Integrand size = 26, antiderivative size = 148 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\frac {105 c^5 x}{2 a^2}+\frac {35 c^5 \cos ^3(e+f x)}{a^2 f}+\frac {105 c^5 \cos (e+f x) \sin (e+f x)}{2 a^2 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\frac {42 c^5 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^2} \] Output:
105/2*c^5*x/a^2+35*c^5*cos(f*x+e)^3/a^2/f+105/2*c^5*cos(f*x+e)*sin(f*x+e)/ a^2/f-2/3*a^4*c^5*cos(f*x+e)^9/f/(a+a*sin(f*x+e))^6+6*a^2*c^5*cos(f*x+e)^7 /f/(a+a*sin(f*x+e))^4+42*c^5*cos(f*x+e)^5/f/(a+a*sin(f*x+e))^2
Time = 12.54 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.86 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^5 \left (256 \sin \left (\frac {1}{2} (e+f x)\right )-128 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-1664 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+630 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+285 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-\cos (3 (e+f x)) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-21 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sin (2 (e+f x))\right )}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10} (a+a \sin (e+f x))^2} \] Input:
Integrate[(c - c*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^2,x]
Output:
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^5*(256*Sin[(e + f*x)/2] - 128*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 1664*Sin[(e + f*x) /2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 630*(e + f*x)*(Cos[(e + f*x) /2] + Sin[(e + f*x)/2])^3 + 285*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - Cos[3*(e + f*x)]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 21 *(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[2*(e + f*x)]))/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(a + a*Sin[e + f*x])^2)
Time = 0.83 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3215, 3042, 3159, 3042, 3159, 3042, 3159, 3042, 3161, 3042, 3115, 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.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^5}{(a \sin (e+f x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^5}{(a \sin (e+f x)+a)^2}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle a^5 c^5 \int \frac {\cos ^{10}(e+f x)}{(\sin (e+f x) a+a)^7}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \int \frac {\cos (e+f x)^{10}}{(\sin (e+f x) a+a)^7}dx\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^5 c^5 \left (-\frac {3 \int \frac {\cos ^8(e+f x)}{(\sin (e+f x) a+a)^5}dx}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {3 \int \frac {\cos (e+f x)^8}{(\sin (e+f x) a+a)^5}dx}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^5 c^5 \left (-\frac {3 \left (-\frac {7 \int \frac {\cos ^6(e+f x)}{(\sin (e+f x) a+a)^3}dx}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {3 \left (-\frac {7 \int \frac {\cos (e+f x)^6}{(\sin (e+f x) a+a)^3}dx}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle a^5 c^5 \left (-\frac {3 \left (-\frac {7 \left (\frac {5 \int \frac {\cos ^4(e+f x)}{\sin (e+f x) a+a}dx}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {3 \left (-\frac {7 \left (\frac {5 \int \frac {\cos (e+f x)^4}{\sin (e+f x) a+a}dx}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle a^5 c^5 \left (-\frac {3 \left (-\frac {7 \left (\frac {5 \left (\frac {\int \cos ^2(e+f x)dx}{a}+\frac {\cos ^3(e+f x)}{3 a f}\right )}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^5 c^5 \left (-\frac {3 \left (-\frac {7 \left (\frac {5 \left (\frac {\int \sin \left (e+f x+\frac {\pi }{2}\right )^2dx}{a}+\frac {\cos ^3(e+f x)}{3 a f}\right )}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^5 c^5 \left (-\frac {3 \left (-\frac {7 \left (\frac {5 \left (\frac {\frac {\int 1dx}{2}+\frac {\sin (e+f x) \cos (e+f x)}{2 f}}{a}+\frac {\cos ^3(e+f x)}{3 a f}\right )}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^5 c^5 \left (-\frac {3 \left (-\frac {7 \left (\frac {5 \left (\frac {\cos ^3(e+f x)}{3 a f}+\frac {\frac {\sin (e+f x) \cos (e+f x)}{2 f}+\frac {x}{2}}{a}\right )}{a^2}+\frac {2 \cos ^5(e+f x)}{a f (a \sin (e+f x)+a)^2}\right )}{a^2}-\frac {2 \cos ^7(e+f x)}{a f (a \sin (e+f x)+a)^4}\right )}{a^2}-\frac {2 \cos ^9(e+f x)}{3 a f (a \sin (e+f x)+a)^6}\right )\) |
Input:
Int[(c - c*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^2,x]
Output:
a^5*c^5*((-2*Cos[e + f*x]^9)/(3*a*f*(a + a*Sin[e + f*x])^6) - (3*((-2*Cos[ e + f*x]^7)/(a*f*(a + a*Sin[e + f*x])^4) - (7*((2*Cos[e + f*x]^5)/(a*f*(a + a*Sin[e + f*x])^2) + (5*(Cos[e + f*x]^3/(3*a*f) + (x/2 + (Cos[e + f*x]*S in[e + f*x])/(2*f))/a))/a^2))/a^2))/a^2)
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 86.33 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.84
| method | result | size |
| parallelrisch | \(-\frac {\left (\left (-1260 f x -1976\right ) \cos \left (3 f x +3 e \right )-4977 \cos \left (2 f x +2 e \right )-282 \cos \left (4 f x +4 e \right )+\cos \left (6 f x +6 e \right )+1727 \sin \left (3 f x +3 e \right )+21 \sin \left (5 f x +5 e \right )+\left (-3780 f x -5928\right ) \cos \left (f x +e \right )-342 \sin \left (f x +e \right )-2646\right ) c^{5}}{24 f \,a^{2} \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right )}\) | \(124\) |
