Integrand size = 34, antiderivative size = 176 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {2 a (A+B) \cos (e+f x)}{9 f (c-c \sin (e+f x))^5}-\frac {a (A+19 B) \cos (e+f x)}{63 c f (c-c \sin (e+f x))^4}-\frac {a (A-2 B) c \cos (e+f x)}{105 f \left (c^2-c^2 \sin (e+f x)\right )^3}-\frac {2 a (A-2 B) c \cos (e+f x)}{315 f \left (c^3-c^3 \sin (e+f x)\right )^2}-\frac {2 a (A-2 B) \cos (e+f x)}{315 f \left (c^5-c^5 \sin (e+f x)\right )} \] Output:
2/9*a*(A+B)*cos(f*x+e)/f/(c-c*sin(f*x+e))^5-1/63*a*(A+19*B)*cos(f*x+e)/c/f /(c-c*sin(f*x+e))^4-1/105*a*(A-2*B)*c*cos(f*x+e)/f/(c^2-c^2*sin(f*x+e))^3- 2/315*a*(A-2*B)*c*cos(f*x+e)/f/(c^3-c^3*sin(f*x+e))^2-2/315*a*(A-2*B)*cos( f*x+e)/f/(c^5-c^5*sin(f*x+e))
Time = 6.83 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.14 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {a \left (315 A \cos \left (e+\frac {f x}{2}\right )-42 (2 A+B) \cos \left (e+\frac {3 f x}{2}\right )+9 A \cos \left (3 e+\frac {7 f x}{2}\right )-18 B \cos \left (3 e+\frac {7 f x}{2}\right )+189 A \sin \left (\frac {f x}{2}\right )+252 B \sin \left (\frac {f x}{2}\right )+210 B \sin \left (2 e+\frac {3 f x}{2}\right )+36 A \sin \left (2 e+\frac {5 f x}{2}\right )-72 B \sin \left (2 e+\frac {5 f x}{2}\right )-A \sin \left (4 e+\frac {9 f x}{2}\right )+2 B \sin \left (4 e+\frac {9 f x}{2}\right )\right )}{1260 c^5 f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \] Input:
Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]) ^5,x]
Output:
(a*(315*A*Cos[e + (f*x)/2] - 42*(2*A + B)*Cos[e + (3*f*x)/2] + 9*A*Cos[3*e + (7*f*x)/2] - 18*B*Cos[3*e + (7*f*x)/2] + 189*A*Sin[(f*x)/2] + 252*B*Sin [(f*x)/2] + 210*B*Sin[2*e + (3*f*x)/2] + 36*A*Sin[2*e + (5*f*x)/2] - 72*B* Sin[2*e + (5*f*x)/2] - A*Sin[4*e + (9*f*x)/2] + 2*B*Sin[4*e + (9*f*x)/2])) /(1260*c^5*f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9 )
Time = 0.86 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.382, Rules used = {3042, 3446, 3042, 3336, 25, 3042, 3229, 3042, 3129, 3042, 3129, 3042, 3127}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a c \int \frac {\cos ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \int \frac {\cos (e+f x)^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6}dx\) |
| \(\Big \downarrow \) 3336 |
| \(\displaystyle a c \left (\frac {\int -\frac {(A+10 B) c+9 B \sin (e+f x) c}{(c-c \sin (e+f x))^4}dx}{9 c^3}+\frac {2 (A+B) \cos (e+f x)}{9 c f (c-c \sin (e+f x))^5}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a c \left (\frac {2 (A+B) \cos (e+f x)}{9 c f (c-c \sin (e+f x))^5}-\frac {\int \frac {(A+10 B) c+9 B \sin (e+f x) c}{(c-c \sin (e+f x))^4}dx}{9 c^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {2 (A+B) \cos (e+f x)}{9 c f (c-c \sin (e+f x))^5}-\frac {\int \frac {(A+10 B) c+9 B \sin (e+f x) c}{(c-c \sin (e+f x))^4}dx}{9 c^3}\right )\) |
| \(\Big \downarrow \) 3229 |
