Integrand size = 41, antiderivative size = 185 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2} \, dx=\frac {(5 A-i B) x}{16 a^3 c^2}+\frac {A+i B}{24 a^3 c^2 f (i-\tan (e+f x))^3}-\frac {3 i A-B}{32 a^3 c^2 f (i-\tan (e+f x))^2}-\frac {3 A}{16 a^3 c^2 f (i-\tan (e+f x))}+\frac {i A+B}{32 a^3 c^2 f (i+\tan (e+f x))^2}+\frac {2 A-i B}{16 a^3 c^2 f (i+\tan (e+f x))} \] Output:
1/16*(5*A-I*B)*x/a^3/c^2+1/24*(A+I*B)/a^3/c^2/f/(I-tan(f*x+e))^3-1/32*(3*I *A-B)/a^3/c^2/f/(I-tan(f*x+e))^2-3/16*A/a^3/c^2/f/(I-tan(f*x+e))+1/32*(I*A +B)/a^3/c^2/f/(I+tan(f*x+e))^2+1/16*(2*A-I*B)/a^3/c^2/f/(I+tan(f*x+e))
Time = 5.66 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.93 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2} \, dx=\frac {\sec ^4(e+f x) (47 A+5 i B+(-14 A+22 i B) \cos (2 (e+f x))-A \cos (4 (e+f x))+5 i B \cos (4 (e+f x))-40 i A \sin (2 (e+f x))-8 B \sin (2 (e+f x))-5 i A \sin (4 (e+f x))-B \sin (4 (e+f x))+12 (5 A-i B) \arctan (\tan (e+f x)) (-i+\tan (e+f x)))}{192 a^3 c^2 f (-i+\tan (e+f x))^3 (i+\tan (e+f x))^2} \] Input:
Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^2),x]
Output:
(Sec[e + f*x]^4*(47*A + (5*I)*B + (-14*A + (22*I)*B)*Cos[2*(e + f*x)] - A* Cos[4*(e + f*x)] + (5*I)*B*Cos[4*(e + f*x)] - (40*I)*A*Sin[2*(e + f*x)] - 8*B*Sin[2*(e + f*x)] - (5*I)*A*Sin[4*(e + f*x)] - B*Sin[4*(e + f*x)] + 12* (5*A - I*B)*ArcTan[Tan[e + f*x]]*(-I + Tan[e + f*x])))/(192*a^3*c^2*f*(-I + Tan[e + f*x])^3*(I + Tan[e + f*x])^2)
Time = 0.68 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.81, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3042, 4071, 27, 86, 2009}
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.
| \(\displaystyle \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2}dx\) |
| \(\Big \downarrow \) 4071 |
| \(\displaystyle \frac {a c \int \frac {A+B \tan (e+f x)}{a^4 c^3 (1-i \tan (e+f x))^3 (i \tan (e+f x)+1)^4}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {A+B \tan (e+f x)}{(1-i \tan (e+f x))^3 (i \tan (e+f x)+1)^4}d\tan (e+f x)}{a^3 c^2 f}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\int \left (-\frac {3 A}{16 (\tan (e+f x)-i)^2}+\frac {5 A-i B}{16 \left (\tan ^2(e+f x)+1\right )}+\frac {i B-2 A}{16 (\tan (e+f x)+i)^2}+\frac {i (3 A+i B)}{16 (\tan (e+f x)-i)^3}-\frac {i (A-i B)}{16 (\tan (e+f x)+i)^3}+\frac {A+i B}{8 (\tan (e+f x)-i)^4}\right )d\tan (e+f x)}{a^3 c^2 f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{16} (5 A-i B) \arctan (\tan (e+f x))+\frac {2 A-i B}{16 (\tan (e+f x)+i)}-\frac {-B+3 i A}{32 (-\tan (e+f x)+i)^2}+\frac {B+i A}{32 (\tan (e+f x)+i)^2}+\frac {A+i B}{24 (-\tan (e+f x)+i)^3}-\frac {3 A}{16 (-\tan (e+f x)+i)}}{a^3 c^2 f}\) |
Input:
Int[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^ 2),x]
Output:
(((5*A - I*B)*ArcTan[Tan[e + f*x]])/16 + (A + I*B)/(24*(I - Tan[e + f*x])^ 3) - ((3*I)*A - B)/(32*(I - Tan[e + f*x])^2) - (3*A)/(16*(I - Tan[e + f*x] )) + (I*A + B)/(32*(I + Tan[e + f*x])^2) + (2*A - I*B)/(16*(I + Tan[e + f* x])))/(a^3*c^2*f)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 = 0.61 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.13
