Integrand size = 40, antiderivative size = 246 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{224 c^2 f (c-c \sin (e+f x))^{13/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{1120 c^3 f (c-c \sin (e+f x))^{11/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{8960 c^4 f (c-c \sin (e+f x))^{9/2}} \] Output:
1/16*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(17/2)+1/5 6*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c/f/(c-c*sin(f*x+e))^(15/2)+1/ 224*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c^2/f/(c-c*sin(f*x+e))^(13/2 )+1/1120*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c^3/f/(c-c*sin(f*x+e))^ (11/2)+1/8960*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c^4/f/(c-c*sin(f*x +e))^(9/2)
Time = 17.41 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.77 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx=\frac {(A+B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}-\frac {4 (3 A+5 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{7/2}}{7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}+\frac {(A+3 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}+\frac {(-A-7 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (a (1+\sin (e+f x)))^{7/2}}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}}+\frac {B \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{7/2}}{4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{17/2}} \] Input:
Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(17/2),x]
Output:
((A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2 ))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) - (4*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(7*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) + ((A + 3*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*S in[e + f*x])^(17/2)) + ((-A - 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7 *(a*(1 + Sin[e + f*x]))^(7/2))/(5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^ 7*(c - c*Sin[e + f*x])^(17/2)) + (B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^ 9*(a*(1 + Sin[e + f*x]))^(7/2))/(4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) ^7*(c - c*Sin[e + f*x])^(17/2))
Time = 1.23 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3451, 3042, 3222, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}}dx\) |
| \(\Big \downarrow \) 3451 |
| \(\displaystyle \frac {(A-3 B) \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{15/2}}dx}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A-3 B) \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{15/2}}dx}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}\) |
| \(\Big \downarrow \) 3222 |
| \(\displaystyle \frac {(A-3 B) \left (\frac {3 \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{13/2}}dx}{14 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\right )}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A-3 B) \left (\frac {3 \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{13/2}}dx}{14 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\right )}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}\) |
| \(\Big \downarrow \) 3222 |
| \(\displaystyle \frac {(A-3 B) \left (\frac {3 \left (\frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{11/2}}dx}{6 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}\right )}{14 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\right )}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A-3 B) \left (\frac {3 \left (\frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{11/2}}dx}{6 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}\right )}{14 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\right )}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}\) |
| \(\Big \downarrow \) 3222 |
| \(\displaystyle \frac {(A-3 B) \left (\frac {3 \left (\frac {\frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{9/2}}dx}{10 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}}{6 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}\right )}{14 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\right )}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A-3 B) \left (\frac {3 \left (\frac {\frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{9/2}}dx}{10 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}}{6 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}\right )}{14 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\right )}{4 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}\) |
| \(\Big \downarrow \) 3221 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}+\frac {(A-3 B) \left (\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {3 \left (\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{80 c f (c-c \sin (e+f x))^{9/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}}{6 c}\right )}{14 c}\right )}{4 c}\) |
Input:
Int[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]) ^(17/2),x]
Output:
((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(16*f*(c - c*Sin[e + f*x ])^(17/2)) + ((A - 3*B)*((Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(14*f*( c - c*Sin[e + f*x])^(15/2)) + (3*((Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2) )/(12*f*(c - c*Sin[e + f*x])^(13/2)) + ((Cos[e + f*x]*(a + a*Sin[e + f*x]) ^(7/2))/(10*f*(c - c*Sin[e + f*x])^(11/2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(80*c*f*(c - c*Sin[e + f*x])^(9/2)))/(6*c)))/(14*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. \(573\) vs. \(2(216)=432\).
