Integrand size = 36, antiderivative size = 197 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}+\frac {a^2 (4 A-9 B) c \cos ^5(e+f x)}{143 f (c-c \sin (e+f x))^8}+\frac {a^2 (4 A-9 B) \cos ^5(e+f x)}{429 f (c-c \sin (e+f x))^7}+\frac {2 a^2 (4 A-9 B) \cos ^5(e+f x)}{3003 c f (c-c \sin (e+f x))^6}+\frac {2 a^2 (4 A-9 B) \cos ^5(e+f x)}{15015 c^2 f (c-c \sin (e+f x))^5} \]
1/13*a^2*(A+B)*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^9+1/143*a^2*(4*A-9*B)*c *cos(f*x+e)^5/f/(c-c*sin(f*x+e))^8+1/429*a^2*(4*A-9*B)*cos(f*x+e)^5/f/(c-c *sin(f*x+e))^7+2/3003*a^2*(4*A-9*B)*cos(f*x+e)^5/c/f/(c-c*sin(f*x+e))^6+2/ 15015*a^2*(4*A-9*B)*cos(f*x+e)^5/c^2/f/(c-c*sin(f*x+e))^5
Time = 10.42 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.59 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=-\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^2 \left (6006 (8 A+7 B) \cos \left (\frac {1}{2} (e+f x)\right )-1716 (11 A+19 B) \cos \left (\frac {3}{2} (e+f x)\right )-15015 B \cos \left (\frac {5}{2} (e+f x)\right )-1144 A \cos \left (\frac {7}{2} (e+f x)\right )+2574 B \cos \left (\frac {7}{2} (e+f x)\right )+52 A \cos \left (\frac {11}{2} (e+f x)\right )-117 B \cos \left (\frac {11}{2} (e+f x)\right )+54912 A \sin \left (\frac {1}{2} (e+f x)\right )+26598 B \sin \left (\frac {1}{2} (e+f x)\right )+24024 A \sin \left (\frac {3}{2} (e+f x)\right )+21021 B \sin \left (\frac {3}{2} (e+f x)\right )-2860 A \sin \left (\frac {5}{2} (e+f x)\right )-8580 B \sin \left (\frac {5}{2} (e+f x)\right )+312 A \sin \left (\frac {9}{2} (e+f x)\right )-702 B \sin \left (\frac {9}{2} (e+f x)\right )-4 A \sin \left (\frac {13}{2} (e+f x)\right )+9 B \sin \left (\frac {13}{2} (e+f x)\right )\right )}{240240 c^7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (-1+\sin (e+f x))^7} \]
-1/240240*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2* (6006*(8*A + 7*B)*Cos[(e + f*x)/2] - 1716*(11*A + 19*B)*Cos[(3*(e + f*x))/ 2] - 15015*B*Cos[(5*(e + f*x))/2] - 1144*A*Cos[(7*(e + f*x))/2] + 2574*B*C os[(7*(e + f*x))/2] + 52*A*Cos[(11*(e + f*x))/2] - 117*B*Cos[(11*(e + f*x) )/2] + 54912*A*Sin[(e + f*x)/2] + 26598*B*Sin[(e + f*x)/2] + 24024*A*Sin[( 3*(e + f*x))/2] + 21021*B*Sin[(3*(e + f*x))/2] - 2860*A*Sin[(5*(e + f*x))/ 2] - 8580*B*Sin[(5*(e + f*x))/2] + 312*A*Sin[(9*(e + f*x))/2] - 702*B*Sin[ (9*(e + f*x))/2] - 4*A*Sin[(13*(e + f*x))/2] + 9*B*Sin[(13*(e + f*x))/2])) /(c^7*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-1 + Sin[e + f*x])^7)
Time = 0.97 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3446, 3042, 3338, 3042, 3151, 3042, 3151, 3042, 3151, 3042, 3150}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^2 c^2 \int \frac {\cos ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^9}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \int \frac {\cos (e+f x)^4 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^9}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^8}dx}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^8}dx}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \left (\frac {3 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^7}dx}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \left (\frac {3 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^7}dx}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \left (\frac {3 \left (\frac {2 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^6}dx}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \left (\frac {3 \left (\frac {2 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^6}dx}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^5}dx}{7 c}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {(4 A-9 B) \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^5}dx}{7 c}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}+\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )}{11 c}+\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}\right )}{13 c}+\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}\right )\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle a^2 c^2 \left (\frac {(A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}+\frac {(4 A-9 B) \left (\frac {\cos ^5(e+f x)}{11 f (c-c \sin (e+f x))^8}+\frac {3 \left (\frac {\cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac {2 \left (\frac {\cos ^5(e+f x)}{35 c f (c-c \sin (e+f x))^5}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}\right )}{11 c}\right )}{13 c}\right )\) |
a^2*c^2*(((A + B)*Cos[e + f*x]^5)/(13*f*(c - c*Sin[e + f*x])^9) + ((4*A - 9*B)*(Cos[e + f*x]^5/(11*f*(c - c*Sin[e + f*x])^8) + (3*(Cos[e + f*x]^5/(9 *f*(c - c*Sin[e + f*x])^7) + (2*(Cos[e + f*x]^5/(7*f*(c - c*Sin[e + f*x])^ 6) + Cos[e + f*x]^5/(35*c*f*(c - c*Sin[e + f*x])^5)))/(9*c)))/(11*c)))/(13 *c))
