Integrand size = 36, antiderivative size = 153 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=-\frac {a^3 (A+6 B) x}{c^3}+\frac {a^3 (A+6 B) \cos (e+f x)}{c^3 f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac {2 a^3 (A+6 B) c \cos ^5(e+f x)}{15 f (c-c \sin (e+f x))^4}+\frac {2 a^3 (A+6 B) c^3 \cos ^3(e+f x)}{3 f \left (c^3-c^3 \sin (e+f x)\right )^2} \] Output:
-a^3*(A+6*B)*x/c^3+a^3*(A+6*B)*cos(f*x+e)/c^3/f+1/5*a^3*(A+B)*c^3*cos(f*x+ e)^7/f/(c-c*sin(f*x+e))^6-2/15*a^3*(A+6*B)*c*cos(f*x+e)^5/f/(c-c*sin(f*x+e ))^4+2/3*a^3*(A+6*B)*c^3*cos(f*x+e)^3/f/(c^3-c^3*sin(f*x+e))^2
Leaf count is larger than twice the leaf count of optimal. \(316\) vs. \(2(153)=306\).
Time = 11.78 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.07 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (24 (A+B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-4 (11 A+21 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-15 (A+6 B) (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+15 B \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+48 (A+B) \sin \left (\frac {1}{2} (e+f x)\right )-8 (11 A+21 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {1}{2} (e+f x)\right )+4 (23 A+93 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3}{15 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^3} \] Input:
Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x ])^3,x]
Output:
(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(24*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - 4*(11*A + 21*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) ^3 - 15*(A + 6*B)*(e + f*x)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + 15*B *Cos[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5 + 48*(A + B)*Sin[(e + f*x)/2] - 8*(11*A + 21*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*Sin[(e + f*x)/2] + 4*(23*A + 93*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4*Sin[( e + f*x)/2])*(1 + Sin[e + f*x])^3)/(15*f*(Cos[(e + f*x)/2] + Sin[(e + f*x) /2])^6*(c - c*Sin[e + f*x])^3)
Time = 0.82 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {3042, 3446, 3042, 3338, 3042, 3159, 3042, 3159, 3042, 3161, 24}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac {(A+6 B) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^5}dx}{5 c}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac {(A+6 B) \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^5}dx}{5 c}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac {(A+6 B) \left (\frac {2 \cos ^5(e+f x)}{3 c f (c-c \sin (e+f x))^4}-\frac {5 \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^3}dx}{3 c^2}\right )}{5 c}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac {(A+6 B) \left (\frac {2 \cos ^5(e+f x)}{3 c f (c-c \sin (e+f x))^4}-\frac {5 \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^3}dx}{3 c^2}\right )}{5 c}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac {(A+6 B) \left (\frac {2 \cos ^5(e+f x)}{3 c f (c-c \sin (e+f x))^4}-\frac {5 \left (\frac {2 \cos ^3(e+f x)}{c f (c-c \sin (e+f x))^2}-\frac {3 \int \frac {\cos ^2(e+f x)}{c-c \sin (e+f x)}dx}{c^2}\right )}{3 c^2}\right )}{5 c}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac {(A+6 B) \left (\frac {2 \cos ^5(e+f x)}{3 c f (c-c \sin (e+f x))^4}-\frac {5 \left (\frac {2 \cos ^3(e+f x)}{c f (c-c \sin (e+f x))^2}-\frac {3 \int \frac {\cos (e+f x)^2}{c-c \sin (e+f x)}dx}{c^2}\right )}{3 c^2}\right )}{5 c}\right )\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac {(A+6 B) \left (\frac {2 \cos ^5(e+f x)}{3 c f (c-c \sin (e+f x))^4}-\frac {5 \left (\frac {2 \cos ^3(e+f x)}{c f (c-c \sin (e+f x))^2}-\frac {3 \left (\frac {\int 1dx}{c}-\frac {\cos (e+f x)}{c f}\right )}{c^2}\right )}{3 c^2}\right )}{5 c}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^3 c^3 \left (\frac {(A+B) \cos ^7(e+f x)}{5 f (c-c \sin (e+f x))^6}-\frac {(A+6 B) \left (\frac {2 \cos ^5(e+f x)}{3 c f (c-c \sin (e+f x))^4}-\frac {5 \left (\frac {2 \cos ^3(e+f x)}{c f (c-c \sin (e+f x))^2}-\frac {3 \left (\frac {x}{c}-\frac {\cos (e+f x)}{c f}\right )}{c^2}\right )}{3 c^2}\right )}{5 c}\right )\) |
Input:
Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^3,x ]
Output:
a^3*c^3*(((A + B)*Cos[e + f*x]^7)/(5*f*(c - c*Sin[e + f*x])^6) - ((A + 6*B )*((2*Cos[e + f*x]^5)/(3*c*f*(c - c*Sin[e + f*x])^4) - (5*((-3*(x/c - Cos[ e + f*x]/(c*f)))/c^2 + (2*Cos[e + f*x]^3)/(c*f*(c - c*Sin[e + f*x])^2)))/( 3*c^2)))/(5*c))
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f *x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + p + 1 ))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] & & NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[g*((g*Cos[e + f*x])^(p - 1)/(b*f*(p - 1))), x] + Si mp[g^2/a Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g}, x ] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
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]))
Time = 1.98 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {2 a^{3} \left (-\frac {2 A +6 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {8 A -8 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {32 A +32 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {40 A +24 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {64 A +64 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}+\frac {B}{1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}-\left (A +6 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,c^{3}}\) | \(157\) |
| default | \(\frac {2 a^{3} \left (-\frac {2 A +6 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {8 A -8 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {32 A +32 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {40 A +24 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {64 A +64 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}+\frac {B}{1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}-\left (A +6 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,c^{3}}\) | \(157\) |
| risch | \(-\frac {a^{3} x A}{c^{3}}-\frac {6 a^{3} x B}{c^{3}}+\frac {B \,a^{3} {\mathrm e}^{i \left (f x +e \right )}}{2 c^{3} f}+\frac {B \,a^{3} {\mathrm e}^{-i \left (f x +e \right )}}{2 c^{3} f}+\frac {-\frac {112 A \,a^{3} {\mathrm e}^{2 i \left (f x +e \right )}}{3}-24 i A \,a^{3} {\mathrm e}^{3 i \left (f x +e \right )}+\frac {56 i A \,a^{3} {\mathrm e}^{i \left (f x +e \right )}}{3}+12 A \,a^{3} {\mathrm e}^{4 i \left (f x +e \right )}-144 B \,a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-104 i B \,a^{3} {\mathrm e}^{3 i \left (f x +e \right )}+88 i B \,a^{3} {\mathrm e}^{i \left (f x +e \right )}+36 B \,a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+\frac {92 a^{3} A}{15}+\frac {124 a^{3} B}{5}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{5} f \,c^{3}}\) | \(223\) |
| parallelrisch | \(-\frac {a^{3} \left (\left (\left (\frac {233}{2}-60 f x \right ) B -10 f x A +24 A \right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (-\frac {33}{2}+30 f x \right ) B -\frac {16 A}{3}+5 f x A \right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (-\frac {243}{10} B -\frac {24}{5} A +6 f x B +f x A \right ) \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\left (\left (-\frac {143}{2}+60 f x \right ) B -\frac {32 A}{3}+10 f x A \right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (-\frac {155}{2}+30 f x \right ) B +5 f x A -12 A \right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (\left (-\frac {11}{2}-6 f x \right ) B -\frac {4 A}{3}-f x A \right ) \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-\frac {B \left (\cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )-\sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )\right )}{2}\right )}{f \,c^{3} \left (5 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+10 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+5 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-\sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )\right )}\) | \(265\) |
