Integrand size = 36, antiderivative size = 115 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {a^2 (A+B) c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac {a^2 (2 A-7 B) c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6}+\frac {a^2 (2 A-7 B) \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5} \] Output:
1/9*a^2*(A+B)*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^7+1/63*a^2*(2*A-7*B)*c*c os(f*x+e)^5/f/(c-c*sin(f*x+e))^6+1/315*a^2*(2*A-7*B)*cos(f*x+e)^5/f/(c-c*s in(f*x+e))^5
Leaf count is larger than twice the leaf count of optimal. \(261\) vs. \(2(115)=230\).
Time = 12.17 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.27 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=-\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^2 \left (315 (2 A+3 B) \cos \left (\frac {1}{2} (e+f x)\right )-63 (4 A+11 B) \cos \left (\frac {3}{2} (e+f x)\right )-315 B \cos \left (\frac {5}{2} (e+f x)\right )-18 A \cos \left (\frac {7}{2} (e+f x)\right )+63 B \cos \left (\frac {7}{2} (e+f x)\right )+882 A \sin \left (\frac {1}{2} (e+f x)\right )+63 B \sin \left (\frac {1}{2} (e+f x)\right )+420 A \sin \left (\frac {3}{2} (e+f x)\right )+105 B \sin \left (\frac {3}{2} (e+f x)\right )-72 A \sin \left (\frac {5}{2} (e+f x)\right )-63 B \sin \left (\frac {5}{2} (e+f x)\right )+2 A \sin \left (\frac {9}{2} (e+f x)\right )-7 B \sin \left (\frac {9}{2} (e+f x)\right )\right )}{2520 c^5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (-1+\sin (e+f x))^5} \] Input:
Integrate[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x ])^5,x]
Output:
-1/2520*(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(3 15*(2*A + 3*B)*Cos[(e + f*x)/2] - 63*(4*A + 11*B)*Cos[(3*(e + f*x))/2] - 3 15*B*Cos[(5*(e + f*x))/2] - 18*A*Cos[(7*(e + f*x))/2] + 63*B*Cos[(7*(e + f *x))/2] + 882*A*Sin[(e + f*x)/2] + 63*B*Sin[(e + f*x)/2] + 420*A*Sin[(3*(e + f*x))/2] + 105*B*Sin[(3*(e + f*x))/2] - 72*A*Sin[(5*(e + f*x))/2] - 63* B*Sin[(5*(e + f*x))/2] + 2*A*Sin[(9*(e + f*x))/2] - 7*B*Sin[(9*(e + f*x))/ 2]))/(c^5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-1 + Sin[e + f*x])^5)
Time = 0.68 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3042, 3446, 3042, 3338, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^2 c^2 \int \frac {\cos ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \int \frac {\cos (e+f x)^4 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle a^2 c^2 \left (\frac {(2 A-7 B) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^6}dx}{9 c}+\frac {(A+B) \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {(2 A-7 B) \int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^6}dx}{9 c}+\frac {(A+B) \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )\) |
| \(\Big \downarrow \) 3151 |
| \(\displaystyle a^2 c^2 \left (\frac {(2 A-7 B) \left (\frac {\int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^5}dx}{7 c}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}+\frac {(A+B) \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c^2 \left (\frac {(2 A-7 B) \left (\frac {\int \frac {\cos (e+f x)^4}{(c-c \sin (e+f x))^5}dx}{7 c}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}+\frac {(A+B) \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}\right )\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle a^2 c^2 \left (\frac {(A+B) \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac {(2 A-7 B) \left (\frac {\cos ^5(e+f x)}{35 c f (c-c \sin (e+f x))^5}+\frac {\cos ^5(e+f x)}{7 f (c-c \sin (e+f x))^6}\right )}{9 c}\right )\) |
Input:
Int[((a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5,x ]
Output:
a^2*c^2*(((A + B)*Cos[e + f*x]^5)/(9*f*(c - c*Sin[e + f*x])^7) + ((2*A - 7 *B)*(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))
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]))
Time = 1.04 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.49
| method | result | size |
| parallelrisch | \(-\frac {2 a^{2} \left (A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+\left (-2 A +B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+\frac {\left (22 A +B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{3}+\left (-8 A +3 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\frac {\left (54 A +B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{5}+\frac {\left (-26 A +11 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{5}+\frac {\left (\frac {118 A}{7}+B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{5}+\frac {\left (-\frac {12 A}{7}+B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{5}+\frac {47 A}{315}-\frac {B}{45}\right )}{f \,c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(171\) |
