Integrand size = 45, antiderivative size = 261 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{15/2}} \, dx=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{15 f (c-i c \tan (e+f x))^{15/2}}-\frac {(4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{195 c f (c-i c \tan (e+f x))^{13/2}}-\frac {(4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{715 c^2 f (c-i c \tan (e+f x))^{11/2}}-\frac {2 (4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{6435 c^3 f (c-i c \tan (e+f x))^{9/2}}-\frac {2 (4 i A-11 B) (a+i a \tan (e+f x))^{7/2}}{45045 c^4 f (c-i c \tan (e+f x))^{7/2}} \]
-1/15*(I*A+B)*(a+I*a*tan(f*x+e))^(7/2)/f/(c-I*c*tan(f*x+e))^(15/2)-1/195*( 4*I*A-11*B)*(a+I*a*tan(f*x+e))^(7/2)/c/f/(c-I*c*tan(f*x+e))^(13/2)-1/715*( 4*I*A-11*B)*(a+I*a*tan(f*x+e))^(7/2)/c^2/f/(c-I*c*tan(f*x+e))^(11/2)-2/643 5*(4*I*A-11*B)*(a+I*a*tan(f*x+e))^(7/2)/c^3/f/(c-I*c*tan(f*x+e))^(9/2)-2/4 5045*(4*I*A-11*B)*(a+I*a*tan(f*x+e))^(7/2)/c^4/f/(c-I*c*tan(f*x+e))^(7/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(577\) vs. \(2(261)=522\).
Time = 19.38 (sec) , antiderivative size = 577, normalized size of antiderivative = 2.21 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{15/2}} \, dx=\frac {\cos ^4(e+f x) \left ((-i A+B) \cos (6 f x) \left (\frac {\cos (3 e)}{224 c^8}+\frac {i \sin (3 e)}{224 c^8}\right )+(-37 i A+23 B) \cos (8 f x) \left (\frac {\cos (5 e)}{2016 c^8}+\frac {i \sin (5 e)}{2016 c^8}\right )+(-49 i A+11 B) \cos (10 f x) \left (\frac {\cos (7 e)}{1584 c^8}+\frac {i \sin (7 e)}{1584 c^8}\right )+(61 A-11 i B) \cos (12 f x) \left (-\frac {i \cos (9 e)}{2288 c^8}+\frac {\sin (9 e)}{2288 c^8}\right )+(73 A-43 i B) \cos (14 f x) \left (-\frac {i \cos (11 e)}{6240 c^8}+\frac {\sin (11 e)}{6240 c^8}\right )+(A-i B) \cos (16 f x) \left (-\frac {i \cos (13 e)}{480 c^8}+\frac {\sin (13 e)}{480 c^8}\right )+(A+i B) \left (\frac {\cos (3 e)}{224 c^8}+\frac {i \sin (3 e)}{224 c^8}\right ) \sin (6 f x)+(37 A+23 i B) \left (\frac {\cos (5 e)}{2016 c^8}+\frac {i \sin (5 e)}{2016 c^8}\right ) \sin (8 f x)+(49 A+11 i B) \left (\frac {\cos (7 e)}{1584 c^8}+\frac {i \sin (7 e)}{1584 c^8}\right ) \sin (10 f x)+(61 A-11 i B) \left (\frac {\cos (9 e)}{2288 c^8}+\frac {i \sin (9 e)}{2288 c^8}\right ) \sin (12 f x)+(73 A-43 i B) \left (\frac {\cos (11 e)}{6240 c^8}+\frac {i \sin (11 e)}{6240 c^8}\right ) \sin (14 f x)+(A-i B) \left (\frac {\cos (13 e)}{480 c^8}+\frac {i \sin (13 e)}{480 c^8}\right ) \sin (16 f x)\right ) \sqrt {\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))} \]
Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan [e + f*x])^(15/2),x]
(Cos[e + f*x]^4*(((-I)*A + B)*Cos[6*f*x]*(Cos[3*e]/(224*c^8) + ((I/224)*Si n[3*e])/c^8) + ((-37*I)*A + 23*B)*Cos[8*f*x]*(Cos[5*e]/(2016*c^8) + ((I/20 16)*Sin[5*e])/c^8) + ((-49*I)*A + 11*B)*Cos[10*f*x]*(Cos[7*e]/(1584*c^8) + ((I/1584)*Sin[7*e])/c^8) + (61*A - (11*I)*B)*Cos[12*f*x]*(((-1/2288*I)*Co s[9*e])/c^8 + Sin[9*e]/(2288*c^8)) + (73*A - (43*I)*B)*Cos[14*f*x]*(((-1/6 240*I)*Cos[11*e])/c^8 + Sin[11*e]/(6240*c^8)) + (A - I*B)*Cos[16*f*x]*(((- 1/480*I)*Cos[13*e])/c^8 + Sin[13*e]/(480*c^8)) + (A + I*B)*(Cos[3*e]/(224* c^8) + ((I/224)*Sin[3*e])/c^8)*Sin[6*f*x] + (37*A + (23*I)*B)*(Cos[5*e]/(2 016*c^8) + ((I/2016)*Sin[5*e])/c^8)*Sin[8*f*x] + (49*A + (11*I)*B)*(Cos[7* e]/(1584*c^8) + ((I/1584)*Sin[7*e])/c^8)*Sin[10*f*x] + (61*A - (11*I)*B)*( Cos[9*e]/(2288*c^8) + ((I/2288)*Sin[9*e])/c^8)*Sin[12*f*x] + (73*A - (43*I )*B)*(Cos[11*e]/(6240*c^8) + ((I/6240)*Sin[11*e])/c^8)*Sin[14*f*x] + (A - I*B)*(Cos[13*e]/(480*c^8) + ((I/480)*Sin[13*e])/c^8)*Sin[16*f*x])*Sqrt[Sec [e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2 )*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f*x])^3*(A*Cos[e + f*x] + B*S in[e + f*x]))
Time = 0.45 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3042, 4071, 87, 55, 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.
