Integrand size = 36, antiderivative size = 205 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {(A+B) \sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {(8 A-3 B) \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {2 (8 A-3 B) \tan (e+f x)}{33 a^3 c^6 f}+\frac {4 (8 A-3 B) \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac {2 (8 A-3 B) \tan ^5(e+f x)}{165 a^3 c^6 f} \] Output:
1/11*(A+B)*sec(f*x+e)^5/a^3/f/(c^2-c^2*sin(f*x+e))^3+1/99*(8*A-3*B)*sec(f* x+e)^5/a^3/f/(c^3-c^3*sin(f*x+e))^2+1/99*(8*A-3*B)*sec(f*x+e)^5/a^3/f/(c^6 -c^6*sin(f*x+e))+2/33*(8*A-3*B)*tan(f*x+e)/a^3/c^6/f+4/99*(8*A-3*B)*tan(f* x+e)^3/a^3/c^6/f+2/165*(8*A-3*B)*tan(f*x+e)^5/a^3/c^6/f
Time = 12.85 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.96 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {1013760 B-3850 (107 A-3 B) \cos (e+f x)+135168 (8 A-3 B) \cos (2 (e+f x))-127330 A \cos (3 (e+f x))+3570 B \cos (3 (e+f x))+819200 A \cos (4 (e+f x))-307200 B \cos (4 (e+f x))+37450 A \cos (5 (e+f x))-1050 B \cos (5 (e+f x))+163840 A \cos (6 (e+f x))-61440 B \cos (6 (e+f x))+22470 A \cos (7 (e+f x))-630 B \cos (7 (e+f x))-16384 A \cos (8 (e+f x))+6144 B \cos (8 (e+f x))+1802240 A \sin (e+f x)-675840 B \sin (e+f x)+247170 A \sin (2 (e+f x))-6930 B \sin (2 (e+f x))+557056 A \sin (3 (e+f x))-208896 B \sin (3 (e+f x))+187250 A \sin (4 (e+f x))-5250 B \sin (4 (e+f x))-163840 A \sin (5 (e+f x))+61440 B \sin (5 (e+f x))+37450 A \sin (6 (e+f x))-1050 B \sin (6 (e+f x))-98304 A \sin (7 (e+f x))+36864 B \sin (7 (e+f x))-3745 A \sin (8 (e+f x))+105 B \sin (8 (e+f x))}{8110080 a^3 c^6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11} \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \] Input:
Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x] )^6),x]
Output:
(1013760*B - 3850*(107*A - 3*B)*Cos[e + f*x] + 135168*(8*A - 3*B)*Cos[2*(e + f*x)] - 127330*A*Cos[3*(e + f*x)] + 3570*B*Cos[3*(e + f*x)] + 819200*A* Cos[4*(e + f*x)] - 307200*B*Cos[4*(e + f*x)] + 37450*A*Cos[5*(e + f*x)] - 1050*B*Cos[5*(e + f*x)] + 163840*A*Cos[6*(e + f*x)] - 61440*B*Cos[6*(e + f *x)] + 22470*A*Cos[7*(e + f*x)] - 630*B*Cos[7*(e + f*x)] - 16384*A*Cos[8*( e + f*x)] + 6144*B*Cos[8*(e + f*x)] + 1802240*A*Sin[e + f*x] - 675840*B*Si n[e + f*x] + 247170*A*Sin[2*(e + f*x)] - 6930*B*Sin[2*(e + f*x)] + 557056* A*Sin[3*(e + f*x)] - 208896*B*Sin[3*(e + f*x)] + 187250*A*Sin[4*(e + f*x)] - 5250*B*Sin[4*(e + f*x)] - 163840*A*Sin[5*(e + f*x)] + 61440*B*Sin[5*(e + f*x)] + 37450*A*Sin[6*(e + f*x)] - 1050*B*Sin[6*(e + f*x)] - 98304*A*Sin [7*(e + f*x)] + 36864*B*Sin[7*(e + f*x)] - 3745*A*Sin[8*(e + f*x)] + 105*B *Sin[8*(e + f*x)])/(8110080*a^3*c^6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2] )^11*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)
Time = 0.86 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.79, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {3042, 3446, 3042, 3338, 3042, 3151, 3042, 3151, 3042, 4254, 2009}
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.
