Integrand size = 40, antiderivative size = 202 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{168 c f (c-c \sin (e+f x))^{13/2}}+\frac {(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{840 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6720 c^3 f (c-c \sin (e+f x))^{9/2}} \] Output:
1/14*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(15/2)+1/1 68*(3*A-11*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c/f/(c-c*sin(f*x+e))^(13/2 )+1/840*(3*A-11*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c^2/f/(c-c*sin(f*x+e) )^(11/2)+1/6720*(3*A-11*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c^3/f/(c-c*si n(f*x+e))^(9/2)
Leaf count is larger than twice the leaf count of optimal. \(442\) vs. \(2(202)=404\).
Time = 17.37 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.19 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {8 (A+B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{7/2}}{7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}-\frac {2 (3 A+5 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{7/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}+\frac {6 (A+3 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (a (1+\sin (e+f x)))^{7/2}}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}+\frac {(-A-7 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (a (1+\sin (e+f x)))^{7/2}}{4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}+\frac {B \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{7/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}} \] Input:
Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(15/2),x]
Output:
(8*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(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])^(15 /2)) - (2*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[ e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[ e + f*x])^(15/2)) + (6*(A + 3*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*( 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])^(15/2)) + ((-A - 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f* x)/2])^7*(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])^(15/2)) + (B*(Cos[(e + f*x)/2] - Sin[(e + f *x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2))
Time = 1.00 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}}dx\) |
| \(\Big \downarrow \) 3451 |
| \(\displaystyle \frac {(3 A-11 B) \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{13/2}}dx}{14 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(3 A-11 B) \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{13/2}}dx}{14 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\) |
| \(\Big \downarrow \) 3222 |
| \(\displaystyle \frac {(3 A-11 B) \left (\frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{11/2}}dx}{6 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}\right )}{14 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(3 A-11 B) \left (\frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{11/2}}dx}{6 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}\right )}{14 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\) |
| \(\Big \downarrow \) 3222 |
| \(\displaystyle \frac {(3 A-11 B) \left (\frac {\frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{9/2}}dx}{10 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}}{6 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}\right )}{14 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(3 A-11 B) \left (\frac {\frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{9/2}}dx}{10 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}}{6 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}\right )}{14 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}\) |
| \(\Big \downarrow \) 3221 |
| \(\displaystyle \frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {(3 A-11 B) \left (\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{12 f (c-c \sin (e+f x))^{13/2}}+\frac {\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{80 c f (c-c \sin (e+f x))^{9/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}}{6 c}\right )}{14 c}\) |
Input:
Int[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]) ^(15/2),x]
Output:
((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(14*f*(c - c*Sin[e + f*x ])^(15/2)) + ((3*A - 11*B)*((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)
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. \(516\) vs. \(2(178)=356\).
Time = 12.12 (sec) , antiderivative size = 517, normalized size of antiderivative = 2.56
| 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}+4 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{4}+10 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}+20\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 )^{6}}{4480 f \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, c^{7}}+\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 (48 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}-96 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}-204 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+252 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}-70 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-168 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+336 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+322 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-490 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-105\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a^{3}}{105 f \left (-1+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) \left (\left (192 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}-384 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}+32 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}+160 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+12 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+64 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{12}-192 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{10}-48 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+416 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-180 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-60 \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^{7}}\) | \(517\) |
| default | \(\text {Expression too large to display}\) | \(720\) |
Input:
int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(15/2),x,meth od=_RETURNVERBOSE)
Output:
1/4480*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+4*cos(1/4*Pi+1/2*f*x+1/2*e)^4+10*cos(1/4*Pi+1/2*f*x+1/2*e)^2+2 0)*a^3/(c*cos(1/4*Pi+1/2*f*x+1/2*e)^2)^(1/2)/c^7*tan(1/4*Pi+1/2*f*x+1/2*e) ^7*sec(1/4*Pi+1/2*f*x+1/2*e)^6+2/105*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*((48*cos(1/2*f*x+1/2*e)^9-96*cos (1/2*f*x+1/2*e)^7-204*cos(1/2*f*x+1/2*e)^5+252*cos(1/2*f*x+1/2*e)^3-70*cos (1/2*f*x+1/2*e))*sin(1/2*f*x+1/2*e)-168*cos(1/2*f*x+1/2*e)^8+336*cos(1/2*f *x+1/2*e)^6+322*cos(1/2*f*x+1/2*e)^4-490*cos(1/2*f*x+1/2*e)^2-105)*sin(1/2 *f*x+1/2*e)^2*a^3/(-1+2*cos(1/2*f*x+1/2*e)^2)/((192*cos(1/2*f*x+1/2*e)^9-3 84*cos(1/2*f*x+1/2*e)^7+32*cos(1/2*f*x+1/2*e)^5+160*cos(1/2*f*x+1/2*e)^3+1 2*cos(1/2*f*x+1/2*e))*sin(1/2*f*x+1/2*e)+64*cos(1/2*f*x+1/2*e)^12-192*cos( 1/2*f*x+1/2*e)^10-48*cos(1/2*f*x+1/2*e)^8+416*cos(1/2*f*x+1/2*e)^6-180*cos (1/2*f*x+1/2*e)^4-60*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^7
Time = 0.12 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.16 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=-\frac {{\left (140 \, B a^{3} \cos \left (f x + e\right )^{4} - 7 \, {\left (27 \, A + 61 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \, {\left (57 \, A + 71 \, B\right )} a^{3} - 7 \, {\left (5 \, {\left (3 \, A + 5 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 4 \, {\left (9 \, A + 7 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{420 \, {\left (7 \, c^{8} f \cos \left (f x + e\right )^{7} - 56 \, c^{8} f \cos \left (f x + e\right )^{5} + 112 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right ) - {\left (c^{8} f \cos \left (f x + e\right )^{7} - 24 \, c^{8} f \cos \left (f x + e\right )^{5} + 80 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(15/2), x, algorithm="fricas")
Output:
-1/420*(140*B*a^3*cos(f*x + e)^4 - 7*(27*A + 61*B)*a^3*cos(f*x + e)^2 + 4* (57*A + 71*B)*a^3 - 7*(5*(3*A + 5*B)*a^3*cos(f*x + e)^2 - 4*(9*A + 7*B)*a^ 3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(7*c^8 *f*cos(f*x + e)^7 - 56*c^8*f*cos(f*x + e)^5 + 112*c^8*f*cos(f*x + e)^3 - 6 4*c^8*f*cos(f*x + e) - (c^8*f*cos(f*x + e)^7 - 24*c^8*f*cos(f*x + e)^5 + 8 0*c^8*f*cos(f*x + e)^3 - 64*c^8*f*cos(f*x + e))*sin(f*x + e))
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(15/2 ),x)
Output:
Timed out
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(15/2), x, algorithm="maxima")
Output:
Timed out
Exception generated. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(15/2), 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 = 52.46 (sec) , antiderivative size = 827, normalized size of antiderivative = 4.09 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Too large to display} \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x)) ^(15/2),x)
Output:
-((c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)* ((B*a^3*exp(e*4i + f*x*4i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*16i)/(3*c^8*f) + (B*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)*16i)/( 3*c^8*f) - (a^3*exp(e*5i + f*x*5i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - ( exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(A*3i + B*5i)*8i)/(3*c^8*f) + (a^3*exp(e* 11i + f*x*11i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1 i)/2))^(1/2)*(A*3i + B*5i)*8i)/(3*c^8*f) - (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)*(27*A + 4 1*B)*16i)/(15*c^8*f) - (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)*(27*A + 41*B)*16i)/(15*c^8* f) + (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)*(A*43i + B*29i)*8i)/(5*c^8*f) - (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)*(A*43i + B*29i)*8i)/(5*c^8*f) + (a^3*exp(e*8i + f*x*8i)*(a + a*((ex p(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(89*A + 82*B) *32i)/(35*c^8*f)))/(exp(e*1i + f*x*1i)*14i - 90*exp(e*2i + f*x*2i) - exp(e *3i + f*x*3i)*350i + 910*exp(e*4i + f*x*4i) + exp(e*5i + f*x*5i)*1638i - 2 002*exp(e*6i + f*x*6i) - exp(e*7i + f*x*7i)*1430i - exp(e*9i + f*x*9i)*143 0i + 2002*exp(e*10i + f*x*10i) + exp(e*11i + f*x*11i)*1638i - 910*exp(e...
\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx =\text {Too large to display} \] Input:
int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(15/2),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)**8 - 8*sin(e + f*x)**7 + 28*sin(e + f*x) **6 - 56*sin(e + f*x)**5 + 70*sin(e + f*x)**4 - 56*sin(e + f*x)**3 + 28*si n(e + f*x)**2 - 8*sin(e + f*x) + 1),x)*b + int((sqrt(sin(e + f*x) + 1)*sqr t( - sin(e + f*x) + 1)*sin(e + f*x)**3)/(sin(e + f*x)**8 - 8*sin(e + f*x)* *7 + 28*sin(e + f*x)**6 - 56*sin(e + f*x)**5 + 70*sin(e + f*x)**4 - 56*sin (e + f*x)**3 + 28*sin(e + f*x)**2 - 8*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 )**8 - 8*sin(e + f*x)**7 + 28*sin(e + f*x)**6 - 56*sin(e + f*x)**5 + 70*si n(e + f*x)**4 - 56*sin(e + f*x)**3 + 28*sin(e + f*x)**2 - 8*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)**8 - 8*sin(e + f*x)**7 + 28*sin(e + f*x)**6 - 56*s in(e + f*x)**5 + 70*sin(e + f*x)**4 - 56*sin(e + f*x)**3 + 28*sin(e + f*x) **2 - 8*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)**8 - 8*sin(e + f*x)**7 + 28* sin(e + f*x)**6 - 56*sin(e + f*x)**5 + 70*sin(e + f*x)**4 - 56*sin(e + f*x )**3 + 28*sin(e + f*x)**2 - 8*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)**8 - 8*si n(e + f*x)**7 + 28*sin(e + f*x)**6 - 56*sin(e + f*x)**5 + 70*sin(e + f*x)* *4 - 56*sin(e + f*x)**3 + 28*sin(e + f*x)**2 - 8*sin(e + f*x) + 1),x)*a...