Integrand size = 40, antiderivative size = 196 \[ \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{40 c f (c-c \sin (e+f x))^{11/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{160 c^2 f (c-c \sin (e+f x))^{9/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{960 c^3 f (c-c \sin (e+f x))^{7/2}} \] Output:
1/12*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/f/(c-c*sin(f*x+e))^(13/2)+1/4 0*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/c/f/(c-c*sin(f*x+e))^(11/2)+1/ 160*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/c^2/f/(c-c*sin(f*x+e))^(9/2) +1/960*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/c^3/f/(c-c*sin(f*x+e))^(7 /2)
Time = 16.56 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.73 \[ \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} (29 A+13 B-15 (A+B) \cos (2 (e+f x))+6 (6 A+7 B) \sin (e+f x)-10 B \sin (3 (e+f x)))}{120 c^6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x))^6 \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(13/2),x]
Output:
(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(29* A + 13*B - 15*(A + B)*Cos[2*(e + f*x)] + 6*(6*A + 7*B)*Sin[e + f*x] - 10*B *Sin[3*(e + f*x)]))/(120*c^6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^6*Sqrt[c - c*Sin[e + f*x]])
Time = 0.98 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 3451, 3042, 3222, 3042, 3222, 3042, 3221}
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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}}dx\) |
| \(\Big \downarrow \) 3451 |
| \(\displaystyle \frac {(A-3 B) \int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{11/2}}dx}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A-3 B) \int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{11/2}}dx}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}\) |
| \(\Big \downarrow \) 3222 |
| \(\displaystyle \frac {(A-3 B) \left (\frac {\int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{9/2}}dx}{5 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{10 f (c-c \sin (e+f x))^{11/2}}\right )}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A-3 B) \left (\frac {\int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{9/2}}dx}{5 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{10 f (c-c \sin (e+f x))^{11/2}}\right )}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}\) |
| \(\Big \downarrow \) 3222 |
| \(\displaystyle \frac {(A-3 B) \left (\frac {\frac {\int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{7/2}}dx}{8 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{9/2}}}{5 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{10 f (c-c \sin (e+f x))^{11/2}}\right )}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A-3 B) \left (\frac {\frac {\int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{7/2}}dx}{8 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{9/2}}}{5 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{10 f (c-c \sin (e+f x))^{11/2}}\right )}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}\) |
| \(\Big \downarrow \) 3221 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {(A-3 B) \left (\frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{10 f (c-c \sin (e+f x))^{11/2}}+\frac {\frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{48 c f (c-c \sin (e+f x))^{7/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{8 f (c-c \sin (e+f x))^{9/2}}}{5 c}\right )}{4 c}\) |
Input:
Int[((a + a*Sin[e + f*x])^(5/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]) ^(13/2),x]
Output:
((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(12*f*(c - c*Sin[e + f*x ])^(13/2)) + ((A - 3*B)*((Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(10*f*( c - c*Sin[e + f*x])^(11/2)) + ((Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/( 8*f*(c - c*Sin[e + f*x])^(9/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2) )/(48*c*f*(c - c*Sin[e + f*x])^(7/2)))/(5*c)))/(4*c)
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*( (c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && Ne Q[m, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*( (c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))), x] + Simp[(m + n + 1)/(a*(2*m + 1) ) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; Free Q[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + 1], 0] && NeQ[m, -2^(-1)] && (SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Sim p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] + Simp[(a*B*(m - n) + A*b*(m + n + 1))/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[ {a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0 ] && (LtQ[m, -2^(-1)] || (ILtQ[m + n, 0] && !SumSimplerQ[n, 1])) && NeQ[2* m + 1, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(494\) vs. \(2(172)=344\).