| derivativedivides | \(\frac {2 c^{5} \left (\frac {\frac {7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{2}+23 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+48 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-\frac {7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+\frac {71}{3}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{3}}+\frac {105 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {48}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{2}}\) | \(138\) |
| default | \(\frac {2 c^{5} \left (\frac {\frac {7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{2}+23 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+48 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-\frac {7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+\frac {71}{3}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{3}}+\frac {105 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {48}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{2}}\) | \(138\) |
| risch | \(\frac {105 c^{5} x}{2 a^{2}}+\frac {7 i c^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{8 a^{2} f}+\frac {95 c^{5} {\mathrm e}^{i \left (f x +e \right )}}{8 a^{2} f}+\frac {95 c^{5} {\mathrm e}^{-i \left (f x +e \right )}}{8 a^{2} f}-\frac {7 i c^{5} {\mathrm e}^{-2 i \left (f x +e \right )}}{8 a^{2} f}+\frac {256 i c^{5} {\mathrm e}^{i \left (f x +e \right )}+160 c^{5} {\mathrm e}^{2 i \left (f x +e \right )}-\frac {416 c^{5}}{3}}{f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}-\frac {c^{5} \cos \left (3 f x +3 e \right )}{12 a^{2} f}\) | \(170\) |
| norman | \(\frac {\frac {494 c^{5}}{3 a f}+\frac {103 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{a f}+\frac {420 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{a}+\frac {840 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{a}+\frac {2625 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{2 a}+\frac {3675 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{2 a}+\frac {2100 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{a}+\frac {2100 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{a}+\frac {3675 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{2 a}+\frac {2625 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{2 a}+\frac {840 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{a}+\frac {420 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{a}+\frac {315 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}}{2 a}+\frac {105 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}}{2 a}+\frac {391 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {2983 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{3 a f}+\frac {5593 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 a f}+\frac {7385 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{3 a f}+\frac {10694 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{3 a f}+\frac {9490 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{3 a f}+\frac {10270 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}{3 a f}+\frac {6596 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{3 a f}+\frac {1655 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}{a f}+\frac {2311 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}}{3 a f}+\frac {323 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}}{a f}+\frac {315 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a}+\frac {105 c^{5} x}{2 a}}{\left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )^{5} a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(574\) |
Input:
int((c-c*sin(f*x+e))^5/(a+sin(f*x+e)*a)^2,x,method=_RETURNVERBOSE)
Output:
-1/24*((-1260*f*x-1976)*cos(3*f*x+3*e)-4977*cos(2*f*x+2*e)-282*cos(4*f*x+4 *e)+cos(6*f*x+6*e)+1727*sin(3*f*x+3*e)+21*sin(5*f*x+5*e)+(-3780*f*x-5928)* cos(f*x+e)-342*sin(f*x+e)-2646)*c^5/f/a^2/(cos(3*f*x+3*e)+3*cos(f*x+e))
Time = 0.14 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.61 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=-\frac {2 \, c^{5} \cos \left (f x + e\right )^{5} + 19 \, c^{5} \cos \left (f x + e\right )^{4} - 106 \, c^{5} \cos \left (f x + e\right )^{3} + 630 \, c^{5} f x - 64 \, c^{5} - 7 \, {\left (45 \, c^{5} f x - 77 \, c^{5}\right )} \cos \left (f x + e\right )^{2} + {\left (315 \, c^{5} f x + 598 \, c^{5}\right )} \cos \left (f x + e\right ) - {\left (2 \, c^{5} \cos \left (f x + e\right )^{4} - 17 \, c^{5} \cos \left (f x + e\right )^{3} - 630 \, c^{5} f x - 123 \, c^{5} \cos \left (f x + e\right )^{2} - 64 \, c^{5} - {\left (315 \, c^{5} f x + 662 \, c^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="fricas")
Output:
-1/6*(2*c^5*cos(f*x + e)^5 + 19*c^5*cos(f*x + e)^4 - 106*c^5*cos(f*x + e)^ 3 + 630*c^5*f*x - 64*c^5 - 7*(45*c^5*f*x - 77*c^5)*cos(f*x + e)^2 + (315*c ^5*f*x + 598*c^5)*cos(f*x + e) - (2*c^5*cos(f*x + e)^4 - 17*c^5*cos(f*x + e)^3 - 630*c^5*f*x - 123*c^5*cos(f*x + e)^2 - 64*c^5 - (315*c^5*f*x + 662* c^5)*cos(f*x + e))*sin(f*x + e))/(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e ) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))
Leaf count of result is larger than twice the leaf count of optimal. 3641 vs. \(2 (144) = 288\).