Time = 15.05 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.33
| 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 )^{8}+4 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}+10 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{4}+20 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}+35\right ) a^{3} \tan \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{7} \sec \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{8}}{17920 f \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, c^{8}}-\frac {2 B \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}\, \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \left (\left (128 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}-256 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+224 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+32 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}-96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}-128 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+416 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-14 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-210 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-35\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a^{3}}{35 f \left (-1+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) \left (\left (128 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{13}-384 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{11}-288 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}+1216 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-392 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-280 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-14 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-448 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}+1344 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}-784 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}-672 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+476 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+84 \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^{8}}\) | \(574\) |
| default | \(\text {Expression too large to display}\) | \(830\) |
Input:
int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(17/2),x,meth od=_RETURNVERBOSE)
Output:
1/17920*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)^8+4*cos(1/4*Pi+1/2*f*x+1/2*e)^6+10*cos(1/4*Pi+1/2*f*x+1/2*e)^4+ 20*cos(1/4*Pi+1/2*f*x+1/2*e)^2+35)*a^3/(c*cos(1/4*Pi+1/2*f*x+1/2*e)^2)^(1/ 2)/c^8*tan(1/4*Pi+1/2*f*x+1/2*e)^7*sec(1/4*Pi+1/2*f*x+1/2*e)^8-2/35*B/f*(( 2*sin(1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)+1)*a)^(1/2)*cos(1/2*f*x+1/2*e)^2*( (128*cos(1/2*f*x+1/2*e)^9-256*cos(1/2*f*x+1/2*e)^7-96*cos(1/2*f*x+1/2*e)^5 +224*cos(1/2*f*x+1/2*e)^3)*sin(1/2*f*x+1/2*e)+32*cos(1/2*f*x+1/2*e)^12-96* cos(1/2*f*x+1/2*e)^10-128*cos(1/2*f*x+1/2*e)^8+416*cos(1/2*f*x+1/2*e)^6-14 *cos(1/2*f*x+1/2*e)^4-210*cos(1/2*f*x+1/2*e)^2-35)*sin(1/2*f*x+1/2*e)^2*a^ 3/(-1+2*cos(1/2*f*x+1/2*e)^2)/((128*cos(1/2*f*x+1/2*e)^13-384*cos(1/2*f*x+ 1/2*e)^11-288*cos(1/2*f*x+1/2*e)^9+1216*cos(1/2*f*x+1/2*e)^7-392*cos(1/2*f *x+1/2*e)^5-280*cos(1/2*f*x+1/2*e)^3-14*cos(1/2*f*x+1/2*e))*sin(1/2*f*x+1/ 2*e)-448*cos(1/2*f*x+1/2*e)^12+1344*cos(1/2*f*x+1/2*e)^10-784*cos(1/2*f*x+ 1/2*e)^8-672*cos(1/2*f*x+1/2*e)^6+476*cos(1/2*f*x+1/2*e)^4+84*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^8
Time = 0.13 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.99 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx=\frac {{\left (35 \, B a^{3} \cos \left (f x + e\right )^{4} - 56 \, {\left (A + 2 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \, {\left (17 \, A + 19 \, B\right )} a^{3} - 4 \, {\left (7 \, {\left (A + 2 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 2 \, {\left (9 \, A + 8 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{140 \, {\left (c^{9} f \cos \left (f x + e\right )^{9} - 32 \, c^{9} f \cos \left (f x + e\right )^{7} + 160 \, c^{9} f \cos \left (f x + e\right )^{5} - 256 \, c^{9} f \cos \left (f x + e\right )^{3} + 128 \, c^{9} f \cos \left (f x + e\right ) + 8 \, {\left (c^{9} f \cos \left (f x + e\right )^{7} - 10 \, c^{9} f \cos \left (f x + e\right )^{5} + 24 \, c^{9} f \cos \left (f x + e\right )^{3} - 16 \, c^{9} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(17/2), x, algorithm="fricas")