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{15/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{15/2}}dx\) |
| \(\Big \downarrow \) 4071 |
| \(\displaystyle \frac {a c \int \frac {(i \tan (e+f x) a+a)^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{17/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {a c \left (\frac {(4 A+11 i B) \int \frac {(i \tan (e+f x) a+a)^{5/2}}{(c-i c \tan (e+f x))^{15/2}}d\tan (e+f x)}{15 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{15 a c (c-i c \tan (e+f x))^{15/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {(4 A+11 i B) \left (\frac {3 \int \frac {(i \tan (e+f x) a+a)^{5/2}}{(c-i c \tan (e+f x))^{13/2}}d\tan (e+f x)}{13 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{13 a c (c-i c \tan (e+f x))^{13/2}}\right )}{15 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{15 a c (c-i c \tan (e+f x))^{15/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {(4 A+11 i B) \left (\frac {3 \left (\frac {2 \int \frac {(i \tan (e+f x) a+a)^{5/2}}{(c-i c \tan (e+f x))^{11/2}}d\tan (e+f x)}{11 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{13 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{13 a c (c-i c \tan (e+f x))^{13/2}}\right )}{15 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{15 a c (c-i c \tan (e+f x))^{15/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {(4 A+11 i B) \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {(i \tan (e+f x) a+a)^{5/2}}{(c-i c \tan (e+f x))^{9/2}}d\tan (e+f x)}{9 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{9 a c (c-i c \tan (e+f x))^{9/2}}\right )}{11 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{13 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{13 a c (c-i c \tan (e+f x))^{13/2}}\right )}{15 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{15 a c (c-i c \tan (e+f x))^{15/2}}\right )}{f}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {a c \left (\frac {(4 A+11 i B) \left (\frac {3 \left (\frac {2 \left (-\frac {i (a+i a \tan (e+f x))^{7/2}}{63 a c^2 (c-i c \tan (e+f x))^{7/2}}-\frac {i (a+i a \tan (e+f x))^{7/2}}{9 a c (c-i c \tan (e+f x))^{9/2}}\right )}{11 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{13 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{13 a c (c-i c \tan (e+f x))^{13/2}}\right )}{15 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{15 a c (c-i c \tan (e+f x))^{15/2}}\right )}{f}\) |
(a*c*(-1/15*((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(a*c*(c - I*c*Tan[e + f*x])^(15/2)) + ((4*A + (11*I)*B)*(((-1/13*I)*(a + I*a*Tan[e + f*x])^(7/2 ))/(a*c*(c - I*c*Tan[e + f*x])^(13/2)) + (3*(((-1/11*I)*(a + I*a*Tan[e + f *x])^(7/2))/(a*c*(c - I*c*Tan[e + f*x])^(11/2)) + (2*(((-1/9*I)*(a + I*a*T an[e + f*x])^(7/2))/(a*c*(c - I*c*Tan[e + f*x])^(9/2)) - ((I/63)*(a + I*a* Tan[e + f*x])^(7/2))/(a*c^2*(c - I*c*Tan[e + f*x])^(7/2))))/(11*c)))/(13*c )))/(15*c)))/f
3.9.28.3.1 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 = 0.38 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.65
| method | result | size |
| risch | \(-\frac {a^{3} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (3003 i A \,{\mathrm e}^{14 i \left (f x +e \right )}+3003 B \,{\mathrm e}^{14 i \left (f x +e \right )}+13860 i A \,{\mathrm e}^{12 i \left (f x +e \right )}+6930 B \,{\mathrm e}^{12 i \left (f x +e \right )}+24570 i A \,{\mathrm e}^{10 i \left (f x +e \right )}+20020 i A \,{\mathrm e}^{8 i \left (f x +e \right )}-10010 B \,{\mathrm e}^{8 i \left (f x +e \right )}+6435 i A \,{\mathrm e}^{6 i \left (f x +e \right )}-6435 B \,{\mathrm e}^{6 i \left (f x +e \right )}\right )}{720720 c^{7} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(169\) |