Time = 13.17 (sec) , antiderivative size = 3641, normalized size of antiderivative = 24.60 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:
integrate((c-c*sin(f*x+e))**5/(a+a*sin(f*x+e))**2,x)
Output:
Piecewise((315*c**5*f*x*tan(e/2 + f*x/2)**9/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2* f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)** 2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 945*c**5*f*x*tan(e/2 + f*x/2) **8/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36*a** 2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/ 2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2) **3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2* f) + 1890*c**5*f*x*tan(e/2 + f*x/2)**7/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18* a**2*f*tan(e/2 + f*x/2)**8 + 36*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan (e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x /2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 1 8*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 3150*c**5*f*x*tan(e/2 + f*x/2)**6/ (6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2*f*tan(e/2 + f*x/2)**8 + 36*a**2*f* tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(e/2 + f*x/2)**6 + 72*a**2*f*tan(e/2 + f*x/2)**5 + 72*a**2*f*tan(e/2 + f*x/2)**4 + 60*a**2*f*tan(e/2 + f*x/2)**3 + 36*a**2*f*tan(e/2 + f*x/2)**2 + 18*a**2*f*tan(e/2 + f*x/2) + 6*a**2*f) + 3780*c**5*f*x*tan(e/2 + f*x/2)**5/(6*a**2*f*tan(e/2 + f*x/2)**9 + 18*a**2 *f*tan(e/2 + f*x/2)**8 + 36*a**2*f*tan(e/2 + f*x/2)**7 + 60*a**2*f*tan(...
Leaf count of result is larger than twice the leaf count of optimal. 1304 vs. \(2 (142) = 284\).
Time = 0.14 (sec) , antiderivative size = 1304, normalized size of antiderivative = 8.81 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \] Input:
integrate((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="maxima")
Output:
1/3*(5*c^5*((75*sin(f*x + e)/(cos(f*x + e) + 1) + 97*sin(f*x + e)^2/(cos(f *x + e) + 1)^2 + 126*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 98*sin(f*x + e) ^4/(cos(f*x + e) + 1)^4 + 63*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 21*sin( f*x + e)^6/(cos(f*x + e) + 1)^6 + 32)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 5*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 7*a^2*sin(f*x + e)^ 3/(cos(f*x + e) + 1)^3 + 7*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5*a^2 *sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 3*a^2*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a^2*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 21*arctan(sin(f*x + e) /(cos(f*x + e) + 1))/a^2) + 2*c^5*((57*sin(f*x + e)/(cos(f*x + e) + 1) + 9 9*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 155*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 153*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 135*sin(f*x + e)^5/(cos( f*x + e) + 1)^5 + 85*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 45*sin(f*x + e) ^7/(cos(f*x + e) + 1)^7 + 15*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 24)/(a^ 2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 6*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 12*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 12*a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 10*a^2*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 6*a^2*sin(f*x + e)^7/(cos(f* x + e) + 1)^7 + 3*a^2*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + a^2*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) + 15*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^ 2) + 40*c^5*((12*sin(f*x + e)/(cos(f*x + e) + 1) + 11*sin(f*x + e)^2/(c...