| norman | \(\frac {\frac {\left (40 a^{3} A +246 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 f c}-\frac {2 \left (304 a^{3} A +1539 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{15 f c}+\frac {2 \left (92 a^{3} A +591 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 f c}-\frac {\left (1972 a^{3} A +9552 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{15 f c}+\frac {\left (112 a^{3} A +748 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{f c}-\frac {4 \left (712 a^{3} A +3297 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{15 f c}+\frac {4 \left (76 a^{3} A +525 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{3 f c}-\frac {\left (2012 a^{3} A +8802 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{15 f c}+\frac {\left (136 a^{3} A +966 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{3 f c}-\frac {2 \left (64 a^{3} A +255 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{3 f c}+\frac {2 \left (4 a^{3} A +29 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{f c}-\frac {\left (4 a^{3} A +12 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{f c}+\frac {a^{3} \left (A +6 B \right ) x}{c}-\frac {52 a^{3} A +282 a^{3} B}{15 f c}-\frac {5 a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c}+\frac {14 a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{c}-\frac {30 a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{c}+\frac {51 a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{c}-\frac {71 a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{c}+\frac {84 a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{c}-\frac {84 a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{c}+\frac {71 a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{c}-\frac {51 a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{c}+\frac {30 a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{c}-\frac {14 a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{c}+\frac {5 a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{c}-\frac {a^{3} \left (A +6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{c}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{4} c^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(772\) |
Input:
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^3,x,method=_RETUR NVERBOSE)
Output:
2/f*a^3/c^3*(-(2*A+6*B)/(tan(1/2*f*x+1/2*e)-1)-1/2*(8*A-8*B)/(tan(1/2*f*x+ 1/2*e)-1)^2-1/5*(32*A+32*B)/(tan(1/2*f*x+1/2*e)-1)^5-1/3*(40*A+24*B)/(tan( 1/2*f*x+1/2*e)-1)^3-1/4*(64*A+64*B)/(tan(1/2*f*x+1/2*e)-1)^4+B/(1+tan(1/2* f*x+1/2*e)^2)-(A+6*B)*arctan(tan(1/2*f*x+1/2*e)))
Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (150) = 300\).
Time = 0.09 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.20 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\frac {15 \, B a^{3} \cos \left (f x + e\right )^{4} + 60 \, {\left (A + 6 \, B\right )} a^{3} f x - 24 \, {\left (A + B\right )} a^{3} - {\left (15 \, {\left (A + 6 \, B\right )} a^{3} f x - {\left (46 \, A + 231 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (45 \, {\left (A + 6 \, B\right )} a^{3} f x + 2 \, {\left (A + 66 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (5 \, {\left (A + 6 \, B\right )} a^{3} f x - 2 \, {\left (6 \, A + 31 \, B\right )} a^{3}\right )} \cos \left (f x + e\right ) - {\left (15 \, B a^{3} \cos \left (f x + e\right )^{3} + 60 \, {\left (A + 6 \, B\right )} a^{3} f x + 24 \, {\left (A + B\right )} a^{3} - {\left (15 \, {\left (A + 6 \, B\right )} a^{3} f x + 2 \, {\left (23 \, A + 108 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (5 \, {\left (A + 6 \, B\right )} a^{3} f x - 2 \, {\left (4 \, A + 29 \, B\right )} a^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left (c^{3} f \cos \left (f x + e\right )^{3} + 3 \, c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f - {\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) - 4 \, c^{3} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^3,x, algori thm="fricas")
Output:
1/15*(15*B*a^3*cos(f*x + e)^4 + 60*(A + 6*B)*a^3*f*x - 24*(A + B)*a^3 - (1 5*(A + 6*B)*a^3*f*x - (46*A + 231*B)*a^3)*cos(f*x + e)^3 - (45*(A + 6*B)*a ^3*f*x + 2*(A + 66*B)*a^3)*cos(f*x + e)^2 + 6*(5*(A + 6*B)*a^3*f*x - 2*(6* A + 31*B)*a^3)*cos(f*x + e) - (15*B*a^3*cos(f*x + e)^3 + 60*(A + 6*B)*a^3* f*x + 24*(A + B)*a^3 - (15*(A + 6*B)*a^3*f*x + 2*(23*A + 108*B)*a^3)*cos(f *x + e)^2 + 6*(5*(A + 6*B)*a^3*f*x - 2*(4*A + 29*B)*a^3)*cos(f*x + e))*sin (f*x + e))/(c^3*f*cos(f*x + e)^3 + 3*c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f* x + e) - 4*c^3*f - (c^3*f*cos(f*x + e)^2 - 2*c^3*f*cos(f*x + e) - 4*c^3*f) *sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 4665 vs. \(2 (143) = 286\).
