Integrand size = 45, antiderivative size = 208 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(3 i A-8 B) (a+i a \tan (e+f x))^{5/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {2 (3 i A-8 B) (a+i a \tan (e+f x))^{5/2}}{693 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 (3 i A-8 B) (a+i a \tan (e+f x))^{5/2}}{3465 c^3 f (c-i c \tan (e+f x))^{5/2}} \]
-1/11*(I*A+B)*(a+I*a*tan(f*x+e))^(5/2)/f/(c-I*c*tan(f*x+e))^(11/2)-1/99*(3 *I*A-8*B)*(a+I*a*tan(f*x+e))^(5/2)/c/f/(c-I*c*tan(f*x+e))^(9/2)-2/693*(3*I *A-8*B)*(a+I*a*tan(f*x+e))^(5/2)/c^2/f/(c-I*c*tan(f*x+e))^(7/2)-2/3465*(3* I*A-8*B)*(a+I*a*tan(f*x+e))^(5/2)/c^3/f/(c-I*c*tan(f*x+e))^(5/2)
Time = 16.38 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.75 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=\frac {a^2 \cos (e+f x) (55 (-24 i A+B) \cos (e+f x)+63 (-8 i A+3 B) \cos (3 (e+f x))-(3 A+8 i B) (55 \sin (e+f x)+63 \sin (3 (e+f x)))) (\cos (8 e+10 f x)+i \sin (8 e+10 f x)) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{13860 c^6 f (\cos (f x)+i \sin (f x))^2} \]
Integrate[((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan [e + f*x])^(11/2),x]
(a^2*Cos[e + f*x]*(55*((-24*I)*A + B)*Cos[e + f*x] + 63*((-8*I)*A + 3*B)*C os[3*(e + f*x)] - (3*A + (8*I)*B)*(55*Sin[e + f*x] + 63*Sin[3*(e + f*x)])) *(Cos[8*e + 10*f*x] + I*Sin[8*e + 10*f*x])*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt [c - I*c*Tan[e + f*x]])/(13860*c^6*f*(Cos[f*x] + I*Sin[f*x])^2)
Time = 0.42 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3042, 4071, 87, 55, 55, 48}
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.
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}}dx\) |
| \(\Big \downarrow \) 4071 |
| \(\displaystyle \frac {a c \int \frac {(i \tan (e+f x) a+a)^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {a c \left (\frac {(3 A+8 i B) \int \frac {(i \tan (e+f x) a+a)^{3/2}}{(c-i c \tan (e+f x))^{11/2}}d\tan (e+f x)}{11 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {(3 A+8 i B) \left (\frac {2 \int \frac {(i \tan (e+f x) a+a)^{3/2}}{(c-i c \tan (e+f x))^{9/2}}d\tan (e+f x)}{9 c}-\frac {i (a+i a \tan (e+f x))^{5/2}}{9 a c (c-i c \tan (e+f x))^{9/2}}\right )}{11 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {(3 A+8 i B) \left (\frac {2 \left (\frac {\int \frac {(i \tan (e+f x) a+a)^{3/2}}{(c-i c \tan (e+f x))^{7/2}}d\tan (e+f x)}{7 c}-\frac {i (a+i a \tan (e+f x))^{5/2}}{7 a c (c-i c \tan (e+f x))^{7/2}}\right )}{9 c}-\frac {i (a+i a \tan (e+f x))^{5/2}}{9 a c (c-i c \tan (e+f x))^{9/2}}\right )}{11 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{f}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {a c \left (\frac {(3 A+8 i B) \left (\frac {2 \left (-\frac {i (a+i a \tan (e+f x))^{5/2}}{35 a c^2 (c-i c \tan (e+f x))^{5/2}}-\frac {i (a+i a \tan (e+f x))^{5/2}}{7 a c (c-i c \tan (e+f x))^{7/2}}\right )}{9 c}-\frac {i (a+i a \tan (e+f x))^{5/2}}{9 a c (c-i c \tan (e+f x))^{9/2}}\right )}{11 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{f}\) |
(a*c*(-1/11*((I*A + B)*(a + I*a*Tan[e + f*x])^(5/2))/(a*c*(c - I*c*Tan[e + f*x])^(11/2)) + ((3*A + (8*I)*B)*(((-1/9*I)*(a + I*a*Tan[e + f*x])^(5/2)) /(a*c*(c - I*c*Tan[e + f*x])^(9/2)) + (2*(((-1/7*I)*(a + I*a*Tan[e + f*x]) ^(5/2))/(a*c*(c - I*c*Tan[e + f*x])^(7/2)) - ((I/35)*(a + I*a*Tan[e + f*x] )^(5/2))/(a*c^2*(c - I*c*Tan[e + f*x])^(5/2))))/(9*c)))/(11*c)))/f
3.9.14.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a*(c/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x], x , Tan[e + f*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]
