Integrand size = 45, antiderivative size = 208 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{13 f (c-i c \tan (e+f x))^{13/2}}-\frac {(3 i A-10 B) (a+i a \tan (e+f x))^{7/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}-\frac {2 (3 i A-10 B) (a+i a \tan (e+f x))^{7/2}}{1287 c^2 f (c-i c \tan (e+f x))^{9/2}}-\frac {2 (3 i A-10 B) (a+i a \tan (e+f x))^{7/2}}{9009 c^3 f (c-i c \tan (e+f x))^{7/2}} \]
-1/13*(I*A+B)*(a+I*a*tan(f*x+e))^(7/2)/f/(c-I*c*tan(f*x+e))^(13/2)-1/143*( 3*I*A-10*B)*(a+I*a*tan(f*x+e))^(7/2)/c/f/(c-I*c*tan(f*x+e))^(11/2)-2/1287* (3*I*A-10*B)*(a+I*a*tan(f*x+e))^(7/2)/c^2/f/(c-I*c*tan(f*x+e))^(9/2)-2/900 9*(3*I*A-10*B)*(a+I*a*tan(f*x+e))^(7/2)/c^3/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. \(495\) vs. \(2(208)=416\).
Time = 19.12 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.38 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=\frac {\cos ^4(e+f x) \left ((-i A+B) \cos (6 f x) \left (\frac {\cos (3 e)}{112 c^7}+\frac {i \sin (3 e)}{112 c^7}\right )+(-15 i A+8 B) \cos (8 f x) \left (\frac {\cos (5 e)}{504 c^7}+\frac {i \sin (5 e)}{504 c^7}\right )+(-30 i A+B) \cos (10 f x) \left (\frac {\cos (7 e)}{792 c^7}+\frac {i \sin (7 e)}{792 c^7}\right )+(25 A-12 i B) \cos (12 f x) \left (-\frac {i \cos (9 e)}{1144 c^7}+\frac {\sin (9 e)}{1144 c^7}\right )+(A-i B) \cos (14 f x) \left (-\frac {i \cos (11 e)}{208 c^7}+\frac {\sin (11 e)}{208 c^7}\right )+(A+i B) \left (\frac {\cos (3 e)}{112 c^7}+\frac {i \sin (3 e)}{112 c^7}\right ) \sin (6 f x)+(15 A+8 i B) \left (\frac {\cos (5 e)}{504 c^7}+\frac {i \sin (5 e)}{504 c^7}\right ) \sin (8 f x)+(30 A+i B) \left (\frac {\cos (7 e)}{792 c^7}+\frac {i \sin (7 e)}{792 c^7}\right ) \sin (10 f x)+(25 A-12 i B) \left (\frac {\cos (9 e)}{1144 c^7}+\frac {i \sin (9 e)}{1144 c^7}\right ) \sin (12 f x)+(A-i B) \left (\frac {\cos (11 e)}{208 c^7}+\frac {i \sin (11 e)}{208 c^7}\right ) \sin (14 f x)\right ) \sqrt {\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))} \]
Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan [e + f*x])^(13/2),x]
(Cos[e + f*x]^4*(((-I)*A + B)*Cos[6*f*x]*(Cos[3*e]/(112*c^7) + ((I/112)*Si n[3*e])/c^7) + ((-15*I)*A + 8*B)*Cos[8*f*x]*(Cos[5*e]/(504*c^7) + ((I/504) *Sin[5*e])/c^7) + ((-30*I)*A + B)*Cos[10*f*x]*(Cos[7*e]/(792*c^7) + ((I/79 2)*Sin[7*e])/c^7) + (25*A - (12*I)*B)*Cos[12*f*x]*(((-1/1144*I)*Cos[9*e])/ c^7 + Sin[9*e]/(1144*c^7)) + (A - I*B)*Cos[14*f*x]*(((-1/208*I)*Cos[11*e]) /c^7 + Sin[11*e]/(208*c^7)) + (A + I*B)*(Cos[3*e]/(112*c^7) + ((I/112)*Sin [3*e])/c^7)*Sin[6*f*x] + (15*A + (8*I)*B)*(Cos[5*e]/(504*c^7) + ((I/504)*S in[5*e])/c^7)*Sin[8*f*x] + (30*A + I*B)*(Cos[7*e]/(792*c^7) + ((I/792)*Sin [7*e])/c^7)*Sin[10*f*x] + (25*A - (12*I)*B)*(Cos[9*e]/(1144*c^7) + ((I/114 4)*Sin[9*e])/c^7)*Sin[12*f*x] + (A - I*B)*(Cos[11*e]/(208*c^7) + ((I/208)* Sin[11*e])/c^7)*Sin[14*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 = 0.44 (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.