| risch | \(\frac {2 i a^{2} \left (420 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+105 i B \,{\mathrm e}^{6 i \left (f x +e \right )}+315 B \,{\mathrm e}^{7 i \left (f x +e \right )}-882 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-630 A \,{\mathrm e}^{5 i \left (f x +e \right )}-63 i B \,{\mathrm e}^{4 i \left (f x +e \right )}-945 B \,{\mathrm e}^{5 i \left (f x +e \right )}+72 i A \,{\mathrm e}^{2 i \left (f x +e \right )}+252 A \,{\mathrm e}^{3 i \left (f x +e \right )}+63 i B \,{\mathrm e}^{2 i \left (f x +e \right )}+693 B \,{\mathrm e}^{3 i \left (f x +e \right )}-2 i A +18 A \,{\mathrm e}^{i \left (f x +e \right )}+7 i B -63 B \,{\mathrm e}^{i \left (f x +e \right )}\right )}{315 f \,c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9}}\) | \(198\) |
| derivativedivides | \(\frac {2 a^{2} \left (-\frac {64 A +64 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {544 A +448 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {404 A +276 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {256 A +256 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {64 A +22 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {480 A +448 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {200 A +104 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {12 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}\right )}{f \,c^{5}}\) | \(205\) |
| default | \(\frac {2 a^{2} \left (-\frac {64 A +64 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {544 A +448 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {404 A +276 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {256 A +256 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {64 A +22 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {480 A +448 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {200 A +104 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {12 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}\right )}{f \,c^{5}}\) | \(205\) |
| norman | \(\frac {\frac {\left (4 a^{2} A -2 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{f c}+\frac {\left (28 a^{2} A -12 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{f c}-\frac {94 a^{2} A -14 a^{2} B}{315 f c}-\frac {2 a^{2} A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{f c}+\frac {\left (24 a^{2} A -14 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35 f c}-\frac {\left (62 a^{2} A +2 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{3 f c}+\frac {\left (352 a^{2} A -142 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{5 f c}-\frac {\left (358 a^{2} A +12 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{5 f c}+\frac {\left (436 a^{2} A -196 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{35 f c}-\frac {\left (802 a^{2} A +28 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{105 f c}+\frac {\left (1724 a^{2} A -714 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{35 f c}+\frac {\left (2936 a^{2} A -1176 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{35 f c}-\frac {\left (4114 a^{2} A +126 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{35 f c}-\frac {\left (4486 a^{2} A +154 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{105 f c}-\frac {\left (31498 a^{2} A +952 a^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{315 f c}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{3} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(490\) |
Input:
int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x,method=_RETUR NVERBOSE)
Output:
-2*a^2*(A*tan(1/2*f*x+1/2*e)^8+(-2*A+B)*tan(1/2*f*x+1/2*e)^7+1/3*(22*A+B)* tan(1/2*f*x+1/2*e)^6+(-8*A+3*B)*tan(1/2*f*x+1/2*e)^5+1/5*(54*A+B)*tan(1/2* f*x+1/2*e)^4+1/5*(-26*A+11*B)*tan(1/2*f*x+1/2*e)^3+1/5*(118/7*A+B)*tan(1/2 *f*x+1/2*e)^2+1/5*(-12/7*A+B)*tan(1/2*f*x+1/2*e)+47/315*A-1/45*B)/f/c^5/(t an(1/2*f*x+1/2*e)-1)^9
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (112) = 224\).