Output:
1/140*(35*B*a^3*cos(f*x + e)^4 - 56*(A + 2*B)*a^3*cos(f*x + e)^2 + 4*(17*A + 19*B)*a^3 - 4*(7*(A + 2*B)*a^3*cos(f*x + e)^2 - 2*(9*A + 8*B)*a^3)*sin( f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^9*f*cos(f* x + e)^9 - 32*c^9*f*cos(f*x + e)^7 + 160*c^9*f*cos(f*x + e)^5 - 256*c^9*f* cos(f*x + e)^3 + 128*c^9*f*cos(f*x + e) + 8*(c^9*f*cos(f*x + e)^7 - 10*c^9 *f*cos(f*x + e)^5 + 24*c^9*f*cos(f*x + e)^3 - 16*c^9*f*cos(f*x + e))*sin(f *x + e))
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(17/2 ),x)
Output:
Timed out
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(17/2), x, algorithm="maxima")
Output:
Timed out
Exception generated. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(17/2), 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 = 54.77 (sec) , antiderivative size = 841, normalized size of antiderivative = 3.42 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx=\text {Too large to display} \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x)) ^(17/2),x)
Output:
((c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*( (8*B*a^3*exp(e*5i + f*x*5i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1 i + f*x*1i)*1i)/2))^(1/2))/(c^9*f) + (8*B*a^3*exp(e*13i + f*x*13i)*(a + a* ((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(c^9*f) - (64*a^3*exp(e*6i + f*x*6i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e* 1i + f*x*1i)*1i)/2))^(1/2)*(A*1i + B*2i))/(5*c^9*f) - (32*a^3*exp(e*7i + f *x*7i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^( 1/2)*(8*A + 11*B))/(5*c^9*f) + (64*a^3*exp(e*12i + f*x*12i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(A*1i + B*2i))/( 5*c^9*f) - (32*a^3*exp(e*11i + f*x*11i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/ 2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(8*A + 11*B))/(5*c^9*f) + (64*a^3*ex p(e*8i + f*x*8i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i) *1i)/2))^(1/2)*(A*13i + B*10i))/(7*c^9*f) - (64*a^3*exp(e*10i + f*x*10i)*( a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(A* 13i + B*10i))/(7*c^9*f) + (16*a^3*exp(e*9i + f*x*9i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(64*A + 53*B))/(7*c^9*f )))/(exp(e*1i + f*x*1i)*16i - 119*exp(e*2i + f*x*2i) - exp(e*3i + f*x*3i)* 544i + 1700*exp(e*4i + f*x*4i) + exp(e*5i + f*x*5i)*3808i - 6188*exp(e*6i + f*x*6i) - exp(e*7i + f*x*7i)*7072i + 4862*exp(e*8i + f*x*8i) + 4862*exp( e*10i + f*x*10i) + exp(e*11i + f*x*11i)*7072i - 6188*exp(e*12i + f*x*12...
\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx =\text {Too large to display} \] Input:
int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(17/2),x)
Output:
(sqrt(c)*sqrt(a)*a**3*( - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**4)/(sin(e + f*x)**9 - 9*sin(e + f*x)**8 + 36*sin(e + f *x)**7 - 84*sin(e + f*x)**6 + 126*sin(e + f*x)**5 - 126*sin(e + f*x)**4 + 84*sin(e + f*x)**3 - 36*sin(e + f*x)**2 + 9*sin(e + f*x) - 1),x)*b - int(( sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**9 - 9*sin(e + f*x)**8 + 36*sin(e + f*x)**7 - 84*sin(e + f*x)**6 + 1 26*sin(e + f*x)**5 - 126*sin(e + f*x)**4 + 84*sin(e + f*x)**3 - 36*sin(e + f*x)**2 + 9*sin(e + f*x) - 1),x)*a - 3*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**9 - 9*sin(e + f*x)**8 + 36*sin(e + f*x)**7 - 84*sin(e + f*x)**6 + 126*sin(e + f*x)**5 - 126*sin( e + f*x)**4 + 84*sin(e + f*x)**3 - 36*sin(e + f*x)**2 + 9*sin(e + f*x) - 1 ),x)*b - 3*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f *x)**2)/(sin(e + f*x)**9 - 9*sin(e + f*x)**8 + 36*sin(e + f*x)**7 - 84*sin (e + f*x)**6 + 126*sin(e + f*x)**5 - 126*sin(e + f*x)**4 + 84*sin(e + f*x) **3 - 36*sin(e + f*x)**2 + 9*sin(e + f*x) - 1),x)*a - 3*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sin(e + f*x)**2)/(sin(e + f*x)**9 - 9* sin(e + f*x)**8 + 36*sin(e + f*x)**7 - 84*sin(e + f*x)**6 + 126*sin(e + f* x)**5 - 126*sin(e + f*x)**4 + 84*sin(e + f*x)**3 - 36*sin(e + f*x)**2 + 9* sin(e + f*x) - 1),x)*b - 3*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x ) + 1)*sin(e + f*x))/(sin(e + f*x)**9 - 9*sin(e + f*x)**8 + 36*sin(e + ...