Integrand size = 36, antiderivative size = 153 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx=-\frac {(A-6 B) c^3 x}{a^3}-\frac {(A-6 B) c^3 \cos (e+f x)}{a^3 f}-\frac {a^3 (A-B) c^3 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^6}+\frac {2 a (A-6 B) c^3 \cos ^5(e+f x)}{15 f (a+a \sin (e+f x))^4}-\frac {2 a^3 (A-6 B) c^3 \cos ^3(e+f x)}{3 f \left (a^3+a^3 \sin (e+f x)\right )^2} \] Output:
-(A-6*B)*c^3*x/a^3-(A-6*B)*c^3*cos(f*x+e)/a^3/f-1/5*a^3*(A-B)*c^3*cos(f*x+ e)^7/f/(a+a*sin(f*x+e))^6+2/15*a*(A-6*B)*c^3*cos(f*x+e)^5/f/(a+a*sin(f*x+e ))^4-2/3*a^3*(A-6*B)*c^3*cos(f*x+e)^3/f/(a^3+a^3*sin(f*x+e))^2
Leaf count is larger than twice the leaf count of optimal. \(308\) vs. \(2(153)=306\).
Time = 11.80 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.01 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (48 (A-B) \sin \left (\frac {1}{2} (e+f x)\right )-24 (A-B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-8 (11 A-21 B) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+4 (11 A-21 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+4 (23 A-93 B) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-15 (A-6 B) (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+15 B \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5\right ) (c-c \sin (e+f x))^3}{15 a^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (1+\sin (e+f x))^3} \] Input:
Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3)/(a + a*Sin[e + f*x ])^3,x]
Output:
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(48*(A - B)*Sin[(e + f*x)/2] - 24*( A - B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 8*(11*A - 21*B)*Sin[(e + f* x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 4*(11*A - 21*B)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + 4*(23*A - 93*B)*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4 - 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)*(c - c*Sin[e + f*x])^3)/(15*a^3*f*(Cos[(e + f*x)/2] - Sin[(e + f* x)/2])^6*(1 + Sin[e + f*x])^3)
Time = 0.82 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.93, 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.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^3 (A+B \sin (e+f x))}{(a \sin (e+f x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^3 (A+B \sin (e+f x))}{(a \sin (e+f x)+a)^3}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(\sin (e+f x) a+a)^6}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6 (A+B \sin (e+f x))}{(\sin (e+f x) a+a)^6}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle a^3 c^3 \left (-\frac {(A-6 B) \int \frac {\cos ^6(e+f x)}{(\sin (e+f x) a+a)^5}dx}{5 a}-\frac {(A-B) \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (-\frac {(A-6 B) \int \frac {\cos (e+f x)^6}{(\sin (e+f x) a+a)^5}dx}{5 a}-\frac {(A-B) \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^3 c^3 \left (-\frac {(A-6 B) \left (-\frac {5 \int \frac {\cos ^4(e+f x)}{(\sin (e+f x) a+a)^3}dx}{3 a^2}-\frac {2 \cos ^5(e+f x)}{3 a f (a \sin (e+f x)+a)^4}\right )}{5 a}-\frac {(A-B) \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (-\frac {(A-6 B) \left (-\frac {5 \int \frac {\cos (e+f x)^4}{(\sin (e+f x) a+a)^3}dx}{3 a^2}-\frac {2 \cos ^5(e+f x)}{3 a f (a \sin (e+f x)+a)^4}\right )}{5 a}-\frac {(A-B) \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^3 c^3 \left (-\frac {(A-6 B) \left (-\frac {5 \left (-\frac {3 \int \frac {\cos ^2(e+f x)}{\sin (e+f x) a+a}dx}{a^2}-\frac {2 \cos ^3(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{3 a^2}-\frac {2 \cos ^5(e+f x)}{3 a f (a \sin (e+f x)+a)^4}\right )}{5 a}-\frac {(A-B) \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (-\frac {(A-6 B) \left (-\frac {5 \left (-\frac {3 \int \frac {\cos (e+f x)^2}{\sin (e+f x) a+a}dx}{a^2}-\frac {2 \cos ^3(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{3 a^2}-\frac {2 \cos ^5(e+f x)}{3 a f (a \sin (e+f x)+a)^4}\right )}{5 a}-\frac {(A-B) \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle a^3 c^3 \left (-\frac {(A-6 B) \left (-\frac {5 \left (-\frac {3 \left (\frac {\int 1dx}{a}+\frac {\cos (e+f x)}{a f}\right )}{a^2}-\frac {2 \cos ^3(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{3 a^2}-\frac {2 \cos ^5(e+f x)}{3 a f (a \sin (e+f x)+a)^4}\right )}{5 a}-\frac {(A-B) \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^3 c^3 \left (-\frac {(A-6 B) \left (-\frac {5 \left (-\frac {3 \left (\frac {\cos (e+f x)}{a f}+\frac {x}{a}\right )}{a^2}-\frac {2 \cos ^3(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{3 a^2}-\frac {2 \cos ^5(e+f x)}{3 a f (a \sin (e+f x)+a)^4}\right )}{5 a}-\frac {(A-B) \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^6}\right )\) |
Input:
Int[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^3)/(a + a*Sin[e + f*x])^3,x ]
Output:
a^3*c^3*(-1/5*((A - B)*Cos[e + f*x]^7)/(f*(a + a*Sin[e + f*x])^6) - ((A - 6*B)*((-2*Cos[e + f*x]^5)/(3*a*f*(a + a*Sin[e + f*x])^4) - (5*((-3*(x/a + Cos[e + f*x]/(a*f)))/a^2 - (2*Cos[e + f*x]^3)/(a*f*(a + a*Sin[e + f*x])^2) ))/(3*a^2)))/(5*a))
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 = 2.06 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {2 c^{3} \left (-\frac {-64 A +64 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {-8 A -8 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 A -6 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {32 A -32 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {40 A -24 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {B}{1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}-\left (A -6 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,a^{3}}\) | \(157\) |
| default | \(\frac {2 c^{3} \left (-\frac {-64 A +64 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {-8 A -8 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 A -6 B}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {32 A -32 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {40 A -24 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {B}{1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}-\left (A -6 B \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,a^{3}}\) | \(157\) |
| risch | \(-\frac {c^{3} x A}{a^{3}}+\frac {6 c^{3} x B}{a^{3}}+\frac {B \,c^{3} {\mathrm e}^{i \left (f x +e \right )}}{2 a^{3} f}+\frac {B \,c^{3} {\mathrm e}^{-i \left (f x +e \right )}}{2 a^{3} f}+\frac {\frac {112 A \,c^{3} {\mathrm e}^{2 i \left (f x +e \right )}}{3}-24 i A \,c^{3} {\mathrm e}^{3 i \left (f x +e \right )}+\frac {56 i A \,c^{3} {\mathrm e}^{i \left (f x +e \right )}}{3}-12 A \,c^{3} {\mathrm e}^{4 i \left (f x +e \right )}-144 B \,c^{3} {\mathrm e}^{2 i \left (f x +e \right )}+104 i B \,c^{3} {\mathrm e}^{3 i \left (f x +e \right )}-88 i B \,c^{3} {\mathrm e}^{i \left (f x +e \right )}+36 B \,c^{3} {\mathrm e}^{4 i \left (f x +e \right )}-\frac {92 A \,c^{3}}{15}+\frac {124 B \,c^{3}}{5}}{f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(223\) |
| parallelrisch | \(-\frac {c^{3} \left (\left (\left (\frac {233}{2}+60 f x \right ) B -10 f x A -24 A \right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (-\frac {33}{2}-30 f x \right ) B +\frac {16 A}{3}+5 f x A \right ) \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (-\frac {243}{10} B +\frac {24}{5} A -6 f x B +f x A \right ) \cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\left (\left (\frac {143}{2}+60 f x \right ) B -\frac {32 A}{3}-10 f x A \right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+\left (\left (\frac {155}{2}+30 f x \right ) B -5 f x A -12 A \right ) \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\left (\frac {11}{2} B -\frac {4}{3} A +f x A -6 f x B \right ) \sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )-\frac {B \left (\cos \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )+\sin \left (\frac {7 f x}{2}+\frac {7 e}{2}\right )\right )}{2}\right )}{f \,a^{3} \left (-5 \sin \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )-10 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-10 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+5 \cos \left (\frac {3 f x}{2}+\frac {3 e}{2}\right )+\cos \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )+\sin \left (\frac {5 f x}{2}+\frac {5 e}{2}\right )\right )}\) | \(260\) |
