(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 + 38115*c**6*f*tan(e
/2 + f*x/2) - 3465*c**6*f) - 224532*A*a**2*tan(e/2 + f*x/2)**6/(3465*c**6*f*tan(e/2 + f*x/2)**11 - 38115*c**6*
f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 + 1143450*c**6*
f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5 - 1143450*c**6
*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 + 38115*c**6*f*
tan(e/2 + f*x/2) - 3465*c**6*f) + 196812*A*a**2*tan(e/2 + f*x/2)**5/(3465*c**6*f*tan(e/2 + f*x/2)**11 - 38115*
c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 + 1143450*
c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5 - 1143450
*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 + 38115*c*
*6*f*tan(e/2 + f*x/2) - 3465*c**6*f) - 162360*A*a**2*tan(e/2 + f*x/2)**4/(3465*c**6*f*tan(e/2 + f*x/2)**11 - 3
8115*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 + 114
3450*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5 - 11
43450*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 + 381
15*c**6*f*tan(e/2 + f*x/2) - 3465*c**6*f) + 67320*A*a**2*tan(e/2 + f*x/2)**3/(3465*c**6*f*tan(e/2 + f*x/2)**11
- 38115*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 +
1143450*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5
- 1143450*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 +
38115*c**6*f*tan(e/2 + f*x/2) - 3465*c**6*f) - 29370*A*a**2*tan(e/2 + f*x/2)**2/(3465*c**6*f*tan(e/2 + f*x/2)
**11 - 38115*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)*
*8 + 1143450*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)
**5 - 1143450*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)*
*2 + 38115*c**6*f*tan(e/2 + f*x/2) - 3465*c**6*f) + 3102*A*a**2*tan(e/2 + f*x/2)/(3465*c**6*f*tan(e/2 + f*x/2)
**11 - 38115*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)*
*8 + 1143450*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)
**5 - 1143450*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)*
*2 + 38115*c**6*f*tan(e/2 + f*x/2) - 3465*c**6*f) - 912*A*a**2/(3465*c**6*f*tan(e/2 + f*x/2)**11 - 38115*c**6*
f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 + 1143450*c**6*
f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5 - 1143450*c**6
*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 + 38115*c**6*f*
tan(e/2 + f*x/2) - 3465*c**6*f) - 6930*B*a**2*tan(e/2 + f*x/2)**9/(3465*c**6*f*tan(e/2 + f*x/2)**11 - 38115*c*
*6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 + 1143450*c*
*6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5 - 1143450*c
**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 + 38115*c**6
*f*tan(e/2 + f*x/2) - 3465*c**6*f) + 2310*B*a**2*tan(e/2 + f*x/2)**8/(3465*c**6*f*tan(e/2 + f*x/2)**11 - 38115
*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 + 1143450
*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5 - 114345
0*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 + 38115*c
**6*f*tan(e/2 + f*x/2) - 3465*c**6*f) - 32340*B*a**2*tan(e/2 + f*x/2)**7/(3465*c**6*f*tan(e/2 + f*x/2)**11 - 3
8115*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 + 114
3450*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5 - 11
43450*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 + 381
15*c**6*f*tan(e/2 + f*x/2) - 3465*c**6*f) + 12012*B*a**2*tan(e/2 + f*x/2)**6/(3465*c**6*f*tan(e/2 + f*x/2)**11
- 38115*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 +
1143450*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5
- 1143450*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 +
38115*c**6*f*tan(e/2 + f*x/2) - 3465*c**6*f) - 44352*B*a**2*tan(e/2 + f*x/2)**5/(3465*c**6*f*tan(e/2 + f*x/2)
**11 - 38115*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)*
*8 + 1143450*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)
**5 - 1143450*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)*
*2 + 38115*c**6*f*tan(e/2 + f*x/2) - 3465*c**6*f) + 7920*B*a**2*tan(e/2 + f*x/2)**4/(3465*c**6*f*tan(e/2 + f*x
/2)**11 - 38115*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/
2)**8 + 1143450*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x
/2)**5 - 1143450*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/
2)**2 + 38115*c**6*f*tan(e/2 + f*x/2) - 3465*c**6*f) - 17820*B*a**2*tan(e/2 + f*x/2)**3/(3465*c**6*f*tan(e/2 +
f*x/2)**11 - 38115*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 +
f*x/2)**8 + 1143450*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 +
f*x/2)**5 - 1143450*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 +
f*x/2)**2 + 38115*c**6*f*tan(e/2 + f*x/2) - 3465*c**6*f) - 220*B*a**2*tan(e/2 + f*x/2)**2/(3465*c**6*f*tan(e/2
+ f*x/2)**11 - 38115*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2
+ f*x/2)**8 + 1143450*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2
+ f*x/2)**5 - 1143450*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2
+ f*x/2)**2 + 38115*c**6*f*tan(e/2 + f*x/2) - 3465*c**6*f) - 1342*B*a**2*tan(e/2 + f*x/2)/(3465*c**6*f*tan(e/2
+ f*x/2)**11 - 38115*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2
+ f*x/2)**8 + 1143450*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2
+ f*x/2)**5 - 1143450*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2
+ f*x/2)**2 + 38115*c**6*f*tan(e/2 + f*x/2) - 3465*c**6*f) + 122*B*a**2/(3465*c**6*f*tan(e/2 + f*x/2)**11 - 38
115*c**6*f*tan(e/2 + f*x/2)**10 + 190575*c**6*f*tan(e/2 + f*x/2)**9 - 571725*c**6*f*tan(e/2 + f*x/2)**8 + 1143
450*c**6*f*tan(e/2 + f*x/2)**7 - 1600830*c**6*f*tan(e/2 + f*x/2)**6 + 1600830*c**6*f*tan(e/2 + f*x/2)**5 - 114
3450*c**6*f*tan(e/2 + f*x/2)**4 + 571725*c**6*f*tan(e/2 + f*x/2)**3 - 190575*c**6*f*tan(e/2 + f*x/2)**2 + 3811
5*c**6*f*tan(e/2 + f*x/2) - 3465*c**6*f), Ne(f, 0)), (x*(A + B*sin(e))*(a*sin(e) + a)**2/(-c*sin(e) + c)**6, T
rue))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2604 vs. \(2 (152) = 304\).