| method | result | size |
| norman | \(\frac {\frac {\left (-i B +5 A \right ) x}{16 a c}-\frac {-i A +B}{6 a c f}+\frac {\left (-i B +5 A \right ) \tan \left (f x +e \right )^{3}}{6 a c f}+\frac {\left (-i B +5 A \right ) \tan \left (f x +e \right )^{5}}{16 a c f}+\frac {3 \left (-i B +5 A \right ) x \tan \left (f x +e \right )^{2}}{16 a c}+\frac {3 \left (-i B +5 A \right ) x \tan \left (f x +e \right )^{4}}{16 a c}+\frac {\left (-i B +5 A \right ) x \tan \left (f x +e \right )^{6}}{16 a c}+\frac {\left (i B +11 A \right ) \tan \left (f x +e \right )}{16 a c f}}{\left (1+\tan \left (f x +e \right )^{2}\right )^{3} a^{2} c}\) | \(209\) |
| risch | \(-\frac {i x B}{16 a^{3} c^{2}}+\frac {5 x A}{16 a^{3} c^{2}}-\frac {{\mathrm e}^{-6 i \left (f x +e \right )} B}{192 a^{3} c^{2} f}+\frac {i {\mathrm e}^{-6 i \left (f x +e \right )} A}{192 a^{3} c^{2} f}-\frac {\cos \left (4 f x +4 e \right ) B}{32 a^{3} c^{2} f}+\frac {i \cos \left (4 f x +4 e \right ) A}{32 a^{3} c^{2} f}+\frac {i \sin \left (4 f x +4 e \right ) B}{64 a^{3} c^{2} f}+\frac {3 \sin \left (4 f x +4 e \right ) A}{64 a^{3} c^{2} f}-\frac {5 \cos \left (2 f x +2 e \right ) B}{64 a^{3} c^{2} f}+\frac {5 i \cos \left (2 f x +2 e \right ) A}{64 a^{3} c^{2} f}-\frac {i \sin \left (2 f x +2 e \right ) B}{64 a^{3} c^{2} f}+\frac {15 \sin \left (2 f x +2 e \right ) A}{64 a^{3} c^{2} f}\) | \(238\) |
| derivativedivides | \(-\frac {3 i A}{32 f \,a^{3} c^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {5 A \arctan \left (\tan \left (f x +e \right )\right )}{16 f \,a^{3} c^{2}}-\frac {i B \arctan \left (\tan \left (f x +e \right )\right )}{16 f \,a^{3} c^{2}}-\frac {i B}{16 f \,a^{3} c^{2} \left (i+\tan \left (f x +e \right )\right )}+\frac {B}{32 f \,a^{3} c^{2} \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B}{24 f \,a^{3} c^{2} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {A}{8 f \,a^{3} c^{2} \left (i+\tan \left (f x +e \right )\right )}-\frac {A}{24 f \,a^{3} c^{2} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {i A}{32 f \,a^{3} c^{2} \left (i+\tan \left (f x +e \right )\right )^{2}}+\frac {3 A}{16 f \,a^{3} c^{2} \left (-i+\tan \left (f x +e \right )\right )}+\frac {B}{32 f \,a^{3} c^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}\) | \(252\) |
| default | \(-\frac {3 i A}{32 f \,a^{3} c^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {5 A \arctan \left (\tan \left (f x +e \right )\right )}{16 f \,a^{3} c^{2}}-\frac {i B \arctan \left (\tan \left (f x +e \right )\right )}{16 f \,a^{3} c^{2}}-\frac {i B}{16 f \,a^{3} c^{2} \left (i+\tan \left (f x +e \right )\right )}+\frac {B}{32 f \,a^{3} c^{2} \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B}{24 f \,a^{3} c^{2} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {A}{8 f \,a^{3} c^{2} \left (i+\tan \left (f x +e \right )\right )}-\frac {A}{24 f \,a^{3} c^{2} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {i A}{32 f \,a^{3} c^{2} \left (i+\tan \left (f x +e \right )\right )^{2}}+\frac {3 A}{16 f \,a^{3} c^{2} \left (-i+\tan \left (f x +e \right )\right )}+\frac {B}{32 f \,a^{3} c^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}\) | \(252\) |