| norman | \(\frac {-\frac {52 A \,c^{3}-282 B \,c^{3}}{15 a f}-\frac {\left (2012 A \,c^{3}-8802 B \,c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{15 a f}-\frac {\left (136 A \,c^{3}-966 B \,c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{3 a f}-\frac {2 \left (64 A \,c^{3}-255 B \,c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{3 a f}-\frac {2 \left (4 A \,c^{3}-29 B \,c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{a f}-\frac {\left (4 A \,c^{3}-12 B \,c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{a f}-\frac {c^{3} \left (A -6 B \right ) x}{a}-\frac {\left (40 A \,c^{3}-246 B \,c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 a f}-\frac {2 \left (304 A \,c^{3}-1539 B \,c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{15 a f}-\frac {2 \left (92 A \,c^{3}-591 B \,c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 a f}-\frac {\left (1972 A \,c^{3}-9552 B \,c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{15 a f}-\frac {\left (112 A \,c^{3}-748 B \,c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{a f}-\frac {4 \left (712 A \,c^{3}-3297 B \,c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{15 a f}-\frac {4 \left (76 A \,c^{3}-525 B \,c^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{3 a f}-\frac {5 c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a}-\frac {14 c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{a}-\frac {30 c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{a}-\frac {51 c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{a}-\frac {71 c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{a}-\frac {84 c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{a}-\frac {84 c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{a}-\frac {71 c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{a}-\frac {51 c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{a}-\frac {30 c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{a}-\frac {14 c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{a}-\frac {5 c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{a}-\frac {c^{3} \left (A -6 B \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{a}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{4} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(774\) |
Input:
int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^3/(a+a*sin(f*x+e))^3,x,method=_RETUR NVERBOSE)
Output:
2/f*c^3/a^3*(-1/4*(-64*A+64*B)/(tan(1/2*f*x+1/2*e)+1)^4-1/2*(-8*A-8*B)/(ta n(1/2*f*x+1/2*e)+1)^2-(2*A-6*B)/(tan(1/2*f*x+1/2*e)+1)-1/5*(32*A-32*B)/(ta n(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+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. 338 vs. \(2 (147) = 294\).
Time = 0.11 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.21 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx=\frac {15 \, B c^{3} \cos \left (f x + e\right )^{4} + 60 \, {\left (A - 6 \, B\right )} c^{3} f x + 24 \, {\left (A - B\right )} c^{3} - {\left (15 \, {\left (A - 6 \, B\right )} c^{3} f x + {\left (46 \, A - 231 \, B\right )} c^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (45 \, {\left (A - 6 \, B\right )} c^{3} f x - 2 \, {\left (A - 66 \, B\right )} c^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (5 \, {\left (A - 6 \, B\right )} c^{3} f x + 2 \, {\left (6 \, A - 31 \, B\right )} c^{3}\right )} \cos \left (f x + e\right ) + {\left (15 \, B c^{3} \cos \left (f x + e\right )^{3} + 60 \, {\left (A - 6 \, B\right )} c^{3} f x - 24 \, {\left (A - B\right )} c^{3} - {\left (15 \, {\left (A - 6 \, B\right )} c^{3} f x - 2 \, {\left (23 \, A - 108 \, B\right )} c^{3}\right )} \cos \left (f x + e\right )^{2} + 6 \, {\left (5 \, {\left (A - 6 \, B\right )} c^{3} f x + 2 \, {\left (4 \, A - 29 \, B\right )} c^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^3/(a+a*sin(f*x+e))^3,x, algori thm="fricas")