| derivativedivides | \(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (22 i B \tan \left (f x +e \right )^{6}+72 i A \tan \left (f x +e \right )^{5}+8 A \tan \left (f x +e \right )^{6}-825 i B \tan \left (f x +e \right )^{4}-198 B \tan \left (f x +e \right )^{5}-780 i A \tan \left (f x +e \right )^{3}-300 A \tan \left (f x +e \right )^{4}-7260 i B \tan \left (f x +e \right )^{2}+2145 B \tan \left (f x +e \right )^{3}-6858 i A \tan \left (f x +e \right )+1455 A \tan \left (f x +e \right )^{2}-407 i B -3663 B \tan \left (f x +e \right )-4243 A \right )}{45045 f \,c^{8} \left (i+\tan \left (f x +e \right )\right )^{9}}\) | \(206\) |
| default | \(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (22 i B \tan \left (f x +e \right )^{6}+72 i A \tan \left (f x +e \right )^{5}+8 A \tan \left (f x +e \right )^{6}-825 i B \tan \left (f x +e \right )^{4}-198 B \tan \left (f x +e \right )^{5}-780 i A \tan \left (f x +e \right )^{3}-300 A \tan \left (f x +e \right )^{4}-7260 i B \tan \left (f x +e \right )^{2}+2145 B \tan \left (f x +e \right )^{3}-6858 i A \tan \left (f x +e \right )+1455 A \tan \left (f x +e \right )^{2}-407 i B -3663 B \tan \left (f x +e \right )-4243 A \right )}{45045 f \,c^{8} \left (i+\tan \left (f x +e \right )\right )^{9}}\) | \(206\) |
| parts | \(\frac {i A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (-4243 i+6858 \tan \left (f x +e \right )+1455 i \tan \left (f x +e \right )^{2}+780 \tan \left (f x +e \right )^{3}-300 i \tan \left (f x +e \right )^{4}-72 \tan \left (f x +e \right )^{5}+8 i \tan \left (f x +e \right )^{6}\right )}{45045 f \,c^{8} \left (i+\tan \left (f x +e \right )\right )^{9}}-\frac {i B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (18 i \tan \left (f x +e \right )^{5}+2 \tan \left (f x +e \right )^{6}-195 i \tan \left (f x +e \right )^{3}-75 \tan \left (f x +e \right )^{4}+333 i \tan \left (f x +e \right )-660 \tan \left (f x +e \right )^{2}-37\right )}{4095 f \,c^{8} \left (i+\tan \left (f x +e \right )\right )^{9}}\) | \(259\) |
int((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(15/2),x, method=_RETURNVERBOSE)
-1/720720*a^3/c^7*(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*(3003*I*A*exp(14*I*(f*x+e))+3003*B*exp(14*I*(f*x+ e))+13860*I*A*exp(12*I*(f*x+e))+6930*B*exp(12*I*(f*x+e))+24570*I*A*exp(10* I*(f*x+e))+20020*I*A*exp(8*I*(f*x+e))-10010*B*exp(8*I*(f*x+e))+6435*I*A*ex p(6*I*(f*x+e))-6435*B*exp(6*I*(f*x+e)))
Time = 0.25 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.64 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{15/2}} \, dx=-\frac {{\left (3003 \, {\left (i \, A + B\right )} a^{3} e^{\left (17 i \, f x + 17 i \, e\right )} + 231 \, {\left (73 i \, A + 43 \, B\right )} a^{3} e^{\left (15 i \, f x + 15 i \, e\right )} + 630 \, {\left (61 i \, A + 11 \, B\right )} a^{3} e^{\left (13 i \, f x + 13 i \, e\right )} + 910 \, {\left (49 i \, A - 11 \, B\right )} a^{3} e^{\left (11 i \, f x + 11 i \, e\right )} + 715 \, {\left (37 i \, A - 23 \, B\right )} a^{3} e^{\left (9 i \, f x + 9 i \, e\right )} + 6435 \, {\left (i \, A - B\right )} a^{3} e^{\left (7 i \, f x + 7 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{720720 \, c^{8} f} \]
integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(15 /2),x, algorithm="fricas")
-1/720720*(3003*(I*A + B)*a^3*e^(17*I*f*x + 17*I*e) + 231*(73*I*A + 43*B)* a^3*e^(15*I*f*x + 15*I*e) + 630*(61*I*A + 11*B)*a^3*e^(13*I*f*x + 13*I*e) + 910*(49*I*A - 11*B)*a^3*e^(11*I*f*x + 11*I*e) + 715*(37*I*A - 23*B)*a^3* e^(9*I*f*x + 9*I*e) + 6435*(I*A - B)*a^3*e^(7*I*f*x + 7*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^8*f)
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{15/2}} \, dx=\text {Timed out} \]