Time = 10.49 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.53
| method | result | size |
| parts | \(\frac {A \sqrt {4}\, \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, \left (\cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}+3 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{4}+6 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}+10\right ) a^{2} \tan \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{5} \sec \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}}{1920 f \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, c^{6}}-\frac {2 B \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \sqrt {\left (2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) a}\, \left (8 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-16 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-22 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+30 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+15\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a^{2}}{15 f \left (-1+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) \left (32 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}-64 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-80 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-48 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+160 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+80 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-40 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+10 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-40 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right ) \sqrt {-\left (2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) c}\, c^{6}}\) | \(495\) |
| default | \(\frac {a^{2} \left (A \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-\frac {1}{2}\right ) \left (\cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}+3 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{4}+6 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}+10\right ) \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}-2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-\frac {5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{2}-\frac {3 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\frac {5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}-\frac {5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{4}+\frac {5 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )}{16}-\frac {5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{4}-\frac {1}{32}\right ) \tan \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{5} \sec \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}-48 \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} B \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \left (\frac {\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{3}+\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{3}-\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {11 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{12}+\frac {5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{4}+\frac {5}{8}\right )\right ) \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}}{15 \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, c^{6} f \left (-1+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) \left (32 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}-64 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-80 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-48 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+160 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+80 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-40 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+10 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-40 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )}\) | \(604\) |
Input:
int((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(13/2),x,meth od=_RETURNVERBOSE)
Output:
1/1920*A/f*4^(1/2)*(a*sin(1/4*Pi+1/2*f*x+1/2*e)^2)^(1/2)*(cos(1/4*Pi+1/2*f *x+1/2*e)^6+3*cos(1/4*Pi+1/2*f*x+1/2*e)^4+6*cos(1/4*Pi+1/2*f*x+1/2*e)^2+10 )*a^2/(c*cos(1/4*Pi+1/2*f*x+1/2*e)^2)^(1/2)/c^6*tan(1/4*Pi+1/2*f*x+1/2*e)^ 5*sec(1/4*Pi+1/2*f*x+1/2*e)^6-2/15*B/f*cos(1/2*f*x+1/2*e)^2*((2*sin(1/2*f* x+1/2*e)*cos(1/2*f*x+1/2*e)+1)*a)^(1/2)*(8*cos(1/2*f*x+1/2*e)^8-16*cos(1/2 *f*x+1/2*e)^6+24*cos(1/2*f*x+1/2*e)^5*sin(1/2*f*x+1/2*e)-22*cos(1/2*f*x+1/ 2*e)^4-24*cos(1/2*f*x+1/2*e)^3*sin(1/2*f*x+1/2*e)+30*cos(1/2*f*x+1/2*e)^2+ 15)*sin(1/2*f*x+1/2*e)^2*a^2/(-1+2*cos(1/2*f*x+1/2*e)^2)/(32*sin(1/2*f*x+1 /2*e)*cos(1/2*f*x+1/2*e)^9-64*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)^7-80*c os(1/2*f*x+1/2*e)^8-48*cos(1/2*f*x+1/2*e)^5*sin(1/2*f*x+1/2*e)+160*cos(1/2 *f*x+1/2*e)^6+80*cos(1/2*f*x+1/2*e)^3*sin(1/2*f*x+1/2*e)-40*cos(1/2*f*x+1/ 2*e)^4+10*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)-40*cos(1/2*f*x+1/2*e)^2-1) /(-(2*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)-1)*c)^(1/2)/c^6
