Integrand size = 36, antiderivative size = 77 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}+\frac {a^3 (A-8 B) c^2 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^7} \] Output:
1/9*a^3*(A+B)*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^8+1/63*a^3*(A-8*B)*c^2*c os(f*x+e)^7/f/(c-c*sin(f*x+e))^7
Leaf count is larger than twice the leaf count of optimal. \(283\) vs. \(2(77)=154\).
Time = 13.22 (sec) , antiderivative size = 283, normalized size of antiderivative = 3.68 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=-\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \left (315 (A-B) \cos \left (\frac {1}{2} (e+f x)\right )-189 (A-B) \cos \left (\frac {3}{2} (e+f x)\right )-63 A \cos \left (\frac {5}{2} (e+f x)\right )+63 B \cos \left (\frac {5}{2} (e+f x)\right )+9 A \cos \left (\frac {7}{2} (e+f x)\right )-9 B \cos \left (\frac {7}{2} (e+f x)\right )+189 A \sin \left (\frac {1}{2} (e+f x)\right )+693 B \sin \left (\frac {1}{2} (e+f x)\right )+105 A \sin \left (\frac {3}{2} (e+f x)\right )+483 B \sin \left (\frac {3}{2} (e+f x)\right )-27 A \sin \left (\frac {5}{2} (e+f x)\right )-225 B \sin \left (\frac {5}{2} (e+f x)\right )-63 B \sin \left (\frac {7}{2} (e+f x)\right )-A \sin \left (\frac {9}{2} (e+f x)\right )+8 B \sin \left (\frac {9}{2} (e+f x)\right )\right )}{504 c^5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (-1+\sin (e+f x))^5} \] Input:
Integrate[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x ])^5,x]
Output:
-1/504*(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*(31 5*(A - B)*Cos[(e + f*x)/2] - 189*(A - B)*Cos[(3*(e + f*x))/2] - 63*A*Cos[( 5*(e + f*x))/2] + 63*B*Cos[(5*(e + f*x))/2] + 9*A*Cos[(7*(e + f*x))/2] - 9 *B*Cos[(7*(e + f*x))/2] + 189*A*Sin[(e + f*x)/2] + 693*B*Sin[(e + f*x)/2] + 105*A*Sin[(3*(e + f*x))/2] + 483*B*Sin[(3*(e + f*x))/2] - 27*A*Sin[(5*(e + f*x))/2] - 225*B*Sin[(5*(e + f*x))/2] - 63*B*Sin[(7*(e + f*x))/2] - A*S in[(9*(e + f*x))/2] + 8*B*Sin[(9*(e + f*x))/2]))/(c^5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6*(-1 + Sin[e + f*x])^5)
Time = 0.54 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3446, 3042, 3338, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5}dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a^3 c^3 \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \int \frac {\cos (e+f x)^6 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^8}dx\) |
| \(\Big \downarrow \) 3338 |
| \(\displaystyle a^3 c^3 \left (\frac {(A-8 B) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7}dx}{9 c}+\frac {(A+B) \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^3 c^3 \left (\frac {(A-8 B) \int \frac {\cos (e+f x)^6}{(c-c \sin (e+f x))^7}dx}{9 c}+\frac {(A+B) \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}\right )\) |
| \(\Big \downarrow \) 3150 |
| \(\displaystyle a^3 c^3 \left (\frac {(A-8 B) \cos ^7(e+f x)}{63 c f (c-c \sin (e+f x))^7}+\frac {(A+B) \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^8}\right )\) |
Input:
Int[((a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5,x ]
Output:
a^3*c^3*(((A + B)*Cos[e + f*x]^7)/(9*f*(c - c*Sin[e + f*x])^8) + ((A - 8*B )*Cos[e + f*x]^7)/(63*c*f*(c - c*Sin[e + f*x])^7))
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_)*((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]))
Leaf count of result is larger than twice the leaf count of optimal. \(172\) vs. \(2(73)=146\).