Time = 14.97 (sec) , antiderivative size = 4665, normalized size of antiderivative = 30.49 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**3,x)
Output:
Piecewise((-15*A*a**3*f*x*tan(e/2 + f*x/2)**7/(15*c**3*f*tan(e/2 + f*x/2)* *7 - 75*c**3*f*tan(e/2 + f*x/2)**6 + 165*c**3*f*tan(e/2 + f*x/2)**5 - 225* c**3*f*tan(e/2 + f*x/2)**4 + 225*c**3*f*tan(e/2 + f*x/2)**3 - 165*c**3*f*t an(e/2 + f*x/2)**2 + 75*c**3*f*tan(e/2 + f*x/2) - 15*c**3*f) + 75*A*a**3*f *x*tan(e/2 + f*x/2)**6/(15*c**3*f*tan(e/2 + f*x/2)**7 - 75*c**3*f*tan(e/2 + f*x/2)**6 + 165*c**3*f*tan(e/2 + f*x/2)**5 - 225*c**3*f*tan(e/2 + f*x/2) **4 + 225*c**3*f*tan(e/2 + f*x/2)**3 - 165*c**3*f*tan(e/2 + f*x/2)**2 + 75 *c**3*f*tan(e/2 + f*x/2) - 15*c**3*f) - 165*A*a**3*f*x*tan(e/2 + f*x/2)**5 /(15*c**3*f*tan(e/2 + f*x/2)**7 - 75*c**3*f*tan(e/2 + f*x/2)**6 + 165*c**3 *f*tan(e/2 + f*x/2)**5 - 225*c**3*f*tan(e/2 + f*x/2)**4 + 225*c**3*f*tan(e /2 + f*x/2)**3 - 165*c**3*f*tan(e/2 + f*x/2)**2 + 75*c**3*f*tan(e/2 + f*x/ 2) - 15*c**3*f) + 225*A*a**3*f*x*tan(e/2 + f*x/2)**4/(15*c**3*f*tan(e/2 + f*x/2)**7 - 75*c**3*f*tan(e/2 + f*x/2)**6 + 165*c**3*f*tan(e/2 + f*x/2)**5 - 225*c**3*f*tan(e/2 + f*x/2)**4 + 225*c**3*f*tan(e/2 + f*x/2)**3 - 165*c **3*f*tan(e/2 + f*x/2)**2 + 75*c**3*f*tan(e/2 + f*x/2) - 15*c**3*f) - 225* A*a**3*f*x*tan(e/2 + f*x/2)**3/(15*c**3*f*tan(e/2 + f*x/2)**7 - 75*c**3*f* tan(e/2 + f*x/2)**6 + 165*c**3*f*tan(e/2 + f*x/2)**5 - 225*c**3*f*tan(e/2 + f*x/2)**4 + 225*c**3*f*tan(e/2 + f*x/2)**3 - 165*c**3*f*tan(e/2 + f*x/2) **2 + 75*c**3*f*tan(e/2 + f*x/2) - 15*c**3*f) + 165*A*a**3*f*x*tan(e/2 + f *x/2)**2/(15*c**3*f*tan(e/2 + f*x/2)**7 - 75*c**3*f*tan(e/2 + f*x/2)**6...
Leaf count of result is larger than twice the leaf count of optimal. 1685 vs. \(2 (150) = 300\).