Time = 0.31 (sec) , antiderivative size = 2604, normalized size of antiderivative = 16.69 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^6,x, algorithm="maxima")

[Out]

-2/3465*(5*A*a^2*(913*sin(f*x + e)/(cos(f*x + e) + 1) - 4565*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 12540*sin(f
*x + e)^3/(cos(f*x + e) + 1)^3 - 25080*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33726*sin(f*x + e)^5/(cos(f*x + e
) + 1)^5 - 33726*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 23100*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 11550*sin(f
*x + e)^8/(cos(f*x + e) + 1)^8 + 3465*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 693*sin(f*x + e)^10/(cos(f*x + e)
+ 1)^10 - 146)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 16
5*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)
^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) +
1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(
f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) - 6*A*a^2*(671*sin(f*x + e)/(co
s(f*x + e) + 1) - 2200*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 6600*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 10890*
sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 15246*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 12936*sin(f*x + e)^6/(cos(f*
x + e) + 1)^6 + 9240*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3465*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 1155*sin
(f*x + e)^9/(cos(f*x + e) + 1)^9 - 61)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(
cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^
4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*
x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x +
e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) - 3*B*a
^2*(671*sin(f*x + e)/(cos(f*x + e) + 1) - 2200*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 6600*sin(f*x + e)^3/(cos(
f*x + e) + 1)^3 - 10890*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 15246*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 1293
6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 9240*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 3465*sin(f*x + e)^8/(cos(f*
x + e) + 1)^8 + 1155*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 61)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) +
55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x +
e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e)
+ 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*s
in(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f
*x + e) + 1)^11) - 2*B*a^2*(341*sin(f*x + e)/(cos(f*x + e) + 1) - 1705*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 5
115*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 6765*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 9471*sin(f*x + e)^5/(cos(
f*x + e) + 1)^5 - 4851*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 3465*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 31)/(c
^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e
)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e)
+ 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*s
in(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(
f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 4*A*a^2*(253*sin(f*x + e)/(cos(f*x + e) + 1) -
1265*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 2640*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 5280*sin(f*x + e)^4/(co
s(f*x + e) + 1)^4 + 5313*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 5313*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2310
*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 1155*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 23)/(c^6 - 11*c^6*sin(f*x +
e)/(cos(f*x + e) + 1) + 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3 + 330*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f
*x + e)^6/(cos(f*x + e) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x
+ e) + 1)^8 - 55*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6
*sin(f*x + e)^11/(cos(f*x + e) + 1)^11) + 8*B*a^2*(253*sin(f*x + e)/(cos(f*x + e) + 1) - 1265*sin(f*x + e)^2/(
cos(f*x + e) + 1)^2 + 2640*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 5280*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 53
13*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 5313*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2310*sin(f*x + e)^7/(cos(f
*x + e) + 1)^7 - 1155*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 23)/(c^6 - 11*c^6*sin(f*x + e)/(cos(f*x + e) + 1)
+ 55*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 165*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 330*c^6*sin(f*x +
e)^4/(cos(f*x + e) + 1)^4 - 462*c^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 462*c^6*sin(f*x + e)^6/(cos(f*x + e
) + 1)^6 - 330*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 165*c^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 55*c^6*
sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 11*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - c^6*sin(f*x + e)^11/(cos(
f*x + e) + 1)^11))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (152) = 304\).