Time = 0.42 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(-\frac {a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (315 i A \,{\mathrm e}^{10 i \left (f x +e \right )}+315 B \,{\mathrm e}^{10 i \left (f x +e \right )}+1155 i A \,{\mathrm e}^{8 i \left (f x +e \right )}+385 B \,{\mathrm e}^{8 i \left (f x +e \right )}+1485 i A \,{\mathrm e}^{6 i \left (f x +e \right )}-495 B \,{\mathrm e}^{6 i \left (f x +e \right )}+693 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-693 B \,{\mathrm e}^{4 i \left (f x +e \right )}\right )}{27720 c^{5} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(156\) |
| derivativedivides | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (6 i A \tan \left (f x +e \right )^{4}-112 i B \tan \left (f x +e \right )^{3}-16 B \tan \left (f x +e \right )^{4}-135 i A \tan \left (f x +e \right )^{2}-42 A \tan \left (f x +e \right )^{3}-427 i \tan \left (f x +e \right ) B +360 B \tan \left (f x +e \right )^{2}-456 i A +273 A \tan \left (f x +e \right )+61 B \right )}{3465 f \,c^{6} \left (i+\tan \left (f x +e \right )\right )^{7}}\) | \(161\) |
| default | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (6 i A \tan \left (f x +e \right )^{4}-112 i B \tan \left (f x +e \right )^{3}-16 B \tan \left (f x +e \right )^{4}-135 i A \tan \left (f x +e \right )^{2}-42 A \tan \left (f x +e \right )^{3}-427 i \tan \left (f x +e \right ) B +360 B \tan \left (f x +e \right )^{2}-456 i A +273 A \tan \left (f x +e \right )+61 B \right )}{3465 f \,c^{6} \left (i+\tan \left (f x +e \right )\right )^{7}}\) | \(161\) |
| parts | \(-\frac {i A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (2 i \tan \left (f x +e \right )^{4}-45 i \tan \left (f x +e \right )^{2}-14 \tan \left (f x +e \right )^{3}-152 i+91 \tan \left (f x +e \right )\right )}{1155 f \,c^{6} \left (i+\tan \left (f x +e \right )\right )^{7}}+\frac {i B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (-61+427 i \tan \left (f x +e \right )-360 \tan \left (f x +e \right )^{2}+112 i \tan \left (f x +e \right )^{3}+16 \tan \left (f x +e \right )^{4}\right )}{3465 f \,c^{6} \left (i+\tan \left (f x +e \right )\right )^{7}}\) | \(217\) |
int((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(11/2),x, method=_RETURNVERBOSE)
-1/27720*a^2/c^5*(a*exp(2*I*(f*x+e))/(exp(2*I*(f*x+e))+1))^(1/2)/(c/(exp(2 *I*(f*x+e))+1))^(1/2)/f*(315*I*A*exp(10*I*(f*x+e))+315*B*exp(10*I*(f*x+e)) +1155*I*A*exp(8*I*(f*x+e))+385*B*exp(8*I*(f*x+e))+1485*I*A*exp(6*I*(f*x+e) )-495*B*exp(6*I*(f*x+e))+693*I*A*exp(4*I*(f*x+e))-693*B*exp(4*I*(f*x+e)))
Time = 0.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=-\frac {{\left (315 \, {\left (i \, A + B\right )} a^{2} e^{\left (13 i \, f x + 13 i \, e\right )} + 70 \, {\left (21 i \, A + 10 \, B\right )} a^{2} e^{\left (11 i \, f x + 11 i \, e\right )} + 110 \, {\left (24 i \, A - B\right )} a^{2} e^{\left (9 i \, f x + 9 i \, e\right )} + 198 \, {\left (11 i \, A - 6 \, B\right )} a^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + 693 \, {\left (i \, A - B\right )} a^{2} e^{\left (5 i \, f x + 5 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{27720 \, c^{6} f} \]
integrate((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(11 /2),x, algorithm="fricas")
-1/27720*(315*(I*A + B)*a^2*e^(13*I*f*x + 13*I*e) + 70*(21*I*A + 10*B)*a^2 *e^(11*I*f*x + 11*I*e) + 110*(24*I*A - B)*a^2*e^(9*I*f*x + 9*I*e) + 198*(1 1*I*A - 6*B)*a^2*e^(7*I*f*x + 7*I*e) + 693*(I*A - B)*a^2*e^(5*I*f*x + 5*I* e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))/(c ^6*f)
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=\text {Timed out} \]