Time = 0.16 (sec) , antiderivative size = 1685, normalized size of antiderivative = 11.01 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^3,x, algori thm="maxima")
Output:
-2/15*(3*B*a^3*((105*sin(f*x + e)/(cos(f*x + e) + 1) - 189*sin(f*x + e)^2/ (cos(f*x + e) + 1)^2 + 200*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 160*sin(f *x + e)^4/(cos(f*x + e) + 1)^4 + 75*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 15*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 24)/(c^3 - 5*c^3*sin(f*x + e)/(co s(f*x + e) + 1) + 11*c^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 15*c^3*sin( f*x + e)^3/(cos(f*x + e) + 1)^3 + 15*c^3*sin(f*x + e)^4/(cos(f*x + e) + 1) ^4 - 11*c^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5*c^3*sin(f*x + e)^6/(co s(f*x + e) + 1)^6 - c^3*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 15*arctan(s in(f*x + e)/(cos(f*x + e) + 1))/c^3) + A*a^3*((95*sin(f*x + e)/(cos(f*x + e) + 1) - 145*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 75*sin(f*x + e)^3/(cos (f*x + e) + 1)^3 - 15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 22)/(c^3 - 5*c ^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*c^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 10*c^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*c^3*sin(f*x + e)^4/ (cos(f*x + e) + 1)^4 - c^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 15*arcta n(sin(f*x + e)/(cos(f*x + e) + 1))/c^3) + 3*B*a^3*((95*sin(f*x + e)/(cos(f *x + e) + 1) - 145*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 75*sin(f*x + e)^3 /(cos(f*x + e) + 1)^3 - 15*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 22)/(c^3 - 5*c^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*c^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 10*c^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*c^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - c^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + ...
Time = 0.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.41 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\frac {\frac {30 \, B a^{3}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} c^{3}} - \frac {15 \, {\left (A a^{3} + 6 \, B a^{3}\right )} {\left (f x + e\right )}}{c^{3}} - \frac {4 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 45 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 30 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 210 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 100 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 420 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 50 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 270 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13 \, A a^{3} + 63 \, B a^{3}\right )}}{c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{5}}}{15 \, f} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^3,x, algori thm="giac")
Output:
1/15*(30*B*a^3/((tan(1/2*f*x + 1/2*e)^2 + 1)*c^3) - 15*(A*a^3 + 6*B*a^3)*( f*x + e)/c^3 - 4*(15*A*a^3*tan(1/2*f*x + 1/2*e)^4 + 45*B*a^3*tan(1/2*f*x + 1/2*e)^4 - 30*A*a^3*tan(1/2*f*x + 1/2*e)^3 - 210*B*a^3*tan(1/2*f*x + 1/2* e)^3 + 100*A*a^3*tan(1/2*f*x + 1/2*e)^2 + 420*B*a^3*tan(1/2*f*x + 1/2*e)^2 - 50*A*a^3*tan(1/2*f*x + 1/2*e) - 270*B*a^3*tan(1/2*f*x + 1/2*e) + 13*A*a ^3 + 63*B*a^3)/(c^3*(tan(1/2*f*x + 1/2*e) - 1)^5))/f
Time = 36.77 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.20 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\frac {\frac {52\,A\,a^3}{15}-\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {40\,A\,a^3}{3}+82\,B\,a^3\right )+\frac {94\,B\,a^3}{5}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (4\,A\,a^3+12\,B\,a^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (8\,A\,a^3+58\,B\,a^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {64\,A\,a^3}{3}+148\,B\,a^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {92\,A\,a^3}{3}+134\,B\,a^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {452\,A\,a^3}{15}+\frac {744\,B\,a^3}{5}\right )}{f\,\left (-c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+5\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-11\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+15\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-15\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+11\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-5\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+c^3\right )}-\frac {2\,a^3\,\mathrm {atan}\left (\frac {2\,a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A+6\,B\right )}{2\,A\,a^3+12\,B\,a^3}\right )\,\left (A+6\,B\right )}{c^3\,f} \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^3,x )
Output:
((52*A*a^3)/15 - tan(e/2 + (f*x)/2)*((40*A*a^3)/3 + 82*B*a^3) + (94*B*a^3) /5 + tan(e/2 + (f*x)/2)^6*(4*A*a^3 + 12*B*a^3) - tan(e/2 + (f*x)/2)^5*(8*A *a^3 + 58*B*a^3) - tan(e/2 + (f*x)/2)^3*((64*A*a^3)/3 + 148*B*a^3) + tan(e /2 + (f*x)/2)^4*((92*A*a^3)/3 + 134*B*a^3) + tan(e/2 + (f*x)/2)^2*((452*A* a^3)/15 + (744*B*a^3)/5))/(f*(11*c^3*tan(e/2 + (f*x)/2)^2 - 15*c^3*tan(e/2 + (f*x)/2)^3 + 15*c^3*tan(e/2 + (f*x)/2)^4 - 11*c^3*tan(e/2 + (f*x)/2)^5 + 5*c^3*tan(e/2 + (f*x)/2)^6 - c^3*tan(e/2 + (f*x)/2)^7 + c^3 - 5*c^3*tan( e/2 + (f*x)/2))) - (2*a^3*atan((2*a^3*tan(e/2 + (f*x)/2)*(A + 6*B))/(2*A*a ^3 + 12*B*a^3))*(A + 6*B))/(c^3*f)
Time = 0.17 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.83 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx=\frac {a^{3} \left (12 a +36 b +30 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a f x +180 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b f x -36 \cos \left (f x +e \right ) b +123 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b +15 a f x +90 b f x -15 \cos \left (f x +e \right ) a f x -90 \cos \left (f x +e \right ) b f x -45 \sin \left (f x +e \right ) a f x -270 \sin \left (f x +e \right ) b f x -126 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b +20 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a +20 a \sin \left (f x +e \right )+15 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b -32 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a -15 \sin \left (f x +e \right )^{4} b +60 \sin \left (f x +e \right )^{3} a -12 \cos \left (f x +e \right ) a +321 \sin \left (f x +e \right )^{3} b -44 \sin \left (f x +e \right )^{2} a -417 \sin \left (f x +e \right )^{2} b -15 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a f x -90 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b f x -15 \sin \left (f x +e \right )^{3} a f x -90 \sin \left (f x +e \right )^{3} b f x +45 \sin \left (f x +e \right )^{2} a f x +270 \sin \left (f x +e \right )^{2} b f x +123 \sin \left (f x +e \right ) b \right )}{15 c^{3} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}-2 \cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )+\sin \left (f x +e \right )^{3}-3 \sin \left (f x +e \right )^{2}+3 \sin \left (f x +e \right )-1\right )} \] Input:
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^3,x)
Output:
(a**3*(15*cos(e + f*x)*sin(e + f*x)**3*b - 15*cos(e + f*x)*sin(e + f*x)**2 *a*f*x - 32*cos(e + f*x)*sin(e + f*x)**2*a - 90*cos(e + f*x)*sin(e + f*x)* *2*b*f*x - 126*cos(e + f*x)*sin(e + f*x)**2*b + 30*cos(e + f*x)*sin(e + f* x)*a*f*x + 20*cos(e + f*x)*sin(e + f*x)*a + 180*cos(e + f*x)*sin(e + f*x)* b*f*x + 123*cos(e + f*x)*sin(e + f*x)*b - 15*cos(e + f*x)*a*f*x - 12*cos(e + f*x)*a - 90*cos(e + f*x)*b*f*x - 36*cos(e + f*x)*b - 15*sin(e + f*x)**4 *b - 15*sin(e + f*x)**3*a*f*x + 60*sin(e + f*x)**3*a - 90*sin(e + f*x)**3* b*f*x + 321*sin(e + f*x)**3*b + 45*sin(e + f*x)**2*a*f*x - 44*sin(e + f*x) **2*a + 270*sin(e + f*x)**2*b*f*x - 417*sin(e + f*x)**2*b - 45*sin(e + f*x )*a*f*x + 20*sin(e + f*x)*a - 270*sin(e + f*x)*b*f*x + 123*sin(e + f*x)*b + 15*a*f*x + 12*a + 90*b*f*x + 36*b))/(15*c**3*f*(cos(e + f*x)*sin(e + f*x )**2 - 2*cos(e + f*x)*sin(e + f*x) + cos(e + f*x) + sin(e + f*x)**3 - 3*si n(e + f*x)**2 + 3*sin(e + f*x) - 1))