| \(\displaystyle a c \left (\frac {2 (A+B) \cos (e+f x)}{9 c f (c-c \sin (e+f x))^5}-\frac {\frac {3}{7} (A-2 B) \int \frac {1}{(c-c \sin (e+f x))^3}dx+\frac {c (A+19 B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}}{9 c^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {2 (A+B) \cos (e+f x)}{9 c f (c-c \sin (e+f x))^5}-\frac {\frac {3}{7} (A-2 B) \int \frac {1}{(c-c \sin (e+f x))^3}dx+\frac {c (A+19 B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}}{9 c^3}\right )\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle a c \left (\frac {2 (A+B) \cos (e+f x)}{9 c f (c-c \sin (e+f x))^5}-\frac {\frac {3}{7} (A-2 B) \left (\frac {2 \int \frac {1}{(c-c \sin (e+f x))^2}dx}{5 c}+\frac {\cos (e+f x)}{5 f (c-c \sin (e+f x))^3}\right )+\frac {c (A+19 B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}}{9 c^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {2 (A+B) \cos (e+f x)}{9 c f (c-c \sin (e+f x))^5}-\frac {\frac {3}{7} (A-2 B) \left (\frac {2 \int \frac {1}{(c-c \sin (e+f x))^2}dx}{5 c}+\frac {\cos (e+f x)}{5 f (c-c \sin (e+f x))^3}\right )+\frac {c (A+19 B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}}{9 c^3}\right )\) |
| \(\Big \downarrow \) 3129 |
| \(\displaystyle a c \left (\frac {2 (A+B) \cos (e+f x)}{9 c f (c-c \sin (e+f x))^5}-\frac {\frac {3}{7} (A-2 B) \left (\frac {2 \left (\frac {\int \frac {1}{c-c \sin (e+f x)}dx}{3 c}+\frac {\cos (e+f x)}{3 f (c-c \sin (e+f x))^2}\right )}{5 c}+\frac {\cos (e+f x)}{5 f (c-c \sin (e+f x))^3}\right )+\frac {c (A+19 B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}}{9 c^3}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {2 (A+B) \cos (e+f x)}{9 c f (c-c \sin (e+f x))^5}-\frac {\frac {3}{7} (A-2 B) \left (\frac {2 \left (\frac {\int \frac {1}{c-c \sin (e+f x)}dx}{3 c}+\frac {\cos (e+f x)}{3 f (c-c \sin (e+f x))^2}\right )}{5 c}+\frac {\cos (e+f x)}{5 f (c-c \sin (e+f x))^3}\right )+\frac {c (A+19 B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}}{9 c^3}\right )\) |
| \(\Big \downarrow \) 3127 |
| \(\displaystyle a c \left (\frac {2 (A+B) \cos (e+f x)}{9 c f (c-c \sin (e+f x))^5}-\frac {\frac {c (A+19 B) \cos (e+f x)}{7 f (c-c \sin (e+f x))^4}+\frac {3}{7} (A-2 B) \left (\frac {\cos (e+f x)}{5 f (c-c \sin (e+f x))^3}+\frac {2 \left (\frac {\cos (e+f x)}{3 c f (c-c \sin (e+f x))}+\frac {\cos (e+f x)}{3 f (c-c \sin (e+f x))^2}\right )}{5 c}\right )}{9 c^3}\right )\) |
Input:
Int[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5,x]
Output:
a*c*((2*(A + B)*Cos[e + f*x])/(9*c*f*(c - c*Sin[e + f*x])^5) - (((A + 19*B )*c*Cos[e + f*x])/(7*f*(c - c*Sin[e + f*x])^4) + (3*(A - 2*B)*(Cos[e + f*x ]/(5*f*(c - c*Sin[e + f*x])^3) + (2*(Cos[e + f*x]/(3*f*(c - c*Sin[e + f*x] )^2) + Cos[e + f*x]/(3*c*f*(c - c*Sin[e + f*x]))))/(5*c)))/7)/(9*c^3))
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b ^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((a + b*Sin[e + f* x])^m/(a*f*(2*m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)) I nt[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Int[cos[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*( (c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(b*c - a*d)*Cos [e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(2*m + 3))), x] + Simp[1/(b^ 3*(2*m + 3)) Int[(a + b*Sin[e + f*x])^(m + 2)*(b*c + 2*a*d*(m + 1) - b*d* (2*m + 3)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -3/2]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin [e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a* d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Result contains complex when optimal does not.