Time = 1.99 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.25
| method | result | size |
| parallelrisch | \(-\frac {2 a^{3} \left (A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+\left (-A +B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+\frac {\left (23 A +5 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{3}+5 \left (-A +B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+\left (11 A +3 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+3 \left (-A +B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+\frac {\left (25 A +3 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{7}+\frac {\left (-A +B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7}+\frac {8 A}{63}-\frac {B}{63}\right )}{f \,c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(173\) |
| derivativedivides | \(\frac {2 a^{3} \left (-\frac {128 A +128 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {928 A +864 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {992 A +800 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {512 A +512 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {680 A +440 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {304 A +144 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {14 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {86 A +26 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{5}}\) | \(205\) |
| default | \(\frac {2 a^{3} \left (-\frac {128 A +128 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {928 A +864 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {992 A +800 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {512 A +512 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {680 A +440 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {304 A +144 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {14 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {86 A +26 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{5}}\) | \(205\) |
| risch | \(\frac {-\frac {2 a^{3} A}{63}+\frac {16 a^{3} B}{63}-10 i A \,a^{3} {\mathrm e}^{5 i \left (f x +e \right )}+2 i A \,a^{3} {\mathrm e}^{7 i \left (f x +e \right )}-\frac {2 i A \,a^{3} {\mathrm e}^{i \left (f x +e \right )}}{7}-\frac {6 A \,a^{3} {\mathrm e}^{2 i \left (f x +e \right )}}{7}-\frac {50 B \,a^{3} {\mathrm e}^{2 i \left (f x +e \right )}}{7}-\frac {10 A \,a^{3} {\mathrm e}^{6 i \left (f x +e \right )}}{3}+2 B \,a^{3} {\mathrm e}^{8 i \left (f x +e \right )}+6 A \,a^{3} {\mathrm e}^{4 i \left (f x +e \right )}-\frac {46 B \,a^{3} {\mathrm e}^{6 i \left (f x +e \right )}}{3}+22 B \,a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+\frac {2 i B \,a^{3} {\mathrm e}^{i \left (f x +e \right )}}{7}+10 i B \,a^{3} {\mathrm e}^{5 i \left (f x +e \right )}+6 i A \,a^{3} {\mathrm e}^{3 i \left (f x +e \right )}-2 i B \,a^{3} {\mathrm e}^{7 i \left (f x +e \right )}-6 i B \,a^{3} {\mathrm e}^{3 i \left (f x +e \right )}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9} f \,c^{5}}\) | \(269\) |
| norman | \(\frac {-\frac {16 a^{3} A -2 a^{3} B}{63 f c}-\frac {2 a^{3} A \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{16}}{f c}+\frac {2 \left (a^{3} A -a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{15}}{f c}+\frac {\left (2 a^{3} A -2 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{7 f c}+\frac {6 \left (3 a^{3} A -3 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{f c}+\frac {10 \left (5 a^{3} A -5 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{7 f c}-\frac {10 \left (7 a^{3} A +a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{14}}{3 f c}+\frac {2 \left (29 a^{3} A -29 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{f c}+\frac {2 \left (125 a^{3} A -125 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{7 f c}-\frac {2 \left (143 a^{3} A +29 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{3 f c}-\frac {2 \left (257 a^{3} A +23 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{63 f c}+\frac {2 \left (277 a^{3} A -277 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{7 f c}+\frac {2 \left (323 a^{3} A -323 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{7 f c}-\frac {2 \left (547 a^{3} A +97 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{21 f c}-\frac {2 \left (683 a^{3} A +157 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{7 f c}-\frac {2 \left (4637 a^{3} A +1019 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{63 f c}-\frac {2 \left (7061 a^{3} A +1661 a^{3} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{63 f c}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{4} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(554\) |
Input:
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x,method=_RETUR NVERBOSE)
Output:
-2*a^3*(A*tan(1/2*f*x+1/2*e)^8+(-A+B)*tan(1/2*f*x+1/2*e)^7+1/3*(23*A+5*B)* tan(1/2*f*x+1/2*e)^6+5*(-A+B)*tan(1/2*f*x+1/2*e)^5+(11*A+3*B)*tan(1/2*f*x+ 1/2*e)^4+3*(-A+B)*tan(1/2*f*x+1/2*e)^3+1/7*(25*A+3*B)*tan(1/2*f*x+1/2*e)^2 +1/7*(-A+B)*tan(1/2*f*x+1/2*e)+8/63*A-1/63*B)/f/c^5/(tan(1/2*f*x+1/2*e)-1) ^9
Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (75) = 150\).