Time = 0.08 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.91 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {{\left (2 \, A - 7 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} - 4 \, {\left (2 \, A - 7 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} - 5 \, {\left (5 \, A + 14 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} - 5 \, {\left (17 \, A + 35 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} + 70 \, {\left (A + B\right )} a^{2} \cos \left (f x + e\right ) + 140 \, {\left (A + B\right )} a^{2} + {\left ({\left (2 \, A - 7 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} + 5 \, {\left (2 \, A - 7 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} - 15 \, {\left (A + 7 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} + 70 \, {\left (A + B\right )} a^{2} \cos \left (f x + e\right ) + 140 \, {\left (A + B\right )} a^{2}\right )} \sin \left (f x + e\right )}{315 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algori thm="fricas")
Output:
1/315*((2*A - 7*B)*a^2*cos(f*x + e)^5 - 4*(2*A - 7*B)*a^2*cos(f*x + e)^4 - 5*(5*A + 14*B)*a^2*cos(f*x + e)^3 - 5*(17*A + 35*B)*a^2*cos(f*x + e)^2 + 70*(A + B)*a^2*cos(f*x + e) + 140*(A + B)*a^2 + ((2*A - 7*B)*a^2*cos(f*x + e)^4 + 5*(2*A - 7*B)*a^2*cos(f*x + e)^3 - 15*(A + 7*B)*a^2*cos(f*x + e)^2 + 70*(A + B)*a^2*cos(f*x + e) + 140*(A + B)*a^2)*sin(f*x + e))/(c^5*f*cos (f*x + e)^5 + 5*c^5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^3 - 20*c^5*f*c os(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f - (c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 3262 vs. \(2 (102) = 204\).
Time = 28.19 (sec) , antiderivative size = 3262, normalized size of antiderivative = 28.37 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))**2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**5,x)
Output:
Piecewise((-630*A*a**2*tan(e/2 + f*x/2)**8/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26 460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690* c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5 *f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) + 1260 *A*a**2*tan(e/2 + f*x/2)**7/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f* tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan( e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f* x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 4620*A*a**2*tan(e/2 + f*x/2)**6/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2 )**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 2 6460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835* c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) + 5040*A*a**2*tan(e/2 + f*x/2)**5/(3 15*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c* *5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f *tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan (e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 6804*A*a**2*tan(e/2 + f*x/2)**4/(315*c**5*f*ta...
Leaf count of result is larger than twice the leaf count of optimal. 2087 vs. \(2 (112) = 224\).
Time = 0.10 (sec) , antiderivative size = 2087, normalized size of antiderivative = 18.15 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algori thm="maxima")
Output:
-2/315*(A*a^2*(432*sin(f*x + e)/(cos(f*x + e) + 1) - 1728*sin(f*x + e)^2/( cos(f*x + e) + 1)^2 + 3612*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 5418*sin( f*x + e)^4/(cos(f*x + e) + 1)^4 + 5040*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 3360*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1260*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 315*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 83)/(c^5 - 9*c^5* sin(f*x + e)/(cos(f*x + e) + 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1) ^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x + e)^4/( cos(f*x + e) + 1)^4 - 126*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5 *sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(f*x + e)^9/( cos(f*x + e) + 1)^9) - 10*A*a^2*(45*sin(f*x + e)/(cos(f*x + e) + 1) - 117* sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 273*sin(f*x + e)^3/(cos(f*x + e) + 1 )^3 - 315*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 315*sin(f*x + e)^5/(cos(f* x + e) + 1)^5 - 147*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 63*sin(f*x + e)^ 7/(cos(f*x + e) + 1)^7 - 5)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f *x + e) + 1)^3 + 126*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin (f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x + e) + 1 )^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(c os(f*x + e) + 1)^8 - c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) - 5*B*a^2...
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (112) = 224\).