Input:
int((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^2,x,method=_R ETURNVERBOSE)
Output:
(1/16*(5*A-I*B)/a/c*x-1/6*(-I*A+B)/a/c/f+1/6*(5*A-I*B)/a/c/f*tan(f*x+e)^3+ 1/16*(5*A-I*B)/a/c/f*tan(f*x+e)^5+3/16*(5*A-I*B)/a/c*x*tan(f*x+e)^2+3/16*( 5*A-I*B)/a/c*x*tan(f*x+e)^4+1/16*(5*A-I*B)/a/c*x*tan(f*x+e)^6+1/16*(I*B+11 *A)/a/c/f*tan(f*x+e))/(1+tan(f*x+e)^2)^3/a^2/c
Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.62 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2} \, dx=\frac {{\left (24 \, {\left (5 \, A - i \, B\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 3 \, {\left (i \, A + B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} - 6 \, {\left (5 i \, A + 3 \, B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} - 12 \, {\left (-5 i \, A + B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 3 \, {\left (-5 i \, A + 3 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 i \, A - 2 \, B\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{384 \, a^{3} c^{2} f} \] Input:
integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^2,x, al gorithm="fricas")
Output:
1/384*(24*(5*A - I*B)*f*x*e^(6*I*f*x + 6*I*e) - 3*(I*A + B)*e^(10*I*f*x + 10*I*e) - 6*(5*I*A + 3*B)*e^(8*I*f*x + 8*I*e) - 12*(-5*I*A + B)*e^(4*I*f*x + 4*I*e) - 3*(-5*I*A + 3*B)*e^(2*I*f*x + 2*I*e) + 2*I*A - 2*B)*e^(-6*I*f* x - 6*I*e)/(a^3*c^2*f)
Time = 0.41 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.44 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2} \, dx=\begin {cases} \frac {\left (\left (33554432 i A a^{12} c^{8} f^{4} e^{6 i e} - 33554432 B a^{12} c^{8} f^{4} e^{6 i e}\right ) e^{- 6 i f x} + \left (251658240 i A a^{12} c^{8} f^{4} e^{8 i e} - 150994944 B a^{12} c^{8} f^{4} e^{8 i e}\right ) e^{- 4 i f x} + \left (1006632960 i A a^{12} c^{8} f^{4} e^{10 i e} - 201326592 B a^{12} c^{8} f^{4} e^{10 i e}\right ) e^{- 2 i f x} + \left (- 503316480 i A a^{12} c^{8} f^{4} e^{14 i e} - 301989888 B a^{12} c^{8} f^{4} e^{14 i e}\right ) e^{2 i f x} + \left (- 50331648 i A a^{12} c^{8} f^{4} e^{16 i e} - 50331648 B a^{12} c^{8} f^{4} e^{16 i e}\right ) e^{4 i f x}\right ) e^{- 12 i e}}{6442450944 a^{15} c^{10} f^{5}} & \text {for}\: a^{15} c^{10} f^{5} e^{12 i e} \neq 0 \\x \left (- \frac {5 A - i B}{16 a^{3} c^{2}} + \frac {\left (A e^{10 i e} + 5 A e^{8 i e} + 10 A e^{6 i e} + 10 A e^{4 i e} + 5 A e^{2 i e} + A - i B e^{10 i e} - 3 i B e^{8 i e} - 2 i B e^{6 i e} + 2 i B e^{4 i e} + 3 i B e^{2 i e} + i B\right ) e^{- 6 i e}}{32 a^{3} c^{2}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (5 A - i B\right )}{16 a^{3} c^{2}} \] Input:
integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))**3/(c-I*c*tan(f*x+e))**2,x)
Output:
Piecewise((((33554432*I*A*a**12*c**8*f**4*exp(6*I*e) - 33554432*B*a**12*c* *8*f**4*exp(6*I*e))*exp(-6*I*f*x) + (251658240*I*A*a**12*c**8*f**4*exp(8*I *e) - 150994944*B*a**12*c**8*f**4*exp(8*I*e))*exp(-4*I*f*x) + (1006632960* I*A*a**12*c**8*f**4*exp(10*I*e) - 201326592*B*a**12*c**8*f**4*exp(10*I*e)) *exp(-2*I*f*x) + (-503316480*I*A*a**12*c**8*f**4*exp(14*I*e) - 301989888*B *a**12*c**8*f**4*exp(14*I*e))*exp(2*I*f*x) + (-50331648*I*A*a**12*c**8*f** 4*exp(16*I*e) - 50331648*B*a**12*c**8*f**4*exp(16*I*e))*exp(4*I*f*x))*exp( -12*I*e)/(6442450944*a**15*c**10*f**5), Ne(a**15*c**10*f**5*exp(12*I*e), 0 )), (x*(-(5*A - I*B)/(16*a**3*c**2) + (A*exp(10*I*e) + 5*A*exp(8*I*e) + 10 *A*exp(6*I*e) + 10*A*exp(4*I*e) + 5*A*exp(2*I*e) + A - I*B*exp(10*I*e) - 3 *I*B*exp(8*I*e) - 2*I*B*exp(6*I*e) + 2*I*B*exp(4*I*e) + 3*I*B*exp(2*I*e) + I*B)*exp(-6*I*e)/(32*a**3*c**2)), True)) + x*(5*A - I*B)/(16*a**3*c**2)
Exception generated. \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^2,x, al gorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.52 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.83 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2} \, dx=-\frac {{\left (-5 i \, A - B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{32 \, a^{3} c^{2} f} - \frac {{\left (5 i \, A + B\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{32 \, a^{3} c^{2} f} + \frac {3 \, {\left (5 \, A - i \, B\right )} \tan \left (f x + e\right )^{4} - 3 \, {\left (5 i \, A + B\right )} \tan \left (f x + e\right )^{3} + 5 \, {\left (5 \, A - i \, B\right )} \tan \left (f x + e\right )^{2} - 5 \, {\left (5 i \, A + B\right )} \tan \left (f x + e\right ) + 8 \, A + 8 i \, B}{48 \, a^{3} c^{2} f {\left (\tan \left (f x + e\right ) + i\right )}^{2} {\left (\tan \left (f x + e\right ) - i\right )}^{3}} \] Input:
integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^2,x, al gorithm="giac")
Output:
-1/32*(-5*I*A - B)*log(tan(f*x + e) + I)/(a^3*c^2*f) - 1/32*(5*I*A + B)*lo g(tan(f*x + e) - I)/(a^3*c^2*f) + 1/48*(3*(5*A - I*B)*tan(f*x + e)^4 - 3*( 5*I*A + B)*tan(f*x + e)^3 + 5*(5*A - I*B)*tan(f*x + e)^2 - 5*(5*I*A + B)*t an(f*x + e) + 8*A + 8*I*B)/(a^3*c^2*f*(tan(f*x + e) + I)^2*(tan(f*x + e) - I)^3)
Time = 5.87 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.12 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2} \, dx=\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {25\,A}{48\,a^3\,c^2}-\frac {B\,5{}\mathrm {i}}{48\,a^3\,c^2}\right )+{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {5\,A}{16\,a^3\,c^2}-\frac {B\,1{}\mathrm {i}}{16\,a^3\,c^2}\right )+{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (\frac {B}{16\,a^3\,c^2}+\frac {A\,5{}\mathrm {i}}{16\,a^3\,c^2}\right )+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {5\,B}{48\,a^3\,c^2}+\frac {A\,25{}\mathrm {i}}{48\,a^3\,c^2}\right )-\frac {B}{6\,a^3\,c^2}+\frac {A\,1{}\mathrm {i}}{6\,a^3\,c^2}}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^5\,1{}\mathrm {i}+{\mathrm {tan}\left (e+f\,x\right )}^4+{\mathrm {tan}\left (e+f\,x\right )}^3\,2{}\mathrm {i}+2\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}+1\right )}-\frac {x\,\left (B+A\,5{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a^3\,c^2} \] Input:
int((A + B*tan(e + f*x))/((a + a*tan(e + f*x)*1i)^3*(c - c*tan(e + f*x)*1i )^2),x)
Output:
(tan(e + f*x)*((25*A)/(48*a^3*c^2) - (B*5i)/(48*a^3*c^2)) + tan(e + f*x)^3 *((5*A)/(16*a^3*c^2) - (B*1i)/(16*a^3*c^2)) + tan(e + f*x)^4*((A*5i)/(16*a ^3*c^2) + B/(16*a^3*c^2)) + tan(e + f*x)^2*((A*25i)/(48*a^3*c^2) + (5*B)/( 48*a^3*c^2)) + (A*1i)/(6*a^3*c^2) - B/(6*a^3*c^2))/(f*(tan(e + f*x)*1i + 2 *tan(e + f*x)^2 + tan(e + f*x)^3*2i + tan(e + f*x)^4 + tan(e + f*x)^5*1i + 1)) - (x*(A*5i + B)*1i)/(16*a^3*c^2)
\[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^2} \, dx=\frac {-8 \left (\int \frac {1}{\tan \left (f x +e \right )^{5}-\tan \left (f x +e \right )^{4} i +2 \tan \left (f x +e \right )^{3}-2 \tan \left (f x +e \right )^{2} i +\tan \left (f x +e \right )-i}d x \right ) \tan \left (f x +e \right )^{4} a f i +8 \left (\int \frac {1}{\tan \left (f x +e \right )^{5}-\tan \left (f x +e \right )^{4} i +2 \tan \left (f x +e \right )^{3}-2 \tan \left (f x +e \right )^{2} i +\tan \left (f x +e \right )-i}d x \right ) \tan \left (f x +e \right )^{4} b f -16 \left (\int \frac {1}{\tan \left (f x +e \right )^{5}-\tan \left (f x +e \right )^{4} i +2 \tan \left (f x +e \right )^{3}-2 \tan \left (f x +e \right )^{2} i +\tan \left (f x +e \right )-i}d x \right ) \tan \left (f x +e \right )^{2} a f i +16 \left (\int \frac {1}{\tan \left (f x +e \right )^{5}-\tan \left (f x +e \right )^{4} i +2 \tan \left (f x +e \right )^{3}-2 \tan \left (f x +e \right )^{2} i +\tan \left (f x +e \right )-i}d x \right ) \tan \left (f x +e \right )^{2} b f -8 \left (\int \frac {1}{\tan \left (f x +e \right )^{5}-\tan \left (f x +e \right )^{4} i +2 \tan \left (f x +e \right )^{3}-2 \tan \left (f x +e \right )^{2} i +\tan \left (f x +e \right )-i}d x \right ) a f i +8 \left (\int \frac {1}{\tan \left (f x +e \right )^{5}-\tan \left (f x +e \right )^{4} i +2 \tan \left (f x +e \right )^{3}-2 \tan \left (f x +e \right )^{2} i +\tan \left (f x +e \right )-i}d x \right ) b f -3 \tan \left (f x +e \right )^{4} b e i -3 \tan \left (f x +e \right )^{4} b f i x -3 \tan \left (f x +e \right )^{3} b i -6 \tan \left (f x +e \right )^{2} b e i -6 \tan \left (f x +e \right )^{2} b f i x -5 \tan \left (f x +e \right ) b i -3 b e i -3 b f i x}{8 a^{3} c^{2} f \left (\tan \left (f x +e \right )^{4}+2 \tan \left (f x +e \right )^{2}+1\right )} \] Input:
int((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^2,x)
Output:
( - 8*int(1/(tan(e + f*x)**5 - tan(e + f*x)**4*i + 2*tan(e + f*x)**3 - 2*t an(e + f*x)**2*i + tan(e + f*x) - i),x)*tan(e + f*x)**4*a*f*i + 8*int(1/(t an(e + f*x)**5 - tan(e + f*x)**4*i + 2*tan(e + f*x)**3 - 2*tan(e + f*x)**2 *i + tan(e + f*x) - i),x)*tan(e + f*x)**4*b*f - 16*int(1/(tan(e + f*x)**5 - tan(e + f*x)**4*i + 2*tan(