| \(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin (e+f x)}{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^6}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle \frac {\int \frac {\sec ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {A+B \sin (e+f x)}{\cos (e+f x)^6 (c-c \sin (e+f x))^3}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle \frac {\frac {(8 A-3 B) \int \frac {\sec ^6(e+f x)}{(c-c \sin (e+f x))^2}dx}{11 c}+\frac {(A+B) \sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(8 A-3 B) \int \frac {1}{\cos (e+f x)^6 (c-c \sin (e+f x))^2}dx}{11 c}+\frac {(A+B) \sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {(8 A-3 B) \left (\frac {7 \int \frac {\sec ^6(e+f x)}{c-c \sin (e+f x)}dx}{9 c}+\frac {\sec ^5(e+f x)}{9 f (c-c \sin (e+f x))^2}\right )}{11 c}+\frac {(A+B) \sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(8 A-3 B) \left (\frac {7 \int \frac {1}{\cos (e+f x)^6 (c-c \sin (e+f x))}dx}{9 c}+\frac {\sec ^5(e+f x)}{9 f (c-c \sin (e+f x))^2}\right )}{11 c}+\frac {(A+B) \sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle \frac {\frac {(8 A-3 B) \left (\frac {7 \left (\frac {6 \int \sec ^6(e+f x)dx}{7 c}+\frac {\sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}\right )}{9 c}+\frac {\sec ^5(e+f x)}{9 f (c-c \sin (e+f x))^2}\right )}{11 c}+\frac {(A+B) \sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {(8 A-3 B) \left (\frac {7 \left (\frac {6 \int \csc \left (e+f x+\frac {\pi }{2}\right )^6dx}{7 c}+\frac {\sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}\right )}{9 c}+\frac {\sec ^5(e+f x)}{9 f (c-c \sin (e+f x))^2}\right )}{11 c}+\frac {(A+B) \sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {(8 A-3 B) \left (\frac {7 \left (\frac {\sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}-\frac {6 \int \left (\tan ^4(e+f x)+2 \tan ^2(e+f x)+1\right )d(-\tan (e+f x))}{7 c f}\right )}{9 c}+\frac {\sec ^5(e+f x)}{9 f (c-c \sin (e+f x))^2}\right )}{11 c}+\frac {(A+B) \sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}}{a^3 c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(A+B) \sec ^5(e+f x)}{11 f (c-c \sin (e+f x))^3}+\frac {(8 A-3 B) \left (\frac {\sec ^5(e+f x)}{9 f (c-c \sin (e+f x))^2}+\frac {7 \left (\frac {\sec ^5(e+f x)}{7 f (c-c \sin (e+f x))}-\frac {6 \left (-\frac {1}{5} \tan ^5(e+f x)-\frac {2}{3} \tan ^3(e+f x)-\tan (e+f x)\right )}{7 c f}\right )}{9 c}\right )}{11 c}}{a^3 c^3}\) |
Input:
Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x ]
Output:
(((A + B)*Sec[e + f*x]^5)/(11*f*(c - c*Sin[e + f*x])^3) + ((8*A - 3*B)*(Se c[e + f*x]^5/(9*f*(c - c*Sin[e + f*x])^2) + (7*(Sec[e + f*x]^5/(7*f*(c - c *Sin[e + f*x])) - (6*(-Tan[e + f*x] - (2*Tan[e + f*x]^3)/3 - Tan[e + f*x]^ 5/5))/(7*c*f)))/(9*c)))/(11*c))/(a^3*c^3)
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]))
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Result contains complex when optimal does not.