Time = 0.29 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.48 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=-\frac {2 \, {\left (315 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 630 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 315 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 2310 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 105 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 2520 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 945 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3402 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 63 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1638 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 693 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1062 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 63 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 108 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 63 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 47 \, A a^{2} - 7 \, B a^{2}\right )}}{315 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}} \] Input:
integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algori thm="giac")
Output:
-2/315*(315*A*a^2*tan(1/2*f*x + 1/2*e)^8 - 630*A*a^2*tan(1/2*f*x + 1/2*e)^ 7 + 315*B*a^2*tan(1/2*f*x + 1/2*e)^7 + 2310*A*a^2*tan(1/2*f*x + 1/2*e)^6 + 105*B*a^2*tan(1/2*f*x + 1/2*e)^6 - 2520*A*a^2*tan(1/2*f*x + 1/2*e)^5 + 94 5*B*a^2*tan(1/2*f*x + 1/2*e)^5 + 3402*A*a^2*tan(1/2*f*x + 1/2*e)^4 + 63*B* a^2*tan(1/2*f*x + 1/2*e)^4 - 1638*A*a^2*tan(1/2*f*x + 1/2*e)^3 + 693*B*a^2 *tan(1/2*f*x + 1/2*e)^3 + 1062*A*a^2*tan(1/2*f*x + 1/2*e)^2 + 63*B*a^2*tan (1/2*f*x + 1/2*e)^2 - 108*A*a^2*tan(1/2*f*x + 1/2*e) + 63*B*a^2*tan(1/2*f* x + 1/2*e) + 47*A*a^2 - 7*B*a^2)/(c^5*f*(tan(1/2*f*x + 1/2*e) - 1)^9)
Time = 36.28 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.88 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=-\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {265\,A\,a^2\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {49\,B\,a^2}{8}-\frac {4967\,A\,a^2}{16}-\frac {89\,A\,a^2\,\cos \left (3\,e+3\,f\,x\right )}{4}-\frac {49\,A\,a^2\,\cos \left (4\,e+4\,f\,x\right )}{16}+\frac {35\,B\,a^2\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {7\,B\,a^2\,\cos \left (3\,e+3\,f\,x\right )}{8}+\frac {7\,B\,a^2\,\cos \left (4\,e+4\,f\,x\right )}{8}-\frac {567\,A\,a^2\,\sin \left (2\,e+2\,f\,x\right )}{8}-\frac {243\,A\,a^2\,\sin \left (3\,e+3\,f\,x\right )}{8}+\frac {45\,A\,a^2\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {63\,B\,a^2\,\sin \left (2\,e+2\,f\,x\right )}{2}+\frac {63\,B\,a^2\,\sin \left (3\,e+3\,f\,x\right )}{8}+\frac {625\,A\,a^2\,\cos \left (e+f\,x\right )}{4}+\frac {35\,B\,a^2\,\cos \left (e+f\,x\right )}{8}+\frac {2205\,A\,a^2\,\sin \left (e+f\,x\right )}{8}-\frac {945\,B\,a^2\,\sin \left (e+f\,x\right )}{8}\right )}{315\,c^5\,f\,\left (\frac {63\,\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}{8}-\frac {21\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}-\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{4}-\frac {9\,\sqrt {2}\,\cos \left (\frac {5\,e}{2}+\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{4}+\frac {9\,\sqrt {2}\,\cos \left (\frac {7\,e}{2}-\frac {\pi }{4}+\frac {7\,f\,x}{2}\right )}{16}+\frac {\sqrt {2}\,\cos \left (\frac {9\,e}{2}+\frac {\pi }{4}+\frac {9\,f\,x}{2}\right )}{16}\right )} \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^2)/(c - c*sin(e + f*x))^5,x )