Time = 0.11 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00 \[ \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {{\left (15 \, {\left (A + B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2 \, {\left (11 \, A + 7 \, B\right )} a^{2} + 2 \, {\left (10 \, B a^{2} \cos \left (f x + e\right )^{2} - {\left (9 \, A + 13 \, B\right )} a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{60 \, {\left (c^{7} f \cos \left (f x + e\right )^{7} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} - 32 \, c^{7} f \cos \left (f x + e\right ) + 2 \, {\left (3 \, c^{7} f \cos \left (f x + e\right )^{5} - 16 \, c^{7} f \cos \left (f x + e\right )^{3} + 16 \, c^{7} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(13/2), x, algorithm="fricas")
Output:
1/60*(15*(A + B)*a^2*cos(f*x + e)^2 - 2*(11*A + 7*B)*a^2 + 2*(10*B*a^2*cos (f*x + e)^2 - (9*A + 13*B)*a^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqr t(-c*sin(f*x + e) + c)/(c^7*f*cos(f*x + e)^7 - 18*c^7*f*cos(f*x + e)^5 + 4 8*c^7*f*cos(f*x + e)^3 - 32*c^7*f*cos(f*x + e) + 2*(3*c^7*f*cos(f*x + e)^5 - 16*c^7*f*cos(f*x + e)^3 + 16*c^7*f*cos(f*x + e))*sin(f*x + e))
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**(5/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(13/2 ),x)
Output:
Timed out
\[ \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {13}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(13/2), x, algorithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(5/2)/(-c*sin(f*x + e) + c)^(13/2), x)
Exception generated. \[ \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(13/2), x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 45.08 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.82 \[ \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\left (A\,29{}\mathrm {i}+B\,13{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,16{}\mathrm {i}}{15\,c^7\,f}-\frac {a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\left (A\,1{}\mathrm {i}+B\,1{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,16{}\mathrm {i}}{c^7\,f}-\frac {32\,a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\left (6\,A+7\,B\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{5\,c^7\,f}+\frac {32\,B\,a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,c^7\,f}\right )}{-858\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}+858\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )-130\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )+2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (7\,e+7\,f\,x\right )+1144\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )-416\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )+24\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )} \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x)) ^(13/2),x)
Output:
((c - c*sin(e + f*x))^(1/2)*((a^2*exp(e*7i + f*x*7i)*(A*29i + B*13i)*(a + a*sin(e + f*x))^(1/2)*16i)/(15*c^7*f) - (a^2*exp(e*7i + f*x*7i)*cos(2*e + 2*f*x)*(A*1i + B*1i)*(a + a*sin(e + f*x))^(1/2)*16i)/(c^7*f) - (32*a^2*exp (e*7i + f*x*7i)*sin(e + f*x)*(6*A + 7*B)*(a + a*sin(e + f*x))^(1/2))/(5*c^ 7*f) + (32*B*a^2*exp(e*7i + f*x*7i)*sin(3*e + 3*f*x)*(a + a*sin(e + f*x))^ (1/2))/(3*c^7*f)))/(858*exp(e*7i + f*x*7i)*cos(3*e + 3*f*x) - 858*cos(e + f*x)*exp(e*7i + f*x*7i) - 130*exp(e*7i + f*x*7i)*cos(5*e + 5*f*x) + 2*exp( e*7i + f*x*7i)*cos(7*e + 7*f*x) + 1144*exp(e*7i + f*x*7i)*sin(2*e + 2*f*x) - 416*exp(e*7i + f*x*7i)*sin(4*e + 4*f*x) + 24*exp(e*7i + f*x*7i)*sin(6*e + 6*f*x))
\[ \int \frac {(a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{13/2}} \, dx =\text {Too large to display} \] Input:
int((a+a*sin(f*x+e))^(5/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(13/2),x)
Output:
(sqrt(c)*sqrt(a)*a**2*( - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**7 - 7*sin(e + f*x)**6 + 21*sin(e + f *x)**5 - 35*sin(e + f*x)**4 + 35*sin(e + f*x)**3 - 21*sin(e + f*x)**2 + 7* sin(e + f*x) - 1),x)*b - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x)**7 - 7*sin(e + f*x)**6 + 21*sin(e + f* x)**5 - 35*sin(e + f*x)**4 + 35*sin(e + f*x)**3 - 21*sin(e + f*x)**2 + 7*s in(e + f*x) - 1),x)*a - 2*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x)**7 - 7*sin(e + f*x)**6 + 21*sin(e + f *x)**5 - 35*sin(e + f*x)**4 + 35*sin(e + f*x)**3 - 21*sin(e + f*x)**2 + 7* sin(e + f*x) - 1),x)*b - 2*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x ) + 1)*sin(e + f*x))/(sin(e + f*x)**7 - 7*sin(e + f*x)**6 + 21*sin(e + f*x )**5 - 35*sin(e + f*x)**4 + 35*sin(e + f*x)**3 - 21*sin(e + f*x)**2 + 7*si n(e + f*x) - 1),x)*a - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x)**7 - 7*sin(e + f*x)**6 + 21*sin(e + f*x)**5 - 35*sin(e + f*x)**4 + 35*sin(e + f*x)**3 - 21*sin(e + f*x)**2 + 7*sin(e + f*x) - 1),x)*b - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1))/ (sin(e + f*x)**7 - 7*sin(e + f*x)**6 + 21*sin(e + f*x)**5 - 35*sin(e + f*x )**4 + 35*sin(e + f*x)**3 - 21*sin(e + f*x)**2 + 7*sin(e + f*x) - 1),x)*a) )/c**7