Time = 1.44 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(-\frac {4 \left (-42 i B a \,{\mathrm e}^{3 i \left (f x +e \right )}+A a -2 B a -189 A a \,{\mathrm e}^{4 i \left (f x +e \right )}+210 B a \,{\mathrm e}^{6 i \left (f x +e \right )}-252 B a \,{\mathrm e}^{4 i \left (f x +e \right )}-36 A a \,{\mathrm e}^{2 i \left (f x +e \right )}+72 B a \,{\mathrm e}^{2 i \left (f x +e \right )}+315 i A a \,{\mathrm e}^{5 i \left (f x +e \right )}-84 i A a \,{\mathrm e}^{3 i \left (f x +e \right )}+9 i A a \,{\mathrm e}^{i \left (f x +e \right )}-18 i B a \,{\mathrm e}^{i \left (f x +e \right )}\right )}{315 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9} f \,c^{5}}\) | \(166\) |
| parallelrisch | \(-\frac {2 \left (A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+\left (-3 A +B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+\left (\frac {25 A}{3}-B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\left (-11 A +3 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\frac {\left (61 A -7 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{5}+\frac {\left (-107 A +29 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{15}+\frac {\left (127 A -9 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{35}+\frac {\left (-23 A +11 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35}+\frac {58 A}{315}-\frac {11 B}{315}\right ) a}{f \,c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(176\) |
| derivativedivides | \(\frac {2 a \left (-\frac {46 A +18 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {128 A +72 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {236 A +168 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {10 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {296 A +248 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {248 A +232 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {128 A +128 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {32 A +32 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\right )}{f \,c^{5}}\) | \(203\) |
| default | \(\frac {2 a \left (-\frac {46 A +18 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {128 A +72 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {236 A +168 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {10 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {296 A +248 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {248 A +232 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {128 A +128 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {32 A +32 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\right )}{f \,c^{5}}\) | \(203\) |
| norman | \(\frac {\frac {\left (34 A a -10 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{f c}-\frac {116 A a -22 B a}{315 f c}-\frac {2 A a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{f c}+\frac {2 \left (3 A a -B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{f c}-\frac {2 \left (31 A a -3 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{3 f c}+\frac {\left (46 A a -22 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35 f c}+\frac {4 \left (241 A a -67 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{15 f c}-\frac {\left (896 A a -102 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{15 f c}+\frac {2 \left (887 A a -269 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{105 f c}-\frac {2 \left (1259 A a -103 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{315 f c}-\frac {4 \left (1909 A a -213 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{105 f c}+\frac {\left (5444 A a -1508 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{105 f c}-\frac {\left (12374 A a -1228 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{315 f c}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{2} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(377\) |
Input:
int((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x,method=_RETURNV ERBOSE)
Output:
-4/315*(-42*I*B*a*exp(3*I*(f*x+e))+A*a-2*B*a-189*A*a*exp(4*I*(f*x+e))+210* B*a*exp(6*I*(f*x+e))-252*B*a*exp(4*I*(f*x+e))-36*A*a*exp(2*I*(f*x+e))+72*B *a*exp(2*I*(f*x+e))+315*I*A*a*exp(5*I*(f*x+e))-84*I*A*a*exp(3*I*(f*x+e))+9 *I*A*a*exp(I*(f*x+e))-18*I*B*a*exp(I*(f*x+e)))/(exp(I*(f*x+e))-I)^9/f/c^5
Time = 0.08 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.73 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=-\frac {2 \, {\left (A - 2 \, B\right )} a \cos \left (f x + e\right )^{5} - 8 \, {\left (A - 2 \, B\right )} a \cos \left (f x + e\right )^{4} - 25 \, {\left (A - 2 \, B\right )} a \cos \left (f x + e\right )^{3} + 5 \, {\left (4 \, A + 13 \, B\right )} a \cos \left (f x + e\right )^{2} - 35 \, {\left (A + B\right )} a \cos \left (f x + e\right ) - 70 \, {\left (A + B\right )} a + {\left (2 \, {\left (A - 2 \, B\right )} a \cos \left (f x + e\right )^{4} + 10 \, {\left (A - 2 \, B\right )} a \cos \left (f x + e\right )^{3} - 15 \, {\left (A - 2 \, B\right )} a \cos \left (f x + e\right )^{2} - 35 \, {\left (A + B\right )} a \cos \left (f x + e\right ) - 70 \, {\left (A + B\right )} a\right )} \sin \left (f x + e\right )}{315 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorith m="fricas")
Output:
-1/315*(2*(A - 2*B)*a*cos(f*x + e)^5 - 8*(A - 2*B)*a*cos(f*x + e)^4 - 25*( A - 2*B)*a*cos(f*x + e)^3 + 5*(4*A + 13*B)*a*cos(f*x + e)^2 - 35*(A + B)*a *cos(f*x + e) - 70*(A + B)*a + (2*(A - 2*B)*a*cos(f*x + e)^4 + 10*(A - 2*B )*a*cos(f*x + e)^3 - 15*(A - 2*B)*a*cos(f*x + e)^2 - 35*(A + B)*a*cos(f*x + e) - 70*(A + B)*a)*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*cos(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. 3232 vs. \(2 (160) = 320\).