Time = 4.31 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(\frac {32 i \left (18 i B \,{\mathrm e}^{i \left (f x +e \right )}-80 i A \,{\mathrm e}^{3 i \left (f x +e \right )}+495 B \,{\mathrm e}^{8 i \left (f x +e \right )}+880 i A \,{\mathrm e}^{7 i \left (f x +e \right )}+528 A \,{\mathrm e}^{6 i \left (f x +e \right )}+30 i B \,{\mathrm e}^{3 i \left (f x +e \right )}-198 B \,{\mathrm e}^{6 i \left (f x +e \right )}-102 i B \,{\mathrm e}^{5 i \left (f x +e \right )}+400 A \,{\mathrm e}^{4 i \left (f x +e \right )}+272 i A \,{\mathrm e}^{5 i \left (f x +e \right )}-150 B \,{\mathrm e}^{4 i \left (f x +e \right )}-48 i A \,{\mathrm e}^{i \left (f x +e \right )}+80 A \,{\mathrm e}^{2 i \left (f x +e \right )}-330 i B \,{\mathrm e}^{7 i \left (f x +e \right )}-30 B \,{\mathrm e}^{2 i \left (f x +e \right )}-8 A +3 B \right )}{495 \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{11} f \,a^{3} c^{6}}\) | \(236\) |
| parallelrisch | \(\frac {-990 A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{15}+\left (2970 A -990 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}+\left (-3630 A +1980 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}+\left (-4950 A -2970 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}+\left (9834 A -1584 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}+\left (66 A +594 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}+\left (-23430 A +1980 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+\left (17490 A -10890 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+\left (4070 A +5280 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+\left (-16434 A -1386 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\left (1334 A -2604 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (7550 A -1470 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (-6130 A +1680 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+\left (470 A -1290 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+\left (510 A +180 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-250 A -30 B}{495 f \,a^{3} c^{6} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\) | \(323\) |
| derivativedivides | \(\frac {-\frac {-\frac {5 A}{32}+\frac {B}{8}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-\frac {A}{8}+\frac {B}{8}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (\frac {A}{16}-\frac {B}{16}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 \left (\frac {37 A}{256}-\frac {21 B}{256}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (\frac {7 A}{32}-\frac {3 B}{16}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (4 A +4 B \right )}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {20 A +20 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {2 \left (53 A +51 B \right )}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {92 A +84 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {\frac {169 A}{4}+\frac {99 B}{4}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {\frac {217 A}{2}+84 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {2 \left (\frac {219 A}{256}+\frac {21 B}{256}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {2 \left (\frac {231 A}{2}+98 B \right )}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {\frac {303 A}{64}+\frac {99 B}{64}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {623 A}{8}+\frac {427 B}{8}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2 \left (\frac {1095 A}{64}+\frac {507 B}{64}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}}{f \,a^{3} c^{6}}\) | \(365\) |
| default | \(\frac {-\frac {-\frac {5 A}{32}+\frac {B}{8}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-\frac {A}{8}+\frac {B}{8}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (\frac {A}{16}-\frac {B}{16}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 \left (\frac {37 A}{256}-\frac {21 B}{256}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (\frac {7 A}{32}-\frac {3 B}{16}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {2 \left (4 A +4 B \right )}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {20 A +20 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {2 \left (53 A +51 B \right )}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {92 A +84 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {\frac {169 A}{4}+\frac {99 B}{4}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {\frac {217 A}{2}+84 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {2 \left (\frac {219 A}{256}+\frac {21 B}{256}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {2 \left (\frac {231 A}{2}+98 B \right )}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {\frac {303 A}{64}+\frac {99 B}{64}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {623 A}{8}+\frac {427 B}{8}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2 \left (\frac {1095 A}{64}+\frac {507 B}{64}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}}{f \,a^{3} c^{6}}\) | \(365\) |
Input:
int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x,method=_RETUR NVERBOSE)
Output:
32/495*I*(18*I*B*exp(I*(f*x+e))-80*I*A*exp(3*I*(f*x+e))+495*B*exp(8*I*(f*x +e))+880*I*A*exp(7*I*(f*x+e))+528*A*exp(6*I*(f*x+e))+30*I*B*exp(3*I*(f*x+e ))-198*B*exp(6*I*(f*x+e))-102*I*B*exp(5*I*(f*x+e))+400*A*exp(4*I*(f*x+e))+ 272*I*A*exp(5*I*(f*x+e))-150*B*exp(4*I*(f*x+e))-48*I*A*exp(I*(f*x+e))+80*A *exp(2*I*(f*x+e))-330*I*B*exp(7*I*(f*x+e))-30*B*exp(2*I*(f*x+e))-8*A+3*B)/ (exp(I*(f*x+e))+I)^5/(exp(I*(f*x+e))-I)^11/f/a^3/c^6
Time = 0.09 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.08 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {16 \, {\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{8} - 72 \, {\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{6} + 30 \, {\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{4} + 7 \, {\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{2} + {\left (48 \, {\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{6} - 40 \, {\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{4} - 14 \, {\left (8 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{2} - 72 \, A + 27 \, B\right )} \sin \left (f x + e\right ) + 27 \, A - 72 \, B}{495 \, {\left (3 \, a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5} - {\left (a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5}\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algori thm="fricas")
Output:
1/495*(16*(8*A - 3*B)*cos(f*x + e)^8 - 72*(8*A - 3*B)*cos(f*x + e)^6 + 30* (8*A - 3*B)*cos(f*x + e)^4 + 7*(8*A - 3*B)*cos(f*x + e)^2 + (48*(8*A - 3*B )*cos(f*x + e)^6 - 40*(8*A - 3*B)*cos(f*x + e)^4 - 14*(8*A - 3*B)*cos(f*x + e)^2 - 72*A + 27*B)*sin(f*x + e) + 27*A - 72*B)/(3*a^3*c^6*f*cos(f*x + e )^7 - 4*a^3*c^6*f*cos(f*x + e)^5 - (a^3*c^6*f*cos(f*x + e)^7 - 4*a^3*c^6*f *cos(f*x + e)^5)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 11011 vs. \(2 (187) = 374\).
Time = 103.02 (sec) , antiderivative size = 11011, normalized size of antiderivative = 53.71 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**6,x)
Output:
Piecewise((-990*A*tan(e/2 + f*x/2)**15/(495*a**3*c**6*f*tan(e/2 + f*x/2)** 16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f* x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e /2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6 *f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a* *3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16 830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)** 4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/ 2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 2970*A*tan( e/2 + f*x/2)**14/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f* tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c **6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 1683 0*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**1 0 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f* x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/ 2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*t an(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6 *f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) - 3630*A*tan(e/2 + f*x/2)**13/(495* a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x...
Leaf count of result is larger than twice the leaf count of optimal. 1387 vs. \(2 (196) = 392\).
Time = 0.09 (sec) , antiderivative size = 1387, normalized size of antiderivative = 6.77 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algori thm="maxima")
Output:
-2/495*(A*(255*sin(f*x + e)/(cos(f*x + e) + 1) + 235*sin(f*x + e)^2/(cos(f *x + e) + 1)^2 - 3065*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3775*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 667*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 821 7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2035*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 8745*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 11715*sin(f*x + e)^9/( cos(f*x + e) + 1)^9 + 33*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 4917*sin( f*x + e)^11/(cos(f*x + e) + 1)^11 - 2475*sin(f*x + e)^12/(cos(f*x + e) + 1 )^12 - 1815*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 + 1485*sin(f*x + e)^14/( cos(f*x + e) + 1)^14 - 495*sin(f*x + e)^15/(cos(f*x + e) + 1)^15 - 125)/(a ^3*c^6 - 6*a^3*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 50*a^3*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 34*a^3*c^6*sin(f*x + e )^5/(cos(f*x + e) + 1)^5 + 66*a^3*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 110*a^3*c^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 110*a^3*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 66*a^3*c^6*sin(f*x + e)^10/(cos(f*x + e) + 1)^ 10 - 34*a^3*c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 50*a^3*c^6*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 10*a^3*c^6*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 10*a^3*c^6*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 + 6*a^3*c^6*sin (f*x + e)^15/(cos(f*x + e) + 1)^15 - a^3*c^6*sin(f*x + e)^16/(cos(f*x + e) + 1)^16) + 3*B*(30*sin(f*x + e)/(cos(f*x + e) + 1) - 215*sin(f*x + e)^...
Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (196) = 392\).
Time = 0.28 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.17 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx =\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x, algori thm="giac")
Output:
-1/63360*(33*(555*A*tan(1/2*f*x + 1/2*e)^4 - 315*B*tan(1/2*f*x + 1/2*e)^4 + 1920*A*tan(1/2*f*x + 1/2*e)^3 - 1020*B*tan(1/2*f*x + 1/2*e)^3 + 2710*A*t an(1/2*f*x + 1/2*e)^2 - 1410*B*tan(1/2*f*x + 1/2*e)^2 + 1760*A*tan(1/2*f*x + 1/2*e) - 900*B*tan(1/2*f*x + 1/2*e) + 463*A - 243*B)/(a^3*c^6*(tan(1/2* f*x + 1/2*e) + 1)^5) + (108405*A*tan(1/2*f*x + 1/2*e)^10 + 10395*B*tan(1/2 *f*x + 1/2*e)^10 - 784080*A*tan(1/2*f*x + 1/2*e)^9 - 5940*B*tan(1/2*f*x + 1/2*e)^9 + 2901195*A*tan(1/2*f*x + 1/2*e)^8 - 79695*B*tan(1/2*f*x + 1/2*e) ^8 - 6652800*A*tan(1/2*f*x + 1/2*e)^7 + 388080*B*tan(1/2*f*x + 1/2*e)^7 + 10407474*A*tan(1/2*f*x + 1/2*e)^6 - 816354*B*tan(1/2*f*x + 1/2*e)^6 - 1143 5424*A*tan(1/2*f*x + 1/2*e)^5 + 1114344*B*tan(1/2*f*x + 1/2*e)^5 + 8949270 *A*tan(1/2*f*x + 1/2*e)^4 - 990990*B*tan(1/2*f*x + 1/2*e)^4 - 4899840*A*ta n(1/2*f*x + 1/2*e)^3 + 609840*B*tan(1/2*f*x + 1/2*e)^3 + 1816265*A*tan(1/2 *f*x + 1/2*e)^2 - 235785*B*tan(1/2*f*x + 1/2*e)^2 - 411664*A*tan(1/2*f*x + 1/2*e) + 56364*B*tan(1/2*f*x + 1/2*e) + 47279*A - 4179*B)/(a^3*c^6*(tan(1 /2*f*x + 1/2*e) - 1)^11))/f
Time = 39.29 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.31 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {2\,\left (\frac {165\,B\,\sin \left (e+f\,x\right )}{4}-\frac {6875\,A\,\cos \left (e+f\,x\right )}{64}-\frac {825\,B\,\cos \left (e+f\,x\right )}{64}-110\,A\,\sin \left (e+f\,x\right )-\frac {495\,B}{8}-66\,A\,\cos \left (2\,e+2\,f\,x\right )-\frac {2125\,A\,\cos \left (3\,e+3\,f\,x\right )}{64}-50\,A\,\cos \left (4\,e+4\,f\,x\right )+\frac {625\,A\,\cos \left (5\,e+5\,f\,x\right )}{64}-10\,A\,\cos \left (6\,e+6\,f\,x\right )+\frac {375\,A\,\cos \left (7\,e+7\,f\,x\right )}{64}+A\,\cos \left (8\,e+8\,f\,x\right )+\frac {99\,B\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {255\,B\,\cos \left (3\,e+3\,f\,x\right )}{64}+\frac {75\,B\,\cos \left (4\,e+4\,f\,x\right )}{4}+\frac {75\,B\,\cos \left (5\,e+5\,f\,x\right )}{64}+\frac {15\,B\,\cos \left (6\,e+6\,f\,x\right )}{4}+\frac {45\,B\,\cos \left (7\,e+7\,f\,x\right )}{64}-\frac {3\,B\,\cos \left (8\,e+8\,f\,x\right )}{8}+\frac {4125\,A\,\sin \left (2\,e+2\,f\,x\right )}{64}-34\,A\,\sin \left (3\,e+3\,f\,x\right )+\frac {3125\,A\,\sin \left (4\,e+4\,f\,x\right )}{64}+10\,A\,\sin \left (5\,e+5\,f\,x\right )+\frac {625\,A\,\sin \left (6\,e+6\,f\,x\right )}{64}+6\,A\,\sin \left (7\,e+7\,f\,x\right )-\frac {125\,A\,\sin \left (8\,e+8\,f\,x\right )}{128}+\frac {495\,B\,\sin \left (2\,e+2\,f\,x\right )}{64}+\frac {51\,B\,\sin \left (3\,e+3\,f\,x\right )}{4}+\frac {375\,B\,\sin \left (4\,e+4\,f\,x\right )}{64}-\frac {15\,B\,\sin \left (5\,e+5\,f\,x\right )}{4}+\frac {75\,B\,\sin \left (6\,e+6\,f\,x\right )}{64}-\frac {9\,B\,\sin \left (7\,e+7\,f\,x\right )}{4}-\frac {15\,B\,\sin \left (8\,e+8\,f\,x\right )}{128}\right )}{495\,a^3\,c^6\,f\,\left (\frac {5\,\cos \left (5\,e+5\,f\,x\right )}{32}-\frac {17\,\cos \left (3\,e+3\,f\,x\right )}{32}-\frac {55\,\cos \left (e+f\,x\right )}{32}+\frac {3\,\cos \left (7\,e+7\,f\,x\right )}{32}+\frac {33\,\sin \left (2\,e+2\,f\,x\right )}{32}+\frac {25\,\sin \left (4\,e+4\,f\,x\right )}{32}+\frac {5\,\sin \left (6\,e+6\,f\,x\right )}{32}-\frac {\sin \left (8\,e+8\,f\,x\right )}{64}\right )} \] Input:
int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^6),x )
Output:
(2*((165*B*sin(e + f*x))/4 - (6875*A*cos(e + f*x))/64 - (825*B*cos(e + f*x ))/64 - 110*A*sin(e + f*x) - (495*B)/8 - 66*A*cos(2*e + 2*f*x) - (2125*A*c os(3*e + 3*f*x))/64 - 50*A*cos(4*e + 4*f*x) + (625*A*cos(5*e + 5*f*x))/64 - 10*A*cos(6*e + 6*f*x) + (375*A*cos(7*e + 7*f*x))/64 + A*cos(8*e + 8*f*x) + (99*B*cos(2*e + 2*f*x))/4 - (255*B*cos(3*e + 3*f*x))/64 + (75*B*cos(4*e + 4*f*x))/4 + (75*B*cos(5*e + 5*f*x))/64 + (15*B*cos(6*e + 6*f*x))/4 + (4 5*B*cos(7*e + 7*f*x))/64 - (3*B*cos(8*e + 8*f*x))/8 + (4125*A*sin(2*e + 2* f*x))/64 - 34*A*sin(3*e + 3*f*x) + (3125*A*sin(4*e + 4*f*x))/64 + 10*A*sin (5*e + 5*f*x) + (625*A*sin(6*e + 6*f*x))/64 + 6*A*sin(7*e + 7*f*x) - (125* A*sin(8*e + 8*f*x))/128 + (495*B*sin(2*e + 2*f*x))/64 + (51*B*sin(3*e + 3* f*x))/4 + (375*B*sin(4*e + 4*f*x))/64 - (15*B*sin(5*e + 5*f*x))/4 + (75*B* sin(6*e + 6*f*x))/64 - (9*B*sin(7*e + 7*f*x))/4 - (15*B*sin(8*e + 8*f*x))/ 128))/(495*a^3*c^6*f*((5*cos(5*e + 5*f*x))/32 - (17*cos(3*e + 3*f*x))/32 - (55*cos(e + f*x))/32 + (3*cos(7*e + 7*f*x))/32 + (33*sin(2*e + 2*f*x))/32 + (25*sin(4*e + 4*f*x))/32 + (5*sin(6*e + 6*f*x))/32 - sin(8*e + 8*f*x)/6 4))
Time = 0.18 (sec) , antiderivative size = 518, normalized size of antiderivative = 2.53 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx=\frac {-125 a -15 b +144 \sin \left (f x +e \right )^{7} b +64 \sin \left (f x +e \right )^{6} a -15 \cos \left (f x +e \right ) b -24 \sin \left (f x +e \right )^{6} b +832 \sin \left (f x +e \right )^{5} a +15 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{7} b +120 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} a -45 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{6} b -40 