Integrand size = 45, antiderivative size = 314 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{17/2}} \, dx=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{17 f (c-i c \tan (e+f x))^{17/2}}-\frac {(5 i A-12 B) (a+i a \tan (e+f x))^{7/2}}{255 c f (c-i c \tan (e+f x))^{15/2}}-\frac {4 (5 i A-12 B) (a+i a \tan (e+f x))^{7/2}}{3315 c^2 f (c-i c \tan (e+f x))^{13/2}}-\frac {4 (5 i A-12 B) (a+i a \tan (e+f x))^{7/2}}{12155 c^3 f (c-i c \tan (e+f x))^{11/2}}-\frac {8 (5 i A-12 B) (a+i a \tan (e+f x))^{7/2}}{109395 c^4 f (c-i c \tan (e+f x))^{9/2}}-\frac {8 (5 i A-12 B) (a+i a \tan (e+f x))^{7/2}}{765765 c^5 f (c-i c \tan (e+f x))^{7/2}} \]
-1/17*(I*A+B)*(a+I*a*tan(f*x+e))^(7/2)/f/(c-I*c*tan(f*x+e))^(17/2)-1/255*( 5*I*A-12*B)*(a+I*a*tan(f*x+e))^(7/2)/c/f/(c-I*c*tan(f*x+e))^(15/2)-4/3315* (5*I*A-12*B)*(a+I*a*tan(f*x+e))^(7/2)/c^2/f/(c-I*c*tan(f*x+e))^(13/2)-4/12 155*(5*I*A-12*B)*(a+I*a*tan(f*x+e))^(7/2)/c^3/f/(c-I*c*tan(f*x+e))^(11/2)- 8/109395*(5*I*A-12*B)*(a+I*a*tan(f*x+e))^(7/2)/c^4/f/(c-I*c*tan(f*x+e))^(9 /2)-8/765765*(5*I*A-12*B)*(a+I*a*tan(f*x+e))^(7/2)/c^5/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. \(655\) vs. \(2(314)=628\).
Time = 19.83 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.09 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{17/2}} \, dx=\frac {\cos ^4(e+f x) \left ((-i A+B) \cos (6 f x) \left (\frac {\cos (3 e)}{448 c^9}+\frac {i \sin (3 e)}{448 c^9}\right )+(-22 i A+15 B) \cos (8 f x) \left (\frac {\cos (5 e)}{2016 c^9}+\frac {i \sin (5 e)}{2016 c^9}\right )+(-145 i A+51 B) \cos (10 f x) \left (\frac {\cos (7 e)}{6336 c^9}+\frac {i \sin (7 e)}{6336 c^9}\right )+(-60 i A+B) \cos (12 f x) \left (\frac {\cos (9 e)}{2288 c^9}+\frac {i \sin (9 e)}{2288 c^9}\right )+(215 A-69 i B) \cos (14 f x) \left (-\frac {i \cos (11 e)}{12480 c^9}+\frac {\sin (11 e)}{12480 c^9}\right )+(50 A-33 i B) \cos (16 f x) \left (-\frac {i \cos (13 e)}{8160 c^9}+\frac {\sin (13 e)}{8160 c^9}\right )+(A-i B) \cos (18 f x) \left (-\frac {i \cos (15 e)}{1088 c^9}+\frac {\sin (15 e)}{1088 c^9}\right )+(A+i B) \left (\frac {\cos (3 e)}{448 c^9}+\frac {i \sin (3 e)}{448 c^9}\right ) \sin (6 f x)+(22 A+15 i B) \left (\frac {\cos (5 e)}{2016 c^9}+\frac {i \sin (5 e)}{2016 c^9}\right ) \sin (8 f x)+(145 A+51 i B) \left (\frac {\cos (7 e)}{6336 c^9}+\frac {i \sin (7 e)}{6336 c^9}\right ) \sin (10 f x)+(60 A+i B) \left (\frac {\cos (9 e)}{2288 c^9}+\frac {i \sin (9 e)}{2288 c^9}\right ) \sin (12 f x)+(215 A-69 i B) \left (\frac {\cos (11 e)}{12480 c^9}+\frac {i \sin (11 e)}{12480 c^9}\right ) \sin (14 f x)+(50 A-33 i B) \left (\frac {\cos (13 e)}{8160 c^9}+\frac {i \sin (13 e)}{8160 c^9}\right ) \sin (16 f x)+(A-i B) \left (\frac {\cos (15 e)}{1088 c^9}+\frac {i \sin (15 e)}{1088 c^9}\right ) \sin (18 f x)\right ) \sqrt {\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))} \]
Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan [e + f*x])^(17/2),x]
(Cos[e + f*x]^4*(((-I)*A + B)*Cos[6*f*x]*(Cos[3*e]/(448*c^9) + ((I/448)*Si n[3*e])/c^9) + ((-22*I)*A + 15*B)*Cos[8*f*x]*(Cos[5*e]/(2016*c^9) + ((I/20 16)*Sin[5*e])/c^9) + ((-145*I)*A + 51*B)*Cos[10*f*x]*(Cos[7*e]/(6336*c^9) + ((I/6336)*Sin[7*e])/c^9) + ((-60*I)*A + B)*Cos[12*f*x]*(Cos[9*e]/(2288*c ^9) + ((I/2288)*Sin[9*e])/c^9) + (215*A - (69*I)*B)*Cos[14*f*x]*(((-1/1248 0*I)*Cos[11*e])/c^9 + Sin[11*e]/(12480*c^9)) + (50*A - (33*I)*B)*Cos[16*f* x]*(((-1/8160*I)*Cos[13*e])/c^9 + Sin[13*e]/(8160*c^9)) + (A - I*B)*Cos[18 *f*x]*(((-1/1088*I)*Cos[15*e])/c^9 + Sin[15*e]/(1088*c^9)) + (A + I*B)*(Co s[3*e]/(448*c^9) + ((I/448)*Sin[3*e])/c^9)*Sin[6*f*x] + (22*A + (15*I)*B)* (Cos[5*e]/(2016*c^9) + ((I/2016)*Sin[5*e])/c^9)*Sin[8*f*x] + (145*A + (51* I)*B)*(Cos[7*e]/(6336*c^9) + ((I/6336)*Sin[7*e])/c^9)*Sin[10*f*x] + (60*A + I*B)*(Cos[9*e]/(2288*c^9) + ((I/2288)*Sin[9*e])/c^9)*Sin[12*f*x] + (215* A - (69*I)*B)*(Cos[11*e]/(12480*c^9) + ((I/12480)*Sin[11*e])/c^9)*Sin[14*f *x] + (50*A - (33*I)*B)*(Cos[13*e]/(8160*c^9) + ((I/8160)*Sin[13*e])/c^9)* Sin[16*f*x] + (A - I*B)*(Cos[15*e]/(1088*c^9) + ((I/1088)*Sin[15*e])/c^9)* Sin[18*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*Sin[e + f*x]))
Time = 0.48 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {3042, 4071, 87, 55, 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.
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{17/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{17/2}}dx\) |
| \(\Big \downarrow \) 4071 |
| \(\displaystyle \frac {a c \int \frac {(i \tan (e+f x) a+a)^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{19/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {a c \left (\frac {(5 A+12 i B) \int \frac {(i \tan (e+f x) a+a)^{5/2}}{(c-i c \tan (e+f x))^{17/2}}d\tan (e+f x)}{17 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{17 a c (c-i c \tan (e+f x))^{17/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {(5 A+12 i B) \left (\frac {4 \int \frac {(i \tan (e+f x) a+a)^{5/2}}{(c-i c \tan (e+f x))^{15/2}}d\tan (e+f x)}{15 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{15 a c (c-i c \tan (e+f x))^{15/2}}\right )}{17 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{17 a c (c-i c \tan (e+f x))^{17/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {(5 A+12 i B) \left (\frac {4 \left (\frac {3 \int \frac {(i \tan (e+f x) a+a)^{5/2}}{(c-i c \tan (e+f x))^{13/2}}d\tan (e+f x)}{13 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{13 a c (c-i c \tan (e+f x))^{13/2}}\right )}{15 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{15 a c (c-i c \tan (e+f x))^{15/2}}\right )}{17 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{17 a c (c-i c \tan (e+f x))^{17/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {(5 A+12 i B) \left (\frac {4 \left (\frac {3 \left (\frac {2 \int \frac {(i \tan (e+f x) a+a)^{5/2}}{(c-i c \tan (e+f x))^{11/2}}d\tan (e+f x)}{11 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{13 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{13 a c (c-i c \tan (e+f x))^{13/2}}\right )}{15 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{15 a c (c-i c \tan (e+f x))^{15/2}}\right )}{17 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{17 a c (c-i c \tan (e+f x))^{17/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {(5 A+12 i B) \left (\frac {4 \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {(i \tan (e+f x) a+a)^{5/2}}{(c-i c \tan (e+f x))^{9/2}}d\tan (e+f x)}{9 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{9 a c (c-i c \tan (e+f x))^{9/2}}\right )}{11 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{13 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{13 a c (c-i c \tan (e+f x))^{13/2}}\right )}{15 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{15 a c (c-i c \tan (e+f x))^{15/2}}\right )}{17 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{17 a c (c-i c \tan (e+f x))^{17/2}}\right )}{f}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {a c \left (\frac {(5 A+12 i B) \left (\frac {4 \left (\frac {3 \left (\frac {2 \left (-\frac {i (a+i a \tan (e+f x))^{7/2}}{63 a c^2 (c-i c \tan (e+f x))^{7/2}}-\frac {i (a+i a \tan (e+f x))^{7/2}}{9 a c (c-i c \tan (e+f x))^{9/2}}\right )}{11 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{13 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{13 a c (c-i c \tan (e+f x))^{13/2}}\right )}{15 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{15 a c (c-i c \tan (e+f x))^{15/2}}\right )}{17 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{17 a c (c-i c \tan (e+f x))^{17/2}}\right )}{f}\) |
(a*c*(-1/17*((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(a*c*(c - I*c*Tan[e + f*x])^(17/2)) + ((5*A + (12*I)*B)*(((-1/15*I)*(a + I*a*Tan[e + f*x])^(7/2 ))/(a*c*(c - I*c*Tan[e + f*x])^(15/2)) + (4*(((-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*Tan[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)))/(17*c)))/f
3.9.29.3.1 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 = 0.37 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(-\frac {a^{3} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (45045 i A \,{\mathrm e}^{16 i \left (f x +e \right )}+45045 B \,{\mathrm e}^{16 i \left (f x +e \right )}+255255 i A \,{\mathrm e}^{14 i \left (f x +e \right )}+153153 B \,{\mathrm e}^{14 i \left (f x +e \right )}+589050 i A \,{\mathrm e}^{12 i \left (f x +e \right )}+117810 B \,{\mathrm e}^{12 i \left (f x +e \right )}+696150 i A \,{\mathrm e}^{10 i \left (f x +e \right )}-139230 B \,{\mathrm e}^{10 i \left (f x +e \right )}+425425 i A \,{\mathrm e}^{8 i \left (f x +e \right )}-255255 B \,{\mathrm e}^{8 i \left (f x +e \right )}+109395 i A \,{\mathrm e}^{6 i \left (f x +e \right )}-109395 B \,{\mathrm e}^{6 i \left (f x +e \right )}\right )}{24504480 c^{8} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(206\) |
| derivativedivides | \(\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (5871 i B +12960 i B \tan \left (f x +e \right )^{4}-96 B \tan \left (f x +e \right )^{7}+11175 i A \tan \left (f x +e \right )^{3}-400 A \tan \left (f x +e \right )^{6}-1860 i A \tan \left (f x +e \right )^{5}+4464 B \tan \left (f x +e \right )^{5}+103165 i A \tan \left (f x +e \right )+5400 A \tan \left (f x +e \right )^{4}-960 i B \tan \left (f x +e \right )^{6}-26820 B \tan \left (f x +e \right )^{3}+109881 i B \tan \left (f x +e \right )^{2}-18030 A \tan \left (f x +e \right )^{2}+40 i A \tan \left (f x +e \right )^{7}+58710 B \tan \left (f x +e \right )+66260 A \right )}{765765 f \,c^{9} \left (i+\tan \left (f x +e \right )\right )^{10}}\) | \(230\) |
| default | \(\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (5871 i B +12960 i B \tan \left (f x +e \right )^{4}-96 B \tan \left (f x +e \right )^{7}+11175 i A \tan \left (f x +e \right )^{3}-400 A \tan \left (f x +e \right )^{6}-1860 i A \tan \left (f x +e \right )^{5}+4464 B \tan \left (f x +e \right )^{5}+103165 i A \tan \left (f x +e \right )+5400 A \tan \left (f x +e \right )^{4}-960 i B \tan \left (f x +e \right )^{6}-26820 B \tan \left (f x +e \right )^{3}+109881 i B \tan \left (f x +e \right )^{2}-18030 A \tan \left (f x +e \right )^{2}+40 i A \tan \left (f x +e \right )^{7}+58710 B \tan \left (f x +e \right )+66260 A \right )}{765765 f \,c^{9} \left (i+\tan \left (f x +e \right )\right )^{10}}\) | \(230\) |