Time = 0.16 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.29 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\frac {\frac {315 \, {\left (f x + e\right )} c^{5}}{a^{2}} + \frac {2 \, {\left (309 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 969 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 1693 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 3027 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 2901 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 3247 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1995 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1173 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 494 \, c^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3} a^{2}}}{6 \, f} \] Input:
integrate((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="giac")
Output:
1/6*(315*(f*x + e)*c^5/a^2 + 2*(309*c^5*tan(1/2*f*x + 1/2*e)^8 + 969*c^5*t an(1/2*f*x + 1/2*e)^7 + 1693*c^5*tan(1/2*f*x + 1/2*e)^6 + 3027*c^5*tan(1/2 *f*x + 1/2*e)^5 + 2901*c^5*tan(1/2*f*x + 1/2*e)^4 + 3247*c^5*tan(1/2*f*x + 1/2*e)^3 + 1995*c^5*tan(1/2*f*x + 1/2*e)^2 + 1173*c^5*tan(1/2*f*x + 1/2*e ) + 494*c^5)/((tan(1/2*f*x + 1/2*e)^3 + tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f *x + 1/2*e) + 1)^3*a^2))/f
Time = 20.00 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.51 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\frac {105\,c^5\,x}{2\,a^2}-\frac {\frac {105\,c^5\,\left (e+f\,x\right )}{2}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {315\,c^5\,\left (e+f\,x\right )}{2}-\frac {c^5\,\left (945\,e+945\,f\,x+2346\right )}{6}\right )-\frac {c^5\,\left (315\,e+315\,f\,x+988\right )}{6}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {315\,c^5\,\left (e+f\,x\right )}{2}-\frac {c^5\,\left (945\,e+945\,f\,x+618\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (315\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (1890\,e+1890\,f\,x+1938\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (315\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (1890\,e+1890\,f\,x+3990\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (525\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3150\,e+3150\,f\,x+3386\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (525\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3150\,e+3150\,f\,x+6494\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (630\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3780\,e+3780\,f\,x+5802\right )}{6}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (630\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3780\,e+3780\,f\,x+6054\right )}{6}\right )}{a^2\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3} \] Input:
int((c - c*sin(e + f*x))^5/(a + a*sin(e + f*x))^2,x)
Output:
(105*c^5*x)/(2*a^2) - ((105*c^5*(e + f*x))/2 + tan(e/2 + (f*x)/2)*((315*c^ 5*(e + f*x))/2 - (c^5*(945*e + 945*f*x + 2346))/6) - (c^5*(315*e + 315*f*x + 988))/6 + tan(e/2 + (f*x)/2)^8*((315*c^5*(e + f*x))/2 - (c^5*(945*e + 9 45*f*x + 618))/6) + tan(e/2 + (f*x)/2)^7*(315*c^5*(e + f*x) - (c^5*(1890*e + 1890*f*x + 1938))/6) + tan(e/2 + (f*x)/2)^2*(315*c^5*(e + f*x) - (c^5*( 1890*e + 1890*f*x + 3990))/6) + tan(e/2 + (f*x)/2)^6*(525*c^5*(e + f*x) - (c^5*(3150*e + 3150*f*x + 3386))/6) + tan(e/2 + (f*x)/2)^3*(525*c^5*(e + f *x) - (c^5*(3150*e + 3150*f*x + 6494))/6) + tan(e/2 + (f*x)/2)^4*(630*c^5* (e + f*x) - (c^5*(3780*e + 3780*f*x + 5802))/6) + tan(e/2 + (f*x)/2)^5*(63 0*c^5*(e + f*x) - (c^5*(3780*e + 3780*f*x + 6054))/6))/(a^2*f*(tan(e/2 + ( f*x)/2) + tan(e/2 + (f*x)/2)^2 + tan(e/2 + (f*x)/2)^3 + 1)^3)
Time = 0.19 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.51 \[ \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx=\frac {c^{5} \left (-2 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{4}+17 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{3}-102 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+315 \cos \left (f x +e \right ) \sin \left (f x +e \right ) f x -391 \cos \left (f x +e \right ) \sin \left (f x +e \right )+315 \cos \left (f x +e \right ) f x -206 \cos \left (f x +e \right )-2 \sin \left (f x +e \right )^{5}+19 \sin \left (f x +e \right )^{4}-119 \sin \left (f x +e \right )^{3}-315 \sin \left (f x +e \right )^{2} f x -865 \sin \left (f x +e \right )^{2}-630 \sin \left (f x +e \right ) f x -391 \sin \left (f x +e \right )-315 f x +206\right )}{6 a^{2} f \left (\cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )-\sin \left (f x +e \right )^{2}-2 \sin \left (f x +e \right )-1\right )} \] Input:
int((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x)
Output:
(c**5*( - 2*cos(e + f*x)*sin(e + f*x)**4 + 17*cos(e + f*x)*sin(e + f*x)**3 - 102*cos(e + f*x)*sin(e + f*x)**2 + 315*cos(e + f*x)*sin(e + f*x)*f*x - 391*cos(e + f*x)*sin(e + f*x) + 315*cos(e + f*x)*f*x - 206*cos(e + f*x) - 2*sin(e + f*x)**5 + 19*sin(e + f*x)**4 - 119*sin(e + f*x)**3 - 315*sin(e + f*x)**2*f*x - 865*sin(e + f*x)**2 - 630*sin(e + f*x)*f*x - 391*sin(e + f* x) - 315*f*x + 206))/(6*a**2*f*(cos(e + f*x)*sin(e + f*x) + cos(e + f*x) - sin(e + f*x)**2 - 2*sin(e + f*x) - 1))