Time = 0.09 (sec) , antiderivative size = 331, normalized size of antiderivative = 4.30 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=-\frac {{\left (A - 8 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} - {\left (4 \, A + 31 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + {\left (19 \, A + 37 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 4 \, {\left (13 \, A + 22 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 28 \, {\left (A + B\right )} a^{3} \cos \left (f x + e\right ) - 56 \, {\left (A + B\right )} a^{3} + {\left ({\left (A - 8 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + {\left (5 \, A + 23 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 12 \, {\left (2 \, A + 5 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 28 \, {\left (A + B\right )} a^{3} \cos \left (f x + e\right ) - 56 \, {\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )}{63 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algori thm="fricas")
Output:
-1/63*((A - 8*B)*a^3*cos(f*x + e)^5 - (4*A + 31*B)*a^3*cos(f*x + e)^4 + (1 9*A + 37*B)*a^3*cos(f*x + e)^3 + 4*(13*A + 22*B)*a^3*cos(f*x + e)^2 - 28*( A + B)*a^3*cos(f*x + e) - 56*(A + B)*a^3 + ((A - 8*B)*a^3*cos(f*x + e)^4 + (5*A + 23*B)*a^3*cos(f*x + e)^3 + 12*(2*A + 5*B)*a^3*cos(f*x + e)^2 - 28* (A + B)*a^3*cos(f*x + e) - 56*(A + B)*a^3)*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. 3262 vs. \(2 (68) = 136\).
Time = 44.16 (sec) , antiderivative size = 3262, normalized size of antiderivative = 42.36 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**5,x)
Output:
Piecewise((-126*A*a**3*tan(e/2 + f*x/2)**8/(63*c**5*f*tan(e/2 + f*x/2)**9 - 567*c**5*f*tan(e/2 + f*x/2)**8 + 2268*c**5*f*tan(e/2 + f*x/2)**7 - 5292* c**5*f*tan(e/2 + f*x/2)**6 + 7938*c**5*f*tan(e/2 + f*x/2)**5 - 7938*c**5*f *tan(e/2 + f*x/2)**4 + 5292*c**5*f*tan(e/2 + f*x/2)**3 - 2268*c**5*f*tan(e /2 + f*x/2)**2 + 567*c**5*f*tan(e/2 + f*x/2) - 63*c**5*f) + 126*A*a**3*tan (e/2 + f*x/2)**7/(63*c**5*f*tan(e/2 + f*x/2)**9 - 567*c**5*f*tan(e/2 + f*x /2)**8 + 2268*c**5*f*tan(e/2 + f*x/2)**7 - 5292*c**5*f*tan(e/2 + f*x/2)**6 + 7938*c**5*f*tan(e/2 + f*x/2)**5 - 7938*c**5*f*tan(e/2 + f*x/2)**4 + 529 2*c**5*f*tan(e/2 + f*x/2)**3 - 2268*c**5*f*tan(e/2 + f*x/2)**2 + 567*c**5* f*tan(e/2 + f*x/2) - 63*c**5*f) - 966*A*a**3*tan(e/2 + f*x/2)**6/(63*c**5* f*tan(e/2 + f*x/2)**9 - 567*c**5*f*tan(e/2 + f*x/2)**8 + 2268*c**5*f*tan(e /2 + f*x/2)**7 - 5292*c**5*f*tan(e/2 + f*x/2)**6 + 7938*c**5*f*tan(e/2 + f *x/2)**5 - 7938*c**5*f*tan(e/2 + f*x/2)**4 + 5292*c**5*f*tan(e/2 + f*x/2)* *3 - 2268*c**5*f*tan(e/2 + f*x/2)**2 + 567*c**5*f*tan(e/2 + f*x/2) - 63*c* *5*f) + 630*A*a**3*tan(e/2 + f*x/2)**5/(63*c**5*f*tan(e/2 + f*x/2)**9 - 56 7*c**5*f*tan(e/2 + f*x/2)**8 + 2268*c**5*f*tan(e/2 + f*x/2)**7 - 5292*c**5 *f*tan(e/2 + f*x/2)**6 + 7938*c**5*f*tan(e/2 + f*x/2)**5 - 7938*c**5*f*tan (e/2 + f*x/2)**4 + 5292*c**5*f*tan(e/2 + f*x/2)**3 - 2268*c**5*f*tan(e/2 + f*x/2)**2 + 567*c**5*f*tan(e/2 + f*x/2) - 63*c**5*f) - 1386*A*a**3*tan(e/ 2 + f*x/2)**4/(63*c**5*f*tan(e/2 + f*x/2)**9 - 567*c**5*f*tan(e/2 + f*x...