Time = 0.36 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.26 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=-\frac {2 \, {\left (3465 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 10395 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 3465 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 41580 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 1155 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 69300 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 16170 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 112266 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 6006 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 98406 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 22176 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 81180 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3960 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 33660 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8910 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 14685 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 110 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1551 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 671 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 456 \, A a^{2} - 61 \, B a^{2}\right )}}{3465 \, c^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{11}} \]

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^6,x, algorithm="giac")

[Out]

-2/3465*(3465*A*a^2*tan(1/2*f*x + 1/2*e)^10 - 10395*A*a^2*tan(1/2*f*x + 1/2*e)^9 + 3465*B*a^2*tan(1/2*f*x + 1/
2*e)^9 + 41580*A*a^2*tan(1/2*f*x + 1/2*e)^8 - 1155*B*a^2*tan(1/2*f*x + 1/2*e)^8 - 69300*A*a^2*tan(1/2*f*x + 1/
2*e)^7 + 16170*B*a^2*tan(1/2*f*x + 1/2*e)^7 + 112266*A*a^2*tan(1/2*f*x + 1/2*e)^6 - 6006*B*a^2*tan(1/2*f*x + 1
/2*e)^6 - 98406*A*a^2*tan(1/2*f*x + 1/2*e)^5 + 22176*B*a^2*tan(1/2*f*x + 1/2*e)^5 + 81180*A*a^2*tan(1/2*f*x +
1/2*e)^4 - 3960*B*a^2*tan(1/2*f*x + 1/2*e)^4 - 33660*A*a^2*tan(1/2*f*x + 1/2*e)^3 + 8910*B*a^2*tan(1/2*f*x + 1
/2*e)^3 + 14685*A*a^2*tan(1/2*f*x + 1/2*e)^2 + 110*B*a^2*tan(1/2*f*x + 1/2*e)^2 - 1551*A*a^2*tan(1/2*f*x + 1/2
*e) + 671*B*a^2*tan(1/2*f*x + 1/2*e) + 456*A*a^2 - 61*B*a^2)/(c^6*f*(tan(1/2*f*x + 1/2*e) - 1)^11)

Mupad [B] (verification not implemented)