Output:
-(2*cos(e/2 + (f*x)/2)*((265*A*a^2*cos(2*e + 2*f*x))/2 - (49*B*a^2)/8 - (4 967*A*a^2)/16 - (89*A*a^2*cos(3*e + 3*f*x))/4 - (49*A*a^2*cos(4*e + 4*f*x) )/16 + (35*B*a^2*cos(2*e + 2*f*x))/4 - (7*B*a^2*cos(3*e + 3*f*x))/8 + (7*B *a^2*cos(4*e + 4*f*x))/8 - (567*A*a^2*sin(2*e + 2*f*x))/8 - (243*A*a^2*sin (3*e + 3*f*x))/8 + (45*A*a^2*sin(4*e + 4*f*x))/16 + (63*B*a^2*sin(2*e + 2* f*x))/2 + (63*B*a^2*sin(3*e + 3*f*x))/8 + (625*A*a^2*cos(e + f*x))/4 + (35 *B*a^2*cos(e + f*x))/8 + (2205*A*a^2*sin(e + f*x))/8 - (945*B*a^2*sin(e + f*x))/8))/(315*c^5*f*((63*2^(1/2)*cos(e/2 + pi/4 + (f*x)/2))/8 - (21*2^(1/ 2)*cos((3*e)/2 - pi/4 + (3*f*x)/2))/4 - (9*2^(1/2)*cos((5*e)/2 + pi/4 + (5 *f*x)/2))/4 + (9*2^(1/2)*cos((7*e)/2 - pi/4 + (7*f*x)/2))/16 + (2^(1/2)*co s((9*e)/2 + pi/4 + (9*f*x)/2))/16))
Time = 0.18 (sec) , antiderivative size = 362, normalized size of antiderivative = 3.15 \[ \int \frac {(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {a^{2} \left (-25 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4} a +102 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} a -7 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3} b -159 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} a -126 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2} b +12 \cos \left (f x +e \right ) \sin \left (f x +e \right ) a -7 \cos \left (f x +e \right ) \sin \left (f x +e \right ) b -70 \cos \left (f x +e \right ) a -21 \sin \left (f x +e \right )^{5} a -14 \sin \left (f x +e \right )^{5} b +107 \sin \left (f x +e \right )^{4} a +63 \sin \left (f x +e \right )^{4} b -219 \sin \left (f x +e \right )^{3} a +49 \sin \left (f x +e \right )^{3} b +331 \sin \left (f x +e \right )^{2} a +189 \sin \left (f x +e \right )^{2} b +12 a \sin \left (f x +e \right )-7 \sin \left (f x +e \right ) b +70 a \right )}{315 c^{5} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}-4 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}+6 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}-4 \cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )+\sin \left (f x +e \right )^{5}-5 \sin \left (f x +e \right )^{4}+10 \sin \left (f x +e \right )^{3}-10 \sin \left (f x +e \right )^{2}+5 \sin \left (f x +e \right )-1\right )} \] Input:
int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x)
Output:
(a**2*( - 25*cos(e + f*x)*sin(e + f*x)**4*a + 102*cos(e + f*x)*sin(e + f*x )**3*a - 7*cos(e + f*x)*sin(e + f*x)**3*b - 159*cos(e + f*x)*sin(e + f*x)* *2*a - 126*cos(e + f*x)*sin(e + f*x)**2*b + 12*cos(e + f*x)*sin(e + f*x)*a - 7*cos(e + f*x)*sin(e + f*x)*b - 70*cos(e + f*x)*a - 21*sin(e + f*x)**5* a - 14*sin(e + f*x)**5*b + 107*sin(e + f*x)**4*a + 63*sin(e + f*x)**4*b - 219*sin(e + f*x)**3*a + 49*sin(e + f*x)**3*b + 331*sin(e + f*x)**2*a + 189 *sin(e + f*x)**2*b + 12*sin(e + f*x)*a - 7*sin(e + f*x)*b + 70*a))/(315*c* *5*f*(cos(e + f*x)*sin(e + f*x)**4 - 4*cos(e + f*x)*sin(e + f*x)**3 + 6*co s(e + f*x)*sin(e + f*x)**2 - 4*cos(e + f*x)*sin(e + f*x) + cos(e + f*x) + sin(e + f*x)**5 - 5*sin(e + f*x)**4 + 10*sin(e + f*x)**3 - 10*sin(e + f*x) **2 + 5*sin(e + f*x) - 1))