Time = 1.09 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.27 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{15/2}} \, dx=\frac {{\left (3003 \, {\left (-i \, A - B\right )} a^{3} \cos \left (\frac {15}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 6930 \, {\left (-2 i \, A - B\right )} a^{3} \cos \left (\frac {13}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 24570 i \, A a^{3} \cos \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 10010 \, {\left (-2 i \, A + B\right )} a^{3} \cos \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 6435 \, {\left (-i \, A + B\right )} a^{3} \cos \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 3003 \, {\left (A - i \, B\right )} a^{3} \sin \left (\frac {15}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 6930 \, {\left (2 \, A - i \, B\right )} a^{3} \sin \left (\frac {13}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 24570 \, A a^{3} \sin \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 10010 \, {\left (2 \, A + i \, B\right )} a^{3} \sin \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 6435 \, {\left (A + i \, B\right )} a^{3} \sin \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a}}{720720 \, c^{\frac {15}{2}} f} \]
integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(15 /2),x, algorithm="maxima")
1/720720*(3003*(-I*A - B)*a^3*cos(15/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 6930*(-2*I*A - B)*a^3*cos(13/2*arctan2(sin(2*f*x + 2*e), cos(2 *f*x + 2*e))) - 24570*I*A*a^3*cos(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 10010*(-2*I*A + B)*a^3*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2 *f*x + 2*e))) + 6435*(-I*A + B)*a^3*cos(7/2*arctan2(sin(2*f*x + 2*e), cos( 2*f*x + 2*e))) + 3003*(A - I*B)*a^3*sin(15/2*arctan2(sin(2*f*x + 2*e), cos (2*f*x + 2*e))) + 6930*(2*A - I*B)*a^3*sin(13/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 24570*A*a^3*sin(11/2*arctan2(sin(2*f*x + 2*e), cos(2* f*x + 2*e))) + 10010*(2*A + I*B)*a^3*sin(9/2*arctan2(sin(2*f*x + 2*e), cos (2*f*x + 2*e))) + 6435*(A + I*B)*a^3*sin(7/2*arctan2(sin(2*f*x + 2*e), cos (2*f*x + 2*e))))*sqrt(a)/(c^(15/2)*f)
\[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{15/2}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {15}{2}}} \,d x } \]
integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(15 /2),x, algorithm="giac")
integrate((B*tan(f*x + e) + A)*(I*a*tan(f*x + e) + a)^(7/2)/(-I*c*tan(f*x + e) + c)^(15/2), x)
Time = 12.60 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.73 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{15/2}} \, dx=-\frac {\sqrt {a+\frac {a\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}\,\left (\frac {a^3\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (2\,A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{72\,c^7\,f}+\frac {a^3\,{\mathrm {e}}^{e\,12{}\mathrm {i}+f\,x\,12{}\mathrm {i}}\,\left (2\,A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{104\,c^7\,f}+\frac {A\,a^3\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,3{}\mathrm {i}}{88\,c^7\,f}+\frac {a^3\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{112\,c^7\,f}+\frac {a^3\,{\mathrm {e}}^{e\,14{}\mathrm {i}+f\,x\,14{}\mathrm {i}}\,\left (A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{240\,c^7\,f}\right )}{\sqrt {c-\frac {c\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}} \]
-((a + (a*sin(e + f*x)*1i)/cos(e + f*x))^(1/2)*((a^3*exp(e*8i + f*x*8i)*(2 *A + B*1i)*1i)/(72*c^7*f) + (a^3*exp(e*12i + f*x*12i)*(2*A - B*1i)*1i)/(10 4*c^7*f) + (A*a^3*exp(e*10i + f*x*10i)*3i)/(88*c^7*f) + (a^3*exp(e*6i + f* x*6i)*(A + B*1i)*1i)/(112*c^7*f) + (a^3*exp(e*14i + f*x*14i)*(A - B*1i)*1i )/(240*c^7*f)))/(c - (c*sin(e + f*x)*1i)/cos(e + f*x))^(1/2)