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}}dx\) |
| \(\Big \downarrow \) 4071 |
| \(\displaystyle \frac {a c \int \frac {(i \tan (e+f x) a+a)^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{15/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {a c \left (\frac {(3 A+10 i B) \int \frac {(i \tan (e+f x) a+a)^{5/2}}{(c-i c \tan (e+f x))^{13/2}}d\tan (e+f x)}{13 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{13 a c (c-i c \tan (e+f x))^{13/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {(3 A+10 i B) \left (\frac {2 \int \frac {(i \tan (e+f x) a+a)^{5/2}}{(c-i c \tan (e+f x))^{11/2}}d\tan (e+f x)}{11 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{13 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{13 a c (c-i c \tan (e+f x))^{13/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {(3 A+10 i B) \left (\frac {2 \left (\frac {\int \frac {(i \tan (e+f x) a+a)^{5/2}}{(c-i c \tan (e+f x))^{9/2}}d\tan (e+f x)}{9 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{9 a c (c-i c \tan (e+f x))^{9/2}}\right )}{11 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{13 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{13 a c (c-i c \tan (e+f x))^{13/2}}\right )}{f}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {a c \left (\frac {(3 A+10 i B) \left (\frac {2 \left (-\frac {i (a+i a \tan (e+f x))^{7/2}}{63 a c^2 (c-i c \tan (e+f x))^{7/2}}-\frac {i (a+i a \tan (e+f x))^{7/2}}{9 a c (c-i c \tan (e+f x))^{9/2}}\right )}{11 c}-\frac {i (a+i a \tan (e+f x))^{7/2}}{11 a c (c-i c \tan (e+f x))^{11/2}}\right )}{13 c}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{13 a c (c-i c \tan (e+f x))^{13/2}}\right )}{f}\) |
(a*c*(-1/13*((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(a*c*(c - I*c*Tan[e + f*x])^(13/2)) + ((3*A + (10*I)*B)*(((-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)))/f
3.9.27.3.1 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 = 0.34 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(-\frac {a^{3} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (693 i A \,{\mathrm e}^{12 i \left (f x +e \right )}+693 B \,{\mathrm e}^{12 i \left (f x +e \right )}+2457 i A \,{\mathrm e}^{10 i \left (f x +e \right )}+819 B \,{\mathrm e}^{10 i \left (f x +e \right )}+3003 i A \,{\mathrm e}^{8 i \left (f x +e \right )}-1001 B \,{\mathrm e}^{8 i \left (f x +e \right )}+1287 i A \,{\mathrm e}^{6 i \left (f x +e \right )}-1287 B \,{\mathrm e}^{6 i \left (f x +e \right )}\right )}{72072 c^{6} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(156\) |
| derivativedivides | \(\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (6 i A \tan \left (f x +e \right )^{5}-160 i B \tan \left (f x +e \right )^{4}-20 B \tan \left (f x +e \right )^{5}-177 i A \tan \left (f x +e \right )^{3}-48 A \tan \left (f x +e \right )^{4}-1643 i B \tan \left (f x +e \right )^{2}+590 B \tan \left (f x +e \right )^{3}-1569 i A \tan \left (f x +e \right )+408 A \tan \left (f x +e \right )^{2}-97 i B -776 B \tan \left (f x +e \right )-930 A \right )}{9009 f \,c^{7} \left (i+\tan \left (f x +e \right )\right )^{8}}\) | \(184\) |
| default | \(\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (6 i A \tan \left (f x +e \right )^{5}-160 i B \tan \left (f x +e \right )^{4}-20 B \tan \left (f x +e \right )^{5}-177 i A \tan \left (f x +e \right )^{3}-48 A \tan \left (f x +e \right )^{4}-1643 i B \tan \left (f x +e \right )^{2}+590 B \tan \left (f x +e \right )^{3}-1569 i A \tan \left (f x +e \right )+408 A \tan \left (f x +e \right )^{2}-97 i B -776 B \tan \left (f x +e \right )-930 A \right )}{9009 f \,c^{7} \left (i+\tan \left (f x +e \right )\right )^{8}}\) | \(184\) |
| parts | \(\frac {i A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (2 i \tan \left (f x +e \right )^{5}-59 i \tan \left (f x +e \right )^{3}-16 \tan \left (f x +e \right )^{4}-523 i \tan \left (f x +e \right )+136 \tan \left (f x +e \right )^{2}-310\right )}{3003 f \,c^{7} \left (i+\tan \left (f x +e \right )\right )^{8}}-\frac {i B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (160 i \tan \left (f x +e \right )^{4}+20 \tan \left (f x +e \right )^{5}+1643 i \tan \left (f x +e \right )^{2}-590 \tan \left (f x +e \right )^{3}+97 i+776 \tan \left (f x +e \right )\right )}{9009 f \,c^{7} \left (i+\tan \left (f x +e \right )\right )^{8}}\) | \(238\) |
int((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(13/2),x, method=_RETURNVERBOSE)