Time = 0.61 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.33 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=\frac {{\left (315 \, {\left (-i \, A - B\right )} a^{2} \cos \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 385 \, {\left (-3 i \, A - B\right )} a^{2} \cos \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 495 \, {\left (-3 i \, A + B\right )} a^{2} \cos \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 693 \, {\left (-i \, A + B\right )} a^{2} \cos \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 315 \, {\left (A - i \, B\right )} a^{2} \sin \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 385 \, {\left (3 \, A - i \, B\right )} a^{2} \sin \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 495 \, {\left (3 \, A + i \, B\right )} a^{2} \sin \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 693 \, {\left (A + i \, B\right )} a^{2} \sin \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a}}{27720 \, c^{\frac {11}{2}} f} \]
integrate((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(11 /2),x, algorithm="maxima")
1/27720*(315*(-I*A - B)*a^2*cos(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 385*(-3*I*A - B)*a^2*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 495*(-3*I*A + B)*a^2*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f *x + 2*e))) + 693*(-I*A + B)*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f *x + 2*e))) + 315*(A - I*B)*a^2*sin(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f *x + 2*e))) + 385*(3*A - I*B)*a^2*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2* f*x + 2*e))) + 495*(3*A + I*B)*a^2*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2 *f*x + 2*e))) + 693*(A + I*B)*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2* f*x + 2*e))))*sqrt(a)/(c^(11/2)*f)
\[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]
integrate((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(11 /2),x, algorithm="giac")
integrate((B*tan(f*x + e) + A)*(I*a*tan(f*x + e) + a)^(5/2)/(-I*c*tan(f*x + e) + c)^(11/2), x)
Time = 12.41 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.40 \[ \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=-\frac {a^2\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,\cos \left (4\,e+4\,f\,x\right )\,693{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,1485{}\mathrm {i}+A\,\cos \left (8\,e+8\,f\,x\right )\,1155{}\mathrm {i}+A\,\cos \left (10\,e+10\,f\,x\right )\,315{}\mathrm {i}-693\,B\,\cos \left (4\,e+4\,f\,x\right )-495\,B\,\cos \left (6\,e+6\,f\,x\right )+385\,B\,\cos \left (8\,e+8\,f\,x\right )+315\,B\,\cos \left (10\,e+10\,f\,x\right )-693\,A\,\sin \left (4\,e+4\,f\,x\right )-1485\,A\,\sin \left (6\,e+6\,f\,x\right )-1155\,A\,\sin \left (8\,e+8\,f\,x\right )-315\,A\,\sin \left (10\,e+10\,f\,x\right )-B\,\sin \left (4\,e+4\,f\,x\right )\,693{}\mathrm {i}-B\,\sin \left (6\,e+6\,f\,x\right )\,495{}\mathrm {i}+B\,\sin \left (8\,e+8\,f\,x\right )\,385{}\mathrm {i}+B\,\sin \left (10\,e+10\,f\,x\right )\,315{}\mathrm {i}\right )}{27720\,c^5\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
-(a^2*((a*(cos(2*e + 2*f*x) + sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1))^(1/2)*(A*cos(4*e + 4*f*x)*693i + A*cos(6*e + 6*f*x)*1485i + A*cos(8* e + 8*f*x)*1155i + A*cos(10*e + 10*f*x)*315i - 693*B*cos(4*e + 4*f*x) - 49 5*B*cos(6*e + 6*f*x) + 385*B*cos(8*e + 8*f*x) + 315*B*cos(10*e + 10*f*x) - 693*A*sin(4*e + 4*f*x) - 1485*A*sin(6*e + 6*f*x) - 1155*A*sin(8*e + 8*f*x ) - 315*A*sin(10*e + 10*f*x) - B*sin(4*e + 4*f*x)*693i - B*sin(6*e + 6*f*x )*495i + B*sin(8*e + 8*f*x)*385i + B*sin(10*e + 10*f*x)*315i))/(27720*c^5* f*((c*(cos(2*e + 2*f*x) - sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1) )^(1/2))