Time = 17.42 (sec) , antiderivative size = 3232, normalized size of antiderivative = 18.36 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**5,x)
Output:
Piecewise((-630*A*a*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 - 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) + 1890*A* a*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) - 5250*A*a*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 + 11 340*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*ta n(e/2 + f*x/2) - 315*c**5*f) + 6930*A*a*tan(e/2 + f*x/2)**5/(315*c**5*f*ta n(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) - 3 15*c**5*f) - 7686*A*a*tan(e/2 + f*x/2)**4/(315*c**5*f*tan(e/2 + f*x/2)*...
Leaf count of result is larger than twice the leaf count of optimal. 1425 vs. \(2 (171) = 342\).
Time = 0.08 (sec) , antiderivative size = 1425, normalized size of antiderivative = 8.10 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorith m="maxima")
Output:
-2/315*(A*a*(432*sin(f*x + e)/(cos(f*x + e) + 1) - 1728*sin(f*x + e)^2/(co s(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*si n(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/(co s(f*x + e) + 1)^4 - 126*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*s in(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/(co s(f*x + e) + 1)^9) - 5*A*a*(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/(co s(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/(cos(f* x + e) + 1)^8 - c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) - 5*B*a*(45*si...
Time = 0.25 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.43 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=-\frac {2 \, {\left (315 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 945 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 315 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 2625 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 315 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 3465 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 945 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3843 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 441 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2247 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 609 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1143 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 81 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 207 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 99 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 58 \, A a - 11 \, B a\right )}}{315 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}} \] Input:
integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorith m="giac")
Output:
-2/315*(315*A*a*tan(1/2*f*x + 1/2*e)^8 - 945*A*a*tan(1/2*f*x + 1/2*e)^7 + 315*B*a*tan(1/2*f*x + 1/2*e)^7 + 2625*A*a*tan(1/2*f*x + 1/2*e)^6 - 315*B*a *tan(1/2*f*x + 1/2*e)^6 - 3465*A*a*tan(1/2*f*x + 1/2*e)^5 + 945*B*a*tan(1/ 2*f*x + 1/2*e)^5 + 3843*A*a*tan(1/2*f*x + 1/2*e)^4 - 441*B*a*tan(1/2*f*x + 1/2*e)^4 - 2247*A*a*tan(1/2*f*x + 1/2*e)^3 + 609*B*a*tan(1/2*f*x + 1/2*e) ^3 + 1143*A*a*tan(1/2*f*x + 1/2*e)^2 - 81*B*a*tan(1/2*f*x + 1/2*e)^2 - 207 *A*a*tan(1/2*f*x + 1/2*e) + 99*B*a*tan(1/2*f*x + 1/2*e) + 58*A*a - 11*B*a) /(c^5*f*(tan(1/2*f*x + 1/2*e) - 1)^9)
Time = 36.55 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.76 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {1357\,A\,a}{4}-\frac {461\,B\,a}{16}-\frac {635\,A\,a\,\cos \left (e+f\,x\right )}{4}+\frac {5\,B\,a\,\cos \left (e+f\,x\right )}{2}-\frac {1575\,A\,a\,\sin \left (e+f\,x\right )}{4}+\frac {945\,B\,a\,\sin \left (e+f\,x\right )}{8}-\frac {625\,A\,a\,\cos \left (2\,e+2\,f\,x\right )}{4}+\frac {121\,A\,a\,\cos \left (3\,e+3\,f\,x\right )}{4}+\frac {7\,A\,a\,\cos \left (4\,e+4\,f\,x\right )}{2}+\frac {95\,B\,a\,\cos \left (2\,e+2\,f\,x\right )}{4}-8\,B\,a\,\cos \left (3\,e+3\,f\,x\right )-\frac {7\,B\,a\,\cos \left (4\,e+4\,f\,x\right )}{16}+\frac {399\,A\,a\,\sin \left (2\,e+2\,f\,x\right )}{4}+\frac {141\,A\,a\,\sin \left (3\,e+3\,f\,x\right )}{4}-\frac {15\,A\,a\,\sin \left (4\,e+4\,f\,x\right )}{4}-\frac {231\,B\,a\,\sin \left (2\,e+2\,f\,x\right )}{8}-\frac {39\,B\,a\,\sin \left (3\,e+3\,f\,x\right )}{8}+\frac {15\,B\,a\,\sin \left (4\,e+4\,f\,x\right )}{16}\right )}{315\,c^5\,f\,\left (\frac {63\,\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}{8}-\frac {21\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}-\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{4}-\frac {9\,\sqrt {2}\,\cos \left (\frac {5\,e}{2}+\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{4}+\frac {9\,\sqrt {2}\,\cos \left (\frac {7\,e}{2}-\frac {\pi }{4}+\frac {7\,f\,x}{2}\right )}{16}+\frac {\sqrt {2}\,\cos \left (\frac {9\,e}{2}+\frac {\pi }{4}+\frac {9\,f\,x}{2}\right )}{16}\right )} \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x)))/(c - c*sin(e + f*x))^5,x)
Output:
(2*cos(e/2 + (f*x)/2)*((1357*A*a)/4 - (461*B*a)/16 - (635*A*a*cos(e + f*x) )/4 + (5*B*a*cos(e + f*x))/2 - (1575*A*a*sin(e + f*x))/4 + (945*B*a*sin(e + f*x))/8 - (625*A*a*cos(2*e + 2*f*x))/4 + (121*A*a*cos(3*e + 3*f*x))/4 + (7*A*a*cos(4*e + 4*f*x))/2 + (95*B*a*cos(2*e + 2*f*x))/4 - 8*B*a*cos(3*e + 3*f*x) - (7*B*a*cos(4*e + 4*f*x))/16 + (399*A*a*sin(2*e + 2*f*x))/4 + (14 1*A*a*sin(3*e + 3*f*x))/4 - (15*A*a*sin(4*e + 4*f*x))/4 - (231*B*a*sin(2*e + 2*f*x))/8 - (39*B*a*sin(3*e + 3*f*x))/8 + (15*B*a*sin(4*e + 4*f*x))/16) )/(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)*cos((9*e)/ 2 + pi/4 + (9*f*x)/2))/16))
Time = 0.19 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.93 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {2 a \left (-35 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9} a -315 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7} a -315 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7} b +315 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6} a +315 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6} b -945 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} a -945 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} b +567 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} a +441 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} b -693 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} a -609 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} b +117 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a +81 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} b -108 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) a -99 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b -23 a +11 b \right )}{315 c^{5} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}-9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+36 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-84 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+126 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-126 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+84 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-36 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )} \] Input:
int((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x)
Output:
(2*a*( - 35*tan((e + f*x)/2)**9*a - 315*tan((e + f*x)/2)**7*a - 315*tan((e + f*x)/2)**7*b + 315*tan((e + f*x)/2)**6*a + 315*tan((e + f*x)/2)**6*b - 945*tan((e + f*x)/2)**5*a - 945*tan((e + f*x)/2)**5*b + 567*tan((e + f*x)/ 2)**4*a + 441*tan((e + f*x)/2)**4*b - 693*tan((e + f*x)/2)**3*a - 609*tan( (e + f*x)/2)**3*b + 117*tan((e + f*x)/2)**2*a + 81*tan((e + f*x)/2)**2*b - 108*tan((e + f*x)/2)*a - 99*tan((e + f*x)/2)*b - 23*a + 11*b))/(315*c**5* f*(tan((e + f*x)/2)**9 - 9*tan((e + f*x)/2)**8 + 36*tan((e + f*x)/2)**7 - 84*tan((e + f*x)/2)**6 + 126*tan((e + f*x)/2)**5 - 126*tan((e + f*x)/2)**4 + 84*tan((e + f*x)/2)**3 - 36*tan((e + f*x)/2)**2 + 9*tan((e + f*x)/2) - 1))