-1/72072*a^3/c^6*(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*(693*I*A*exp(12*I*(f*x+e))+693*B*exp(12*I*(f*x+e)) +2457*I*A*exp(10*I*(f*x+e))+819*B*exp(10*I*(f*x+e))+3003*I*A*exp(8*I*(f*x+ e))-1001*B*exp(8*I*(f*x+e))+1287*I*A*exp(6*I*(f*x+e))-1287*B*exp(6*I*(f*x+ e)))
Time = 0.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=-\frac {{\left (693 \, {\left (i \, A + B\right )} a^{3} e^{\left (15 i \, f x + 15 i \, e\right )} + 126 \, {\left (25 i \, A + 12 \, B\right )} a^{3} e^{\left (13 i \, f x + 13 i \, e\right )} + 182 \, {\left (30 i \, A - B\right )} a^{3} e^{\left (11 i \, f x + 11 i \, e\right )} + 286 \, {\left (15 i \, A - 8 \, B\right )} a^{3} e^{\left (9 i \, f x + 9 i \, e\right )} + 1287 \, {\left (i \, A - B\right )} a^{3} e^{\left (7 i \, f x + 7 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{72072 \, c^{7} f} \]
integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(13 /2),x, algorithm="fricas")
-1/72072*(693*(I*A + B)*a^3*e^(15*I*f*x + 15*I*e) + 126*(25*I*A + 12*B)*a^ 3*e^(13*I*f*x + 13*I*e) + 182*(30*I*A - B)*a^3*e^(11*I*f*x + 11*I*e) + 286 *(15*I*A - 8*B)*a^3*e^(9*I*f*x + 9*I*e) + 1287*(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^7*f)
Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=\text {Timed out} \]
Time = 0.67 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.33 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=\frac {{\left (693 \, {\left (-i \, A - B\right )} a^{3} \cos \left (\frac {13}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 819 \, {\left (-3 i \, A - B\right )} a^{3} \cos \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1001 \, {\left (-3 i \, A + B\right )} a^{3} \cos \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1287 \, {\left (-i \, A + B\right )} a^{3} \cos \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 693 \, {\left (A - i \, B\right )} a^{3} \sin \left (\frac {13}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 819 \, {\left (3 \, A - i \, B\right )} a^{3} \sin \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1001 \, {\left (3 \, A + i \, B\right )} a^{3} \sin \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1287 \, {\left (A + i \, B\right )} a^{3} \sin \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a}}{72072 \, c^{\frac {13}{2}} f} \]
integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(13 /2),x, algorithm="maxima")
1/72072*(693*(-I*A - B)*a^3*cos(13/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 819*(-3*I*A - B)*a^3*cos(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f* x + 2*e))) + 1001*(-3*I*A + B)*a^3*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2 *f*x + 2*e))) + 1287*(-I*A + B)*a^3*cos(7/2*arctan2(sin(2*f*x + 2*e), cos( 2*f*x + 2*e))) + 693*(A - I*B)*a^3*sin(13/2*arctan2(sin(2*f*x + 2*e), cos( 2*f*x + 2*e))) + 819*(3*A - I*B)*a^3*sin(11/2*arctan2(sin(2*f*x + 2*e), co s(2*f*x + 2*e))) + 1001*(3*A + I*B)*a^3*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1287*(A + I*B)*a^3*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt(a)/(c^(13/2)*f)
\[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {13}{2}}} \,d x } \]
integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(13 /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)^(13/2), x)
Time = 13.01 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.80 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx=-\frac {\sqrt {a+\frac {a\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}\,\left (\frac {a^3\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\left (3\,A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{72\,c^6\,f}+\frac {a^3\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\left (3\,A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{88\,c^6\,f}+\frac {a^3\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{56\,c^6\,f}+\frac {a^3\,{\mathrm {e}}^{e\,12{}\mathrm {i}+f\,x\,12{}\mathrm {i}}\,\left (A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{104\,c^6\,f}\right )}{\sqrt {c-\frac {c\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}} \]
-((a + (a*sin(e + f*x)*1i)/cos(e + f*x))^(1/2)*((a^3*exp(e*8i + f*x*8i)*(3 *A + B*1i)*1i)/(72*c^6*f) + (a^3*exp(e*10i + f*x*10i)*(3*A - B*1i)*1i)/(88 *c^6*f) + (a^3*exp(e*6i + f*x*6i)*(A + B*1i)*1i)/(56*c^6*f) + (a^3*exp(e*1 2i + f*x*12i)*(A - B*1i)*1i)/(104*c^6*f)))/(c - (c*sin(e + f*x)*1i)/cos(e + f*x))^(1/2)