| parts | \(\frac {i A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (8 i \tan \left (f x +e \right )^{7}-372 i \tan \left (f x +e \right )^{5}-80 \tan \left (f x +e \right )^{6}+2235 i \tan \left (f x +e \right )^{3}+1080 \tan \left (f x +e \right )^{4}+20633 i \tan \left (f x +e \right )-3606 \tan \left (f x +e \right )^{2}+13252\right )}{153153 f \,c^{9} \left (i+\tan \left (f x +e \right )\right )^{10}}-\frac {i B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (320 i \tan \left (f x +e \right )^{6}+32 \tan \left (f x +e \right )^{7}-4320 i \tan \left (f x +e \right )^{4}-1488 \tan \left (f x +e \right )^{5}-36627 i \tan \left (f x +e \right )^{2}+8940 \tan \left (f x +e \right )^{3}-1957 i-19570 \tan \left (f x +e \right )\right )}{255255 f \,c^{9} \left (i+\tan \left (f x +e \right )\right )^{10}}\) | \(280\) |
int((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(17/2),x, method=_RETURNVERBOSE)
-1/24504480*a^3/c^8*(a*exp(2*I*(f*x+e))/(exp(2*I*(f*x+e))+1))^(1/2)/(c/(ex p(2*I*(f*x+e))+1))^(1/2)/f*(45045*I*A*exp(16*I*(f*x+e))+45045*B*exp(16*I*( f*x+e))+255255*I*A*exp(14*I*(f*x+e))+153153*B*exp(14*I*(f*x+e))+589050*I*A *exp(12*I*(f*x+e))+117810*B*exp(12*I*(f*x+e))+696150*I*exp(10*I*(f*x+e))*A -139230*B*exp(10*I*(f*x+e))+425425*I*exp(8*I*(f*x+e))*A-255255*B*exp(8*I*( f*x+e))+109395*I*A*exp(6*I*(f*x+e))-109395*B*exp(6*I*(f*x+e)))
Time = 0.24 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.60 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{17/2}} \, dx=-\frac {{\left (45045 \, {\left (i \, A + B\right )} a^{3} e^{\left (19 i \, f x + 19 i \, e\right )} + 6006 \, {\left (50 i \, A + 33 \, B\right )} a^{3} e^{\left (17 i \, f x + 17 i \, e\right )} + 3927 \, {\left (215 i \, A + 69 \, B\right )} a^{3} e^{\left (15 i \, f x + 15 i \, e\right )} + 21420 \, {\left (60 i \, A - B\right )} a^{3} e^{\left (13 i \, f x + 13 i \, e\right )} + 7735 \, {\left (145 i \, A - 51 \, B\right )} a^{3} e^{\left (11 i \, f x + 11 i \, e\right )} + 24310 \, {\left (22 i \, A - 15 \, B\right )} a^{3} e^{\left (9 i \, f x + 9 i \, e\right )} + 109395 \, {\left (i \, A - B\right )} a^{3} e^{\left (7 i \, f x + 7 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{24504480 \, c^{9} f} \]
integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(17 /2),x, algorithm="fricas")
-1/24504480*(45045*(I*A + B)*a^3*e^(19*I*f*x + 19*I*e) + 6006*(50*I*A + 33 *B)*a^3*e^(17*I*f*x + 17*I*e) + 3927*(215*I*A + 69*B)*a^3*e^(15*I*f*x + 15 *I*e) + 21420*(60*I*A - B)*a^3*e^(13*I*f*x + 13*I*e) + 7735*(145*I*A - 51* B)*a^3*e^(11*I*f*x + 11*I*e) + 24310*(22*I*A - 15*B)*a^3*e^(9*I*f*x + 9*I* e) + 109395*(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^9*f)
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{17/2}} \, dx=\text {Timed out} \]
Time = 0.56 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.31 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{17/2}} \, dx=\frac {{\left (45045 \, {\left (-i \, A - B\right )} a^{3} \cos \left (\frac {17}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 51051 \, {\left (-5 i \, A - 3 \, 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 ) + 117810 \, {\left (-5 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 ) + 139230 \, {\left (-5 i \, A + B\right )} 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 ) + 85085 \, {\left (-5 i \, A + 3 \, 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 ) + 109395 \, {\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 ) + 45045 \, {\left (A - i \, B\right )} a^{3} \sin \left (\frac {17}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 51051 \, {\left (5 \, A - 3 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 ) + 117810 \, {\left (5 \, 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 ) + 139230 \, {\left (5 \, A + i \, B\right )} 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 ) + 85085 \, {\left (5 \, A + 3 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 ) + 109395 \, {\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}}{24504480 \, c^{\frac {17}{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))^(17 /2),x, algorithm="maxima")
1/24504480*(45045*(-I*A - B)*a^3*cos(17/2*arctan2(sin(2*f*x + 2*e), cos(2* f*x + 2*e))) + 51051*(-5*I*A - 3*B)*a^3*cos(15/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 117810*(-5*I*A - B)*a^3*cos(13/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 139230*(-5*I*A + B)*a^3*cos(11/2*arctan2(sin(2 *f*x + 2*e), cos(2*f*x + 2*e))) + 85085*(-5*I*A + 3*B)*a^3*cos(9/2*arctan2 (sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 109395*(-I*A + B)*a^3*cos(7/2*arct an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 45045*(A - I*B)*a^3*sin(17/2*ar ctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 51051*(5*A - 3*I*B)*a^3*sin(1 5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 117810*(5*A - I*B)*a^3* sin(13/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 139230*(5*A + I*B) *a^3*sin(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 85085*(5*A + 3*I*B)*a^3*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 109395*( A + I*B)*a^3*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt(a) /(c^(17/2)*f)
\[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{17/2}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {17}{2}}} \,d x } \]
integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(17 /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)^(17/2), x)
Time = 13.23 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.73 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{17/2}} \, dx=-\frac {\sqrt {a+\frac {a\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}\,\left (\frac {a^3\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (5\,A+B\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{288\,c^8\,f}+\frac {a^3\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\left (5\,A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{176\,c^8\,f}+\frac {a^3\,{\mathrm {e}}^{e\,12{}\mathrm {i}+f\,x\,12{}\mathrm {i}}\,\left (5\,A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{208\,c^8\,f}+\frac {a^3\,{\mathrm {e}}^{e\,14{}\mathrm {i}+f\,x\,14{}\mathrm {i}}\,\left (5\,A-B\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{480\,c^8\,f}+\frac {a^3\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{224\,c^8\,f}+\frac {a^3\,{\mathrm {e}}^{e\,16{}\mathrm {i}+f\,x\,16{}\mathrm {i}}\,\left (A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{544\,c^8\,f}\right )}{\sqrt {c-\frac {c\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}} \]
-((a + (a*sin(e + f*x)*1i)/cos(e + f*x))^(1/2)*((a^3*exp(e*8i + f*x*8i)*(5 *A + B*3i)*1i)/(288*c^8*f) + (a^3*exp(e*10i + f*x*10i)*(5*A + B*1i)*1i)/(1 76*c^8*f) + (a^3*exp(e*12i + f*x*12i)*(5*A - B*1i)*1i)/(208*c^8*f) + (a^3* exp(e*14i + f*x*14i)*(5*A - B*3i)*1i)/(480*c^8*f) + (a^3*exp(e*6i + f*x*6i )*(A + B*1i)*1i)/(224*c^8*f) + (a^3*exp(e*16i + f*x*16i)*(A - B*1i)*1i)/(5 44*c^8*f)))/(c - (c*sin(e + f*x)*1i)/cos(e + f*x))^(1/2)