3.1.37.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Simp[Simplify[m + p + 1]/(a*Simpl ify[2*m + p + 1]) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x] , x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simpli fy[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(2*m + p + 1) )), x] + Simp[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[m + p], 0 ]) && NeQ[2*m + p + 1, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin [e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a* d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Result contains complex when optimal does not.
Time = 1.56 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.26
| method | result | size |
| risch | \(-\frac {4 i a^{2} \left (4 i A +2860 i A \,{\mathrm e}^{4 i \left (f x +e \right )}+15015 B \,{\mathrm e}^{9 i \left (f x +e \right )}+21021 i B \,{\mathrm e}^{8 i \left (f x +e \right )}-48048 A \,{\mathrm e}^{7 i \left (f x +e \right )}-312 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-42042 B \,{\mathrm e}^{7 i \left (f x +e \right )}-9 i B +18876 A \,{\mathrm e}^{5 i \left (f x +e \right )}-26598 i B \,{\mathrm e}^{6 i \left (f x +e \right )}+32604 B \,{\mathrm e}^{5 i \left (f x +e \right )}+702 i B \,{\mathrm e}^{2 i \left (f x +e \right )}+1144 A \,{\mathrm e}^{3 i \left (f x +e \right )}+24024 i A \,{\mathrm e}^{8 i \left (f x +e \right )}-2574 B \,{\mathrm e}^{3 i \left (f x +e \right )}+8580 i B \,{\mathrm e}^{4 i \left (f x +e \right )}-52 A \,{\mathrm e}^{i \left (f x +e \right )}-54912 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+117 B \,{\mathrm e}^{i \left (f x +e \right )}\right )}{15015 f \,c^{7} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{13}}\) | \(248\) |
| parallelrisch | \(-\frac {2 a^{2} \left (A \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-4 A +B \right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (18 A -B \right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-40 A +7 B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {\left (391 A -31 B \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {2 \left (-244 A +39 B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {18 \left (202 A -17 B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35}+\frac {2 \left (-1276 A +211 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35}+\frac {\left (\frac {923 A}{3}-22 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7}+\frac {\left (-\frac {1636 A}{3}+107 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35}+\frac {\left (-41 B +1986 A \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{385}+\frac {\left (-\frac {608 A}{3}+71 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{385}+\frac {1763 A}{15015}-\frac {71 B}{5005}\right )}{f \,c^{7} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}\) | \(258\) |
| derivativedivides | \(\frac {2 a^{2} \left (-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {16 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {256 A +256 B}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {560 A +208 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {8320 A +7680 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {120 A +30 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {4480 A +4352 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {10560 A +8256 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {1536 A +1536 B}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {4320 A +2568 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {1816 A +884 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {7744 A +5368 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {10896 A +9360 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\right )}{f \,c^{7}}\) | \(293\) |
| default | \(\frac {2 a^{2} \left (-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {16 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {256 A +256 B}{13 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{13}}-\frac {560 A +208 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {8320 A +7680 B}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {120 A +30 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {4480 A +4352 B}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {10560 A +8256 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {1536 A +1536 B}{12 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{12}}-\frac {4320 A +2568 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {1816 A +884 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {7744 A +5368 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {10896 A +9360 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\right )}{f \,c^{7}}\) | \(293\) |
-4/15015*I*a^2*(4*I*A+2860*I*A*exp(4*I*(f*x+e))+15015*B*exp(9*I*(f*x+e))+2 1021*I*B*exp(8*I*(f*x+e))-48048*A*exp(7*I*(f*x+e))-312*I*A*exp(2*I*(f*x+e) )-42042*B*exp(7*I*(f*x+e))-9*I*B+18876*A*exp(5*I*(f*x+e))-26598*I*B*exp(6* I*(f*x+e))+32604*B*exp(5*I*(f*x+e))+702*I*B*exp(2*I*(f*x+e))+1144*A*exp(3* I*(f*x+e))+24024*I*A*exp(8*I*(f*x+e))-2574*B*exp(3*I*(f*x+e))+8580*I*B*exp (4*I*(f*x+e))-52*A*exp(I*(f*x+e))-54912*I*A*exp(6*I*(f*x+e))+117*B*exp(I*( f*x+e)))/f/c^7/(exp(I*(f*x+e))-I)^13
Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (192) = 384\).