Leaf count of result is larger than twice the leaf count of optimal. 2701 vs. \(2 (75) = 150\).
Time = 0.13 (sec) , antiderivative size = 2701, normalized size of antiderivative = 35.08 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algori thm="maxima")
Output:
-2/315*(A*a^3*(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) - 15*A*a^3*(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^3...
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (75) = 150\).
Time = 0.30 (sec) , antiderivative size = 285, normalized size of antiderivative = 3.70 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=-\frac {2 \, {\left (63 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 63 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 63 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 483 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 105 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 315 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 315 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 693 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 189 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 189 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 189 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 225 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 27 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 9 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 8 \, A a^{3} - B a^{3}\right )}}{63 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}} \] Input:
integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algori thm="giac")
Output:
-2/63*(63*A*a^3*tan(1/2*f*x + 1/2*e)^8 - 63*A*a^3*tan(1/2*f*x + 1/2*e)^7 + 63*B*a^3*tan(1/2*f*x + 1/2*e)^7 + 483*A*a^3*tan(1/2*f*x + 1/2*e)^6 + 105* B*a^3*tan(1/2*f*x + 1/2*e)^6 - 315*A*a^3*tan(1/2*f*x + 1/2*e)^5 + 315*B*a^ 3*tan(1/2*f*x + 1/2*e)^5 + 693*A*a^3*tan(1/2*f*x + 1/2*e)^4 + 189*B*a^3*ta n(1/2*f*x + 1/2*e)^4 - 189*A*a^3*tan(1/2*f*x + 1/2*e)^3 + 189*B*a^3*tan(1/ 2*f*x + 1/2*e)^3 + 225*A*a^3*tan(1/2*f*x + 1/2*e)^2 + 27*B*a^3*tan(1/2*f*x + 1/2*e)^2 - 9*A*a^3*tan(1/2*f*x + 1/2*e) + 9*B*a^3*tan(1/2*f*x + 1/2*e) + 8*A*a^3 - B*a^3)/(c^5*f*(tan(1/2*f*x + 1/2*e) - 1)^9)
Time = 36.43 (sec) , antiderivative size = 346, normalized size of antiderivative = 4.49 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {1013\,A\,a^3}{16}+\frac {149\,B\,a^3}{16}-\frac {113\,A\,a^3\,\cos \left (2\,e+2\,f\,x\right )}{4}+\frac {37\,A\,a^3\,\cos \left (3\,e+3\,f\,x\right )}{8}+\frac {7\,A\,a^3\,\cos \left (4\,e+4\,f\,x\right )}{16}-\frac {41\,B\,a^3\,\cos \left (2\,e+2\,f\,x\right )}{4}+\frac {19\,B\,a^3\,\cos \left (3\,e+3\,f\,x\right )}{8}+\frac {7\,B\,a^3\,\cos \left (4\,e+4\,f\,x\right )}{16}+\frac {63\,A\,a^3\,\sin \left (2\,e+2\,f\,x\right )}{8}+\frac {9\,A\,a^3\,\sin \left (3\,e+3\,f\,x\right )}{2}-\frac {9\,A\,a^3\,\sin \left (4\,e+4\,f\,x\right )}{16}-\frac {63\,B\,a^3\,\sin \left (2\,e+2\,f\,x\right )}{8}-\frac {9\,B\,a^3\,\sin \left (3\,e+3\,f\,x\right )}{2}+\frac {9\,B\,a^3\,\sin \left (4\,e+4\,f\,x\right )}{16}-\frac {257\,A\,a^3\,\cos \left (e+f\,x\right )}{8}-\frac {23\,B\,a^3\,\cos \left (e+f\,x\right )}{8}-\frac {63\,A\,a^3\,\sin \left (e+f\,x\right )}{2}+\frac {63\,B\,a^3\,\sin \left (e+f\,x\right )}{2}\right )}{63\,c^5\,f\,\left (\frac {63\,\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}{8}-\frac {21\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}-\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{4}-\frac {9\,\sqrt {2}\,\cos \left (\frac {5\,e}{2}+\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{4}+\frac {9\,\sqrt {2}\,\cos \left (\frac {7\,e}{2}-\frac {\pi }{4}+\frac {7\,f\,x}{2}\right )}{16}+\frac {\sqrt {2}\,\cos \left (\frac {9\,e}{2}+\frac {\pi }{4}+\frac {9\,f\,x}{2}\right )}{16}\right )} \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3)/(c - c*sin(e + f*x))^5,x )
Output:
(2*cos(e/2 + (f*x)/2)*((1013*A*a^3)/16 + (149*B*a^3)/16 - (113*A*a^3*cos(2 *e + 2*f*x))/4 + (37*A*a^3*cos(3*e + 3*f*x))/8 + (7*A*a^3*cos(4*e + 4*f*x) )/16 - (41*B*a^3*cos(2*e + 2*f*x))/4 + (19*B*a^3*cos(3*e + 3*f*x))/8 + (7* B*a^3*cos(4*e + 4*f*x))/16 + (63*A*a^3*sin(2*e + 2*f*x))/8 + (9*A*a^3*sin( 3*e + 3*f*x))/2 - (9*A*a^3*sin(4*e + 4*f*x))/16 - (63*B*a^3*sin(2*e + 2*f* x))/8 - (9*B*a^3*sin(3*e + 3*f*x))/2 + (9*B*a^3*sin(4*e + 4*f*x))/16 - (25 7*A*a^3*cos(e + f*x))/8 - (23*B*a^3*cos(e + f*x))/8 - (63*A*a^3*sin(e + f* x))/2 + (63*B*a^3*sin(e + f*x))/2))/(63*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 = 0.17 (sec) , antiderivative size = 339, normalized size of antiderivative = 4.40 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {2 a^{3} \left (-7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9} a -189 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7} a -63 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7} b +105 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6} a -105 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6} b -567 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} a -315 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} b +189 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} a -189 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} b -399 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} a -189 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} b +27 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a -27 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} b -54 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) a -9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b -a +b \right )}{63 c^{5} f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}-9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+36 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-84 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+126 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-126 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+84 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-36 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )} \] Input:
int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x)
Output:
(2*a**3*( - 7*tan((e + f*x)/2)**9*a - 189*tan((e + f*x)/2)**7*a - 63*tan(( e + f*x)/2)**7*b + 105*tan((e + f*x)/2)**6*a - 105*tan((e + f*x)/2)**6*b - 567*tan((e + f*x)/2)**5*a - 315*tan((e + f*x)/2)**5*b + 189*tan((e + f*x) /2)**4*a - 189*tan((e + f*x)/2)**4*b - 399*tan((e + f*x)/2)**3*a - 189*tan ((e + f*x)/2)**3*b + 27*tan((e + f*x)/2)**2*a - 27*tan((e + f*x)/2)**2*b - 54*tan((e + f*x)/2)*a - 9*tan((e + f*x)/2)*b - a + b))/(63*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))