Time = 15.72 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.71 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx=\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {38163\,A\,a^2}{8}-\frac {1283\,B\,a^2}{8}-\frac {11931\,A\,a^2\,\cos \left (2\,e+2\,f\,x\right )}{4}+\frac {9609\,A\,a^2\,\cos \left (3\,e+3\,f\,x\right )}{16}+\frac {1383\,A\,a^2\,\cos \left (4\,e+4\,f\,x\right )}{8}-\frac {225\,A\,a^2\,\cos \left (5\,e+5\,f\,x\right )}{16}+\frac {631\,B\,a^2\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {1583\,B\,a^2\,\cos \left (3\,e+3\,f\,x\right )}{32}-\frac {223\,B\,a^2\,\cos \left (4\,e+4\,f\,x\right )}{8}+\frac {45\,B\,a^2\,\cos \left (5\,e+5\,f\,x\right )}{32}+1386\,A\,a^2\,\sin \left (2\,e+2\,f\,x\right )+\frac {14949\,A\,a^2\,\sin \left (3\,e+3\,f\,x\right )}{16}-\frac {561\,A\,a^2\,\sin \left (4\,e+4\,f\,x\right )}{4}-\frac {231\,A\,a^2\,\sin \left (5\,e+5\,f\,x\right )}{16}-\frac {3003\,B\,a^2\,\sin \left (2\,e+2\,f\,x\right )}{8}-\frac {4653\,B\,a^2\,\sin \left (3\,e+3\,f\,x\right )}{32}+\frac {209\,B\,a^2\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {77\,B\,a^2\,\sin \left (5\,e+5\,f\,x\right )}{32}-2091\,A\,a^2\,\cos \left (e+f\,x\right )+\frac {281\,B\,a^2\,\cos \left (e+f\,x\right )}{16}-\frac {22869\,A\,a^2\,\sin \left (e+f\,x\right )}{4}+\frac {23331\,B\,a^2\,\sin \left (e+f\,x\right )}{16}\right )}{3465\,c^6\,f\,\left (\frac {231\,\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}{16}-\frac {165\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}-\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{16}-\frac {165\,\sqrt {2}\,\cos \left (\frac {5\,e}{2}+\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{32}+\frac {55\,\sqrt {2}\,\cos \left (\frac {7\,e}{2}-\frac {\pi }{4}+\frac {7\,f\,x}{2}\right )}{32}+\frac {11\,\sqrt {2}\,\cos \left (\frac {9\,e}{2}+\frac {\pi }{4}+\frac {9\,f\,x}{2}\right )}{32}-\frac {\sqrt {2}\,\cos \left (\frac {11\,e}{2}-\frac {\pi }{4}+\frac {11\,f\,x}{2}\right )}{32}\right )} \]

[In]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^2)/(c - c*sin(e + f*x))^6,x)

[Out]

(2*cos(e/2 + (f*x)/2)*((38163*A*a^2)/8 - (1283*B*a^2)/8 - (11931*A*a^2*cos(2*e + 2*f*x))/4 + (9609*A*a^2*cos(3
*e + 3*f*x))/16 + (1383*A*a^2*cos(4*e + 4*f*x))/8 - (225*A*a^2*cos(5*e + 5*f*x))/16 + (631*B*a^2*cos(2*e + 2*f
*x))/4 - (1583*B*a^2*cos(3*e + 3*f*x))/32 - (223*B*a^2*cos(4*e + 4*f*x))/8 + (45*B*a^2*cos(5*e + 5*f*x))/32 +
1386*A*a^2*sin(2*e + 2*f*x) + (14949*A*a^2*sin(3*e + 3*f*x))/16 - (561*A*a^2*sin(4*e + 4*f*x))/4 - (231*A*a^2*
sin(5*e + 5*f*x))/16 - (3003*B*a^2*sin(2*e + 2*f*x))/8 - (4653*B*a^2*sin(3*e + 3*f*x))/32 + (209*B*a^2*sin(4*e
+ 4*f*x))/16 + (77*B*a^2*sin(5*e + 5*f*x))/32 - 2091*A*a^2*cos(e + f*x) + (281*B*a^2*cos(e + f*x))/16 - (2286
9*A*a^2*sin(e + f*x))/4 + (23331*B*a^2*sin(e + f*x))/16))/(3465*c^6*f*((231*2^(1/2)*cos(e/2 + pi/4 + (f*x)/2))
/16 - (165*2^(1/2)*cos((3*e)/2 - pi/4 + (3*f*x)/2))/16 - (165*2^(1/2)*cos((5*e)/2 + pi/4 + (5*f*x)/2))/32 + (5
5*2^(1/2)*cos((7*e)/2 - pi/4 + (7*f*x)/2))/32 + (11*2^(1/2)*cos((9*e)/2 + pi/4 + (9*f*x)/2))/32 - (2^(1/2)*cos
((11*e)/2 - pi/4 + (11*f*x)/2))/32))