Time = 0.27 (sec) , antiderivative size = 475, normalized size of antiderivative = 2.41 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\frac {2 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{7} - 12 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{6} - 49 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} + 70 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} + 105 \, {\left (7 \, A + 20 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} + 105 \, {\left (25 \, A + 51 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2310 \, {\left (A + B\right )} a^{2} \cos \left (f x + e\right ) - 4620 \, {\left (A + B\right )} a^{2} + {\left (2 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{6} + 14 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} - 35 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} - 105 \, {\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} + 105 \, {\left (3 \, A + 29 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2310 \, {\left (A + B\right )} a^{2} \cos \left (f x + e\right ) - 4620 \, {\left (A + B\right )} a^{2}\right )} \sin \left (f x + e\right )}{15015 \, {\left (c^{7} f \cos \left (f x + e\right )^{7} + 7 \, c^{7} f \cos \left (f x + e\right )^{6} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} - 56 \, c^{7} f \cos \left (f x + e\right )^{4} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} + 112 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f - {\left (c^{7} f \cos \left (f x + e\right )^{6} - 6 \, c^{7} f \cos \left (f x + e\right )^{5} - 24 \, c^{7} f \cos \left (f x + e\right )^{4} + 32 \, c^{7} f \cos \left (f x + e\right )^{3} + 80 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f\right )} \sin \left (f x + e\right )\right )}} \]
1/15015*(2*(4*A - 9*B)*a^2*cos(f*x + e)^7 - 12*(4*A - 9*B)*a^2*cos(f*x + e )^6 - 49*(4*A - 9*B)*a^2*cos(f*x + e)^5 + 70*(4*A - 9*B)*a^2*cos(f*x + e)^ 4 + 105*(7*A + 20*B)*a^2*cos(f*x + e)^3 + 105*(25*A + 51*B)*a^2*cos(f*x + e)^2 - 2310*(A + B)*a^2*cos(f*x + e) - 4620*(A + B)*a^2 + (2*(4*A - 9*B)*a ^2*cos(f*x + e)^6 + 14*(4*A - 9*B)*a^2*cos(f*x + e)^5 - 35*(4*A - 9*B)*a^2 *cos(f*x + e)^4 - 105*(4*A - 9*B)*a^2*cos(f*x + e)^3 + 105*(3*A + 29*B)*a^ 2*cos(f*x + e)^2 - 2310*(A + B)*a^2*cos(f*x + e) - 4620*(A + B)*a^2)*sin(f *x + e))/(c^7*f*cos(f*x + e)^7 + 7*c^7*f*cos(f*x + e)^6 - 18*c^7*f*cos(f*x + e)^5 - 56*c^7*f*cos(f*x + e)^4 + 48*c^7*f*cos(f*x + e)^3 + 112*c^7*f*co s(f*x + e)^2 - 32*c^7*f*cos(f*x + e) - 64*c^7*f - (c^7*f*cos(f*x + e)^6 - 6*c^7*f*cos(f*x + e)^5 - 24*c^7*f*cos(f*x + e)^4 + 32*c^7*f*cos(f*x + e)^3 + 80*c^7*f*cos(f*x + e)^2 - 32*c^7*f*cos(f*x + e) - 64*c^7*f)*sin(f*x + e ))
Leaf count of result is larger than twice the leaf count of optimal. 6669 vs. \(2 (178) = 356\).