Output:
1/15*(15*B*c^3*cos(f*x + e)^4 + 60*(A - 6*B)*c^3*f*x + 24*(A - B)*c^3 - (1 5*(A - 6*B)*c^3*f*x + (46*A - 231*B)*c^3)*cos(f*x + e)^3 - (45*(A - 6*B)*c ^3*f*x - 2*(A - 66*B)*c^3)*cos(f*x + e)^2 + 6*(5*(A - 6*B)*c^3*f*x + 2*(6* A - 31*B)*c^3)*cos(f*x + e) + (15*B*c^3*cos(f*x + e)^3 + 60*(A - 6*B)*c^3* f*x - 24*(A - B)*c^3 - (15*(A - 6*B)*c^3*f*x - 2*(23*A - 108*B)*c^3)*cos(f *x + e)^2 + 6*(5*(A - 6*B)*c^3*f*x + 2*(4*A - 29*B)*c^3)*cos(f*x + e))*sin (f*x + e))/(a^3*f*cos(f*x + e)^3 + 3*a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f* x + e) - 4*a^3*f + (a^3*f*cos(f*x + e)^2 - 2*a^3*f*cos(f*x + e) - 4*a^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 = 14.58 (sec) , antiderivative size = 4665, normalized size of antiderivative = 30.49 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))**3/(a+a*sin(f*x+e))**3,x)
Output:
Piecewise((-15*A*c**3*f*x*tan(e/2 + f*x/2)**7/(15*a**3*f*tan(e/2 + f*x/2)* *7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*f*tan(e/2 + f*x/2)**5 + 225* a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*t an(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 75*A*c**3*f *x*tan(e/2 + f*x/2)**6/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2) **4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e/2 + f*x/2)**2 + 75 *a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 165*A*c**3*f*x*tan(e/2 + f*x/2)**5 /(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3 *f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e /2 + f*x/2)**3 + 165*a**3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/ 2) + 15*a**3*f) - 225*A*c**3*f*x*tan(e/2 + f*x/2)**4/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6 + 165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a **3*f*tan(e/2 + f*x/2)**2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 225* A*c**3*f*x*tan(e/2 + f*x/2)**3/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f* tan(e/2 + f*x/2)**6 + 165*a**3*f*tan(e/2 + f*x/2)**5 + 225*a**3*f*tan(e/2 + f*x/2)**4 + 225*a**3*f*tan(e/2 + f*x/2)**3 + 165*a**3*f*tan(e/2 + f*x/2) **2 + 75*a**3*f*tan(e/2 + f*x/2) + 15*a**3*f) - 165*A*c**3*f*x*tan(e/2 + f *x/2)**2/(15*a**3*f*tan(e/2 + f*x/2)**7 + 75*a**3*f*tan(e/2 + f*x/2)**6...
Leaf count of result is larger than twice the leaf count of optimal. 1679 vs. \(2 (147) = 294\).
Time = 0.16 (sec) , antiderivative size = 1679, normalized size of antiderivative = 10.97 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^3/(a+a*sin(f*x+e))^3,x, algori thm="maxima")
Output:
2/15*(3*B*c^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 + 1 5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 24)/(a^3 + 5*a^3*sin(f*x + e)/(cos (f*x + e) + 1) + 11*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 15*a^3*sin(f *x + e)^3/(cos(f*x + e) + 1)^3 + 15*a^3*sin(f*x + e)^4/(cos(f*x + e) + 1)^ 4 + 11*a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 5*a^3*sin(f*x + e)^6/(cos (f*x + e) + 1)^6 + a^3*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 15*arctan(si n(f*x + e)/(cos(f*x + e) + 1))/a^3) - A*c^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)/(a^3 + 5*a^ 3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e)^4/( cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 15*arctan (sin(f*x + e)/(cos(f*x + e) + 1))/a^3) + 3*B*c^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)/(a^3 + 5*a^3*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^3*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*sin(f*x + e )^4/(cos(f*x + e) + 1)^4 + a^3*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 1...