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} a +45 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b +15 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} b -200 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a -40 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{7} a -384 \sin \left (f x +e \right )^{7} a -15 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b -120 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a -120 a \sin \left (f x +e \right )-75 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b +40 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a +75 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} b +200 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a +270 \sin \left (f x +e \right )^{4} b -400 \sin \left (f x +e \right )^{3} a -312 \sin \left (f x +e \right )^{5} b -720 \sin \left (f x +e \right )^{4} a +40 \cos \left (f x +e \right ) a +150 \sin \left (f x +e \right )^{3} b +680 \sin \left (f x +e \right )^{2} a -255 \sin \left (f x +e \right )^{2} b +128 \sin \left (f x +e \right )^{8} a -48 \sin \left (f x +e \right )^{8} b +45 \sin \left (f x +e \right ) b}{495 \cos \left (f x +e \right ) a^{3} c^{6} f \left (\sin \left (f x +e \right )^{7}-3 \sin \left (f x +e \right )^{6}+\sin \left (f x +e \right )^{5}+5 \sin \left (f x +e \right )^{4}-5 \sin \left (f x +e \right )^{3}-\sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )-1\right )} \] Input:
int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^6,x)
Output:
( - 40*cos(e + f*x)*sin(e + f*x)**7*a + 15*cos(e + f*x)*sin(e + f*x)**7*b + 120*cos(e + f*x)*sin(e + f*x)**6*a - 45*cos(e + f*x)*sin(e + f*x)**6*b - 40*cos(e + f*x)*sin(e + f*x)**5*a + 15*cos(e + f*x)*sin(e + f*x)**5*b - 2 00*cos(e + f*x)*sin(e + f*x)**4*a + 75*cos(e + f*x)*sin(e + f*x)**4*b + 20 0*cos(e + f*x)*sin(e + f*x)**3*a - 75*cos(e + f*x)*sin(e + f*x)**3*b + 40* cos(e + f*x)*sin(e + f*x)**2*a - 15*cos(e + f*x)*sin(e + f*x)**2*b - 120*c os(e + f*x)*sin(e + f*x)*a + 45*cos(e + f*x)*sin(e + f*x)*b + 40*cos(e + f *x)*a - 15*cos(e + f*x)*b + 128*sin(e + f*x)**8*a - 48*sin(e + f*x)**8*b - 384*sin(e + f*x)**7*a + 144*sin(e + f*x)**7*b + 64*sin(e + f*x)**6*a - 24 *sin(e + f*x)**6*b + 832*sin(e + f*x)**5*a - 312*sin(e + f*x)**5*b - 720*s in(e + f*x)**4*a + 270*sin(e + f*x)**4*b - 400*sin(e + f*x)**3*a + 150*sin (e + f*x)**3*b + 680*sin(e + f*x)**2*a - 255*sin(e + f*x)**2*b - 120*sin(e + f*x)*a + 45*sin(e + f*x)*b - 125*a - 15*b)/(495*cos(e + f*x)*a**3*c**6* f*(sin(e + f*x)**7 - 3*sin(e + f*x)**6 + sin(e + f*x)**5 + 5*sin(e + f*x)* *4 - 5*sin(e + f*x)**3 - sin(e + f*x)**2 + 3*sin(e + f*x) - 1))