Time = 71.06 (sec) , antiderivative size = 6669, normalized size of antiderivative = 33.85 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\text {Too large to display} \]
Piecewise((-30030*A*a**2*tan(e/2 + f*x/2)**12/(15015*c**7*f*tan(e/2 + f*x/ 2)**13 - 195195*c**7*f*tan(e/2 + f*x/2)**12 + 1171170*c**7*f*tan(e/2 + f*x /2)**11 - 4294290*c**7*f*tan(e/2 + f*x/2)**10 + 10735725*c**7*f*tan(e/2 + f*x/2)**9 - 19324305*c**7*f*tan(e/2 + f*x/2)**8 + 25765740*c**7*f*tan(e/2 + f*x/2)**7 - 25765740*c**7*f*tan(e/2 + f*x/2)**6 + 19324305*c**7*f*tan(e/ 2 + f*x/2)**5 - 10735725*c**7*f*tan(e/2 + f*x/2)**4 + 4294290*c**7*f*tan(e /2 + f*x/2)**3 - 1171170*c**7*f*tan(e/2 + f*x/2)**2 + 195195*c**7*f*tan(e/ 2 + f*x/2) - 15015*c**7*f) + 120120*A*a**2*tan(e/2 + f*x/2)**11/(15015*c** 7*f*tan(e/2 + f*x/2)**13 - 195195*c**7*f*tan(e/2 + f*x/2)**12 + 1171170*c* *7*f*tan(e/2 + f*x/2)**11 - 4294290*c**7*f*tan(e/2 + f*x/2)**10 + 10735725 *c**7*f*tan(e/2 + f*x/2)**9 - 19324305*c**7*f*tan(e/2 + f*x/2)**8 + 257657 40*c**7*f*tan(e/2 + f*x/2)**7 - 25765740*c**7*f*tan(e/2 + f*x/2)**6 + 1932 4305*c**7*f*tan(e/2 + f*x/2)**5 - 10735725*c**7*f*tan(e/2 + f*x/2)**4 + 42 94290*c**7*f*tan(e/2 + f*x/2)**3 - 1171170*c**7*f*tan(e/2 + f*x/2)**2 + 19 5195*c**7*f*tan(e/2 + f*x/2) - 15015*c**7*f) - 540540*A*a**2*tan(e/2 + f*x /2)**10/(15015*c**7*f*tan(e/2 + f*x/2)**13 - 195195*c**7*f*tan(e/2 + f*x/2 )**12 + 1171170*c**7*f*tan(e/2 + f*x/2)**11 - 4294290*c**7*f*tan(e/2 + f*x /2)**10 + 10735725*c**7*f*tan(e/2 + f*x/2)**9 - 19324305*c**7*f*tan(e/2 + f*x/2)**8 + 25765740*c**7*f*tan(e/2 + f*x/2)**7 - 25765740*c**7*f*tan(e/2 + f*x/2)**6 + 19324305*c**7*f*tan(e/2 + f*x/2)**5 - 10735725*c**7*f*tan...
Leaf count of result is larger than twice the leaf count of optimal. 3120 vs. \(2 (192) = 384\).
Time = 0.34 (sec) , antiderivative size = 3120, normalized size of antiderivative = 15.84 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=\text {Too large to display} \]
-2/45045*(2*A*a^2*(4771*sin(f*x + e)/(cos(f*x + e) + 1) - 28626*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 74932*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 18 7330*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 265122*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 353496*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 276276*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 207207*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 75075*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 30030*sin(f*x + e)^10/(cos(f* x + e) + 1)^10 - 367)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c ^7*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*sin(f* x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1) ^6 - 1716*c^7*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^ 8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 286 *c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f *x + e) + 1)^11 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f *x + e)^13/(cos(f*x + e) + 1)^13) + 4*B*a^2*(4771*sin(f*x + e)/(cos(f*x + e) + 1) - 28626*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 74932*sin(f*x + e)^3 /(cos(f*x + e) + 1)^3 - 187330*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 26512 2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 353496*sin(f*x + e)^6/(cos(f*x + e ) + 1)^6 + 276276*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 207207*sin(f*x + e )^8/(cos(f*x + e) + 1)^8 + 75075*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - ...
Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (192) = 384\).
Time = 0.40 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.14 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=-\frac {2 \, {\left (15015 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} - 60060 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 15015 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 270270 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 15015 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 600600 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 105105 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 1174173 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 93093 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 1465464 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 234234 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 1559844 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 131274 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1094808 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 181038 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 659945 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 47190 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 233948 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 45903 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 77454 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1599 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7904 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2769 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1763 \, A a^{2} - 213 \, B a^{2}\right )}}{15015 \, c^{7} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{13}} \]
-2/15015*(15015*A*a^2*tan(1/2*f*x + 1/2*e)^12 - 60060*A*a^2*tan(1/2*f*x + 1/2*e)^11 + 15015*B*a^2*tan(1/2*f*x + 1/2*e)^11 + 270270*A*a^2*tan(1/2*f*x + 1/2*e)^10 - 15015*B*a^2*tan(1/2*f*x + 1/2*e)^10 - 600600*A*a^2*tan(1/2* f*x + 1/2*e)^9 + 105105*B*a^2*tan(1/2*f*x + 1/2*e)^9 + 1174173*A*a^2*tan(1 /2*f*x + 1/2*e)^8 - 93093*B*a^2*tan(1/2*f*x + 1/2*e)^8 - 1465464*A*a^2*tan (1/2*f*x + 1/2*e)^7 + 234234*B*a^2*tan(1/2*f*x + 1/2*e)^7 + 1559844*A*a^2* tan(1/2*f*x + 1/2*e)^6 - 131274*B*a^2*tan(1/2*f*x + 1/2*e)^6 - 1094808*A*a ^2*tan(1/2*f*x + 1/2*e)^5 + 181038*B*a^2*tan(1/2*f*x + 1/2*e)^5 + 659945*A *a^2*tan(1/2*f*x + 1/2*e)^4 - 47190*B*a^2*tan(1/2*f*x + 1/2*e)^4 - 233948* A*a^2*tan(1/2*f*x + 1/2*e)^3 + 45903*B*a^2*tan(1/2*f*x + 1/2*e)^3 + 77454* A*a^2*tan(1/2*f*x + 1/2*e)^2 - 1599*B*a^2*tan(1/2*f*x + 1/2*e)^2 - 7904*A* a^2*tan(1/2*f*x + 1/2*e) + 2769*B*a^2*tan(1/2*f*x + 1/2*e) + 1763*A*a^2 - 213*B*a^2)/(c^7*f*(tan(1/2*f*x + 1/2*e) - 1)^13)
Time = 14.89 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.54 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx=-\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {994249\,A\,a^2}{32}-\frac {63639\,B\,a^2}{32}-\frac {1609013\,A\,a^2\,\cos \left (2\,e+2\,f\,x\right )}{64}+\frac {85687\,A\,a^2\,\cos \left (3\,e+3\,f\,x\right )}{16}+\frac {79591\,A\,a^2\,\cos \left (4\,e+4\,f\,x\right )}{32}-\frac {5261\,A\,a^2\,\cos \left (5\,e+5\,f\,x\right )}{16}-\frac {1771\,A\,a^2\,\cos \left (6\,e+6\,f\,x\right )}{64}+\frac {140553\,B\,a^2\,\cos \left (2\,e+2\,f\,x\right )}{64}-\frac {4431\,B\,a^2\,\cos \left (3\,e+3\,f\,x\right )}{8}-\frac {10161\,B\,a^2\,\cos \left (4\,e+4\,f\,x\right )}{32}+36\,B\,a^2\,\cos \left (5\,e+5\,f\,x\right )+\frac {231\,B\,a^2\,\cos \left (6\,e+6\,f\,x\right )}{64}+\frac {636207\,A\,a^2\,\sin \left (2\,e+2\,f\,x\right )}{64}+\frac {309309\,A\,a^2\,\sin \left (3\,e+3\,f\,x\right )}{32}-\frac {7007\,A\,a^2\,\sin \left (4\,e+4\,f\,x\right )}{4}-\frac {12389\,A\,a^2\,\sin \left (5\,e+5\,f\,x\right )}{32}+\frac {1755\,A\,a^2\,\sin \left (6\,e+6\,f\,x\right )}{64}-\frac {121407\,B\,a^2\,\sin \left (2\,e+2\,f\,x\right )}{64}-\frac {39039\,B\,a^2\,\sin \left (3\,e+3\,f\,x\right )}{32}+\frac {3003\,B\,a^2\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {1599\,B\,a^2\,\sin \left (5\,e+5\,f\,x\right )}{32}-\frac {195\,B\,a^2\,\sin \left (6\,e+6\,f\,x\right )}{64}-\frac {93221\,A\,a^2\,\cos \left (e+f\,x\right )}{8}+\frac {3291\,B\,a^2\,\cos \left (e+f\,x\right )}{8}-\frac {704847\,A\,a^2\,\sin \left (e+f\,x\right )}{16}+\frac {125697\,B\,a^2\,\sin \left (e+f\,x\right )}{16}\right )}{15015\,c^7\,f\,\left (\frac {1287\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}-\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{64}-\frac {429\,\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}{16}+\frac {715\,\sqrt {2}\,\cos \left (\frac {5\,e}{2}+\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{64}-\frac {143\,\sqrt {2}\,\cos \left (\frac {7\,e}{2}-\frac {\pi }{4}+\frac {7\,f\,x}{2}\right )}{32}-\frac {39\,\sqrt {2}\,\cos \left (\frac {9\,e}{2}+\frac {\pi }{4}+\frac {9\,f\,x}{2}\right )}{32}+\frac {13\,\sqrt {2}\,\cos \left (\frac {11\,e}{2}-\frac {\pi }{4}+\frac {11\,f\,x}{2}\right )}{64}+\frac {\sqrt {2}\,\cos \left (\frac {13\,e}{2}+\frac {\pi }{4}+\frac {13\,f\,x}{2}\right )}{64}\right )} \]
-(2*cos(e/2 + (f*x)/2)*((994249*A*a^2)/32 - (63639*B*a^2)/32 - (1609013*A* a^2*cos(2*e + 2*f*x))/64 + (85687*A*a^2*cos(3*e + 3*f*x))/16 + (79591*A*a^ 2*cos(4*e + 4*f*x))/32 - (5261*A*a^2*cos(5*e + 5*f*x))/16 - (1771*A*a^2*co s(6*e + 6*f*x))/64 + (140553*B*a^2*cos(2*e + 2*f*x))/64 - (4431*B*a^2*cos( 3*e + 3*f*x))/8 - (10161*B*a^2*cos(4*e + 4*f*x))/32 + 36*B*a^2*cos(5*e + 5 *f*x) + (231*B*a^2*cos(6*e + 6*f*x))/64 + (636207*A*a^2*sin(2*e + 2*f*x))/ 64 + (309309*A*a^2*sin(3*e + 3*f*x))/32 - (7007*A*a^2*sin(4*e + 4*f*x))/4 - (12389*A*a^2*sin(5*e + 5*f*x))/32 + (1755*A*a^2*sin(6*e + 6*f*x))/64 - ( 121407*B*a^2*sin(2*e + 2*f*x))/64 - (39039*B*a^2*sin(3*e + 3*f*x))/32 + (3 003*B*a^2*sin(4*e + 4*f*x))/16 + (1599*B*a^2*sin(5*e + 5*f*x))/32 - (195*B *a^2*sin(6*e + 6*f*x))/64 - (93221*A*a^2*cos(e + f*x))/8 + (3291*B*a^2*cos (e + f*x))/8 - (704847*A*a^2*sin(e + f*x))/16 + (125697*B*a^2*sin(e + f*x) )/16))/(15015*c^7*f*((1287*2^(1/2)*cos((3*e)/2 - pi/4 + (3*f*x)/2))/64 - ( 429*2^(1/2)*cos(e/2 + pi/4 + (f*x)/2))/16 + (715*2^(1/2)*cos((5*e)/2 + pi/ 4 + (5*f*x)/2))/64 - (143*2^(1/2)*cos((7*e)/2 - pi/4 + (7*f*x)/2))/32 - (3 9*2^(1/2)*cos((9*e)/2 + pi/4 + (9*f*x)/2))/32 + (13*2^(1/2)*cos((11*e)/2 - pi/4 + (11*f*x)/2))/64 + (2^(1/2)*cos((13*e)/2 + pi/4 + (13*f*x)/2))/64))