Time = 0.24 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx=\frac {\frac {30 \, B c^{3}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} a^{3}} - \frac {15 \, {\left (A c^{3} - 6 \, B c^{3}\right )} {\left (f x + e\right )}}{a^{3}} - \frac {4 \, {\left (15 \, A c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 45 \, B c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, A c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 210 \, B c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 100 \, A c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 420 \, B c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 50 \, A c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 270 \, B c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13 \, A c^{3} - 63 \, B c^{3}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{15 \, f} \] Input:
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^3/(a+a*sin(f*x+e))^3,x, algori thm="giac")
Output:
1/15*(30*B*c^3/((tan(1/2*f*x + 1/2*e)^2 + 1)*a^3) - 15*(A*c^3 - 6*B*c^3)*( f*x + e)/a^3 - 4*(15*A*c^3*tan(1/2*f*x + 1/2*e)^4 - 45*B*c^3*tan(1/2*f*x + 1/2*e)^4 + 30*A*c^3*tan(1/2*f*x + 1/2*e)^3 - 210*B*c^3*tan(1/2*f*x + 1/2* e)^3 + 100*A*c^3*tan(1/2*f*x + 1/2*e)^2 - 420*B*c^3*tan(1/2*f*x + 1/2*e)^2 + 50*A*c^3*tan(1/2*f*x + 1/2*e) - 270*B*c^3*tan(1/2*f*x + 1/2*e) + 13*A*c ^3 - 63*B*c^3)/(a^3*(tan(1/2*f*x + 1/2*e) + 1)^5))/f
Time = 38.88 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.18 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx=-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {40\,A\,c^3}{3}-82\,B\,c^3\right )+\frac {52\,A\,c^3}{15}-\frac {94\,B\,c^3}{5}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (4\,A\,c^3-12\,B\,c^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (8\,A\,c^3-58\,B\,c^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {64\,A\,c^3}{3}-148\,B\,c^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {92\,A\,c^3}{3}-134\,B\,c^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {452\,A\,c^3}{15}-\frac {744\,B\,c^3}{5}\right )}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+5\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+11\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+15\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+15\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+11\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+5\,a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a^3\right )}-\frac {2\,c^3\,\mathrm {atan}\left (\frac {2\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A-6\,B\right )}{2\,A\,c^3-12\,B\,c^3}\right )\,\left (A-6\,B\right )}{a^3\,f} \] Input:
int(((A + B*sin(e + f*x))*(c - c*sin(e + f*x))^3)/(a + a*sin(e + f*x))^3,x )
Output:
- (tan(e/2 + (f*x)/2)*((40*A*c^3)/3 - 82*B*c^3) + (52*A*c^3)/15 - (94*B*c^ 3)/5 + tan(e/2 + (f*x)/2)^6*(4*A*c^3 - 12*B*c^3) + tan(e/2 + (f*x)/2)^5*(8 *A*c^3 - 58*B*c^3) + tan(e/2 + (f*x)/2)^3*((64*A*c^3)/3 - 148*B*c^3) + tan (e/2 + (f*x)/2)^4*((92*A*c^3)/3 - 134*B*c^3) + tan(e/2 + (f*x)/2)^2*((452* A*c^3)/15 - (744*B*c^3)/5))/(f*(11*a^3*tan(e/2 + (f*x)/2)^2 + 15*a^3*tan(e /2 + (f*x)/2)^3 + 15*a^3*tan(e/2 + (f*x)/2)^4 + 11*a^3*tan(e/2 + (f*x)/2)^ 5 + 5*a^3*tan(e/2 + (f*x)/2)^6 + a^3*tan(e/2 + (f*x)/2)^7 + a^3 + 5*a^3*ta n(e/2 + (f*x)/2))) - (2*c^3*atan((2*c^3*tan(e/2 + (f*x)/2)*(A - 6*B))/(2*A *c^3 - 12*B*c^3))*(A - 6*B))/(a^3*f)
Time = 0.19 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.84 \[ \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx=\frac {c^{3} \left (-12 a +36 b -30 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a f x +180 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b f x -36 \cos \left (f x +e \right ) b -123 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b +15 a f x -90 b f x -15 \cos \left (f x +e \right ) a f x +90 \cos \left (f x +e \right ) b f x +45 \sin \left (f x +e \right ) a f x -270 \sin \left (f x +e \right ) b f x -126 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b +20 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a +20 a \sin \left (f x +e \right )-15 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b +32 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a -15 \sin \left (f x +e \right )^{4} b +60 \sin \left (f x +e \right )^{3} a +12 \cos \left (f x +e \right ) a -321 \sin \left (f x +e \right )^{3} b +44 \sin \left (f x +e \right )^{2} a -417 \sin \left (f x +e \right )^{2} b -15 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a f x +90 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b f x +15 \sin \left (f x +e \right )^{3} a f x -90 \sin \left (f x +e \right )^{3} b f x +45 \sin \left (f x +e \right )^{2} a f x -270 \sin \left (f x +e \right )^{2} b f x -123 \sin \left (f x +e \right ) b \right )}{15 a^{3} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+2 \cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )-\sin \left (f x +e \right )^{3}-3 \sin \left (f x +e \right )^{2}-3 \sin \left (f x +e \right )-1\right )} \] Input:
int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^3/(a+a*sin(f*x+e))^3,x)
Output:
(c**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*co s(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*a**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 *sin(e + f*x)**2 - 3*sin(e + f*x) - 1))