Integrand size = 40, antiderivative size = 96 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}+\frac {(A-9 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{80 c f (c-c \sin (e+f x))^{9/2}} \] Output:
1/10*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(11/2)+1/8 0*(A-9*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c/f/(c-c*sin(f*x+e))^(9/2)
Leaf count is larger than twice the leaf count of optimal. \(434\) vs. \(2(96)=192\).
Time = 17.29 (sec) , antiderivative size = 434, normalized size of antiderivative = 4.52 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {8 (A+B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{7/2}}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{11/2}}+\frac {(-3 A-5 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{11/2}}+\frac {2 (A+3 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{11/2}}+\frac {(-A-7 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (a (1+\sin (e+f x)))^{7/2}}{2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{11/2}}+\frac {B \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{11/2}} \] Input:
Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(11/2),x]
Output:
(8*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(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])^(11 /2)) + ((-3*A - 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(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])^(11/2)) + (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*Sin[e + f*x])^(11/2)) + ((-A - 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2] )^7*(a*(1 + Sin[e + f*x]))^(7/2))/(2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2 ])^7*(c - c*Sin[e + f*x])^(11/2)) + (B*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2 ])^9*(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])^(11/2))
Time = 0.56 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3042, 3451, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}}dx\) |
| \(\Big \downarrow \) 3451 |
| \(\displaystyle \frac {(A-9 B) \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{9/2}}dx}{10 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(A-9 B) \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{9/2}}dx}{10 c}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}\) |
| \(\Big \downarrow \) 3221 |
| \(\displaystyle \frac {(A-9 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{80 c f (c-c \sin (e+f x))^{9/2}}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{10 f (c-c \sin (e+f x))^{11/2}}\) |
Input:
Int[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x]) ^(11/2),x]
Output:
((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(10*f*(c - c*Sin[e + f*x ])^(11/2)) + ((A - 9*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(80*c*f*( c - c*Sin[e + f*x])^(9/2))
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_)*((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. \(413\) vs. \(2(84)=168\).
Time = 9.45 (sec) , antiderivative size = 414, normalized size of antiderivative = 4.31
| method | result | size |
| parts | \(\frac {A \sqrt {4}\, \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, a^{3} \left (\tan \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{7}+4 \tan \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{7} \sec \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )}{160 f \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, c^{5}}-\frac {2 B \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \sqrt {\left (2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) a}\, \left (8 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-8 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+20 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-10 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-20 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-5\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a^{3}}{5 f \left (-1+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) \left (16 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+32 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-32 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-32 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-8 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-8 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+1\right ) \sqrt {-\left (2 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) c}\, c^{5}}\) | \(414\) |
| default | \(\frac {16 \left (\left (\sec \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}+\frac {1}{4}\right ) A \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-\frac {1}{2}\right ) \left (\frac {\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{2}+\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{4}-\frac {\sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )}{4}+\frac {3 \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 )^{7}-\sin \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} B \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{2}-\frac {5 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )}{4}-\frac {5 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{2}-\frac {5}{8}\right )\right ) a^{3} \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}}{5 \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, c^{5} f \left (-1+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right ) \left (16 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+32 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{5} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-32 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-32 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-8 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-8 \sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+1\right )}\) | \(464\) |
Input:
int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2),x,meth od=_RETURNVERBOSE)
Output:
1/160*A/f*4^(1/2)*(a*sin(1/4*Pi+1/2*f*x+1/2*e)^2)^(1/2)*a^3/(c*cos(1/4*Pi+ 1/2*f*x+1/2*e)^2)^(1/2)/c^5*(tan(1/4*Pi+1/2*f*x+1/2*e)^7+4*tan(1/4*Pi+1/2* f*x+1/2*e)^7*sec(1/4*Pi+1/2*f*x+1/2*e)^2)-2/5*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)^5 *sin(1/2*f*x+1/2*e)-8*cos(1/2*f*x+1/2*e)^3*sin(1/2*f*x+1/2*e)+20*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)-20*cos(1/2*f*x+1/2*e) ^2-5)*sin(1/2*f*x+1/2*e)^2*a^3/(-1+2*cos(1/2*f*x+1/2*e)^2)/(16*cos(1/2*f*x +1/2*e)^8+32*cos(1/2*f*x+1/2*e)^5*sin(1/2*f*x+1/2*e)-32*cos(1/2*f*x+1/2*e) ^6-32*cos(1/2*f*x+1/2*e)^3*sin(1/2*f*x+1/2*e)-8*cos(1/2*f*x+1/2*e)^4-8*sin (1/2*f*x+1/2*e)*cos(1/2*f*x+1/2*e)+24*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^5
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (84) = 168\).
Time = 0.10 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.07 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {{\left (10 \, B a^{3} \cos \left (f x + e\right )^{4} - 5 \, {\left (A + 7 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 2 \, {\left (3 \, A + 13 \, B\right )} a^{3} - 5 \, {\left ({\left (A - B\right )} a^{3} \cos \left (f x + e\right )^{2} - 2 \, {\left (A - B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{10 \, {\left (5 \, c^{6} f \cos \left (f x + e\right )^{5} - 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 16 \, c^{6} f \cos \left (f x + e\right ) - {\left (c^{6} f \cos \left (f x + e\right )^{5} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} + 16 \, c^{6} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2), x, algorithm="fricas")
Output:
1/10*(10*B*a^3*cos(f*x + e)^4 - 5*(A + 7*B)*a^3*cos(f*x + e)^2 + 2*(3*A + 13*B)*a^3 - 5*((A - B)*a^3*cos(f*x + e)^2 - 2*(A - B)*a^3)*sin(f*x + e))*s qrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(5*c^6*f*cos(f*x + e)^5 - 20*c^6*f*cos(f*x + e)^3 + 16*c^6*f*cos(f*x + e) - (c^6*f*cos(f*x + e)^5 - 12*c^6*f*cos(f*x + e)^3 + 16*c^6*f*cos(f*x + e))*sin(f*x + e))
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(11/2 ),x)
Output:
Timed out
\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2), x, algorithm="maxima")
Output:
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(11/2), x)
Exception generated. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2), 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
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{11/2}} \,d x \] Input:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x)) ^(11/2),x)
Output:
int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x)) ^(11/2), x)
\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx =\text {Too large to display} \] Input:
int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(11/2),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)**6 - 6*sin(e + f*x)**5 + 15*sin(e + f*x) **4 - 20*sin(e + f*x)**3 + 15*sin(e + f*x)**2 - 6*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)/(s in(e + f*x)**6 - 6*sin(e + f*x)**5 + 15*sin(e + f*x)**4 - 20*sin(e + f*x)* *3 + 15*sin(e + f*x)**2 - 6*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)**6 - 6*s in(e + f*x)**5 + 15*sin(e + f*x)**4 - 20*sin(e + f*x)**3 + 15*sin(e + f*x) **2 - 6*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)**6 - 6*sin(e + f*x)**5 + 15* sin(e + f*x)**4 - 20*sin(e + f*x)**3 + 15*sin(e + f*x)**2 - 6*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)**6 - 6*sin(e + f*x)**5 + 15*sin(e + f*x)**4 - 20 *sin(e + f*x)**3 + 15*sin(e + f*x)**2 - 6*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)**6 - 6*sin(e + f*x)**5 + 15*sin(e + f*x)**4 - 20*sin(e + f*x)**3 + 15*s in(e + f*x)**2 - 6*sin(e + f*x) + 1),x)*a + int((sqrt(sin(e + f*x) + 1)*sq rt( - sin(e + f*x) + 1)*sin(e + f*x))/(sin(e + f*x)**6 - 6*sin(e + f*x)**5 + 15*sin(e + f*x)**4 - 20*sin(e + f*x)**3 + 15*sin(e + f*x)**2 - 6*sin(e + f*x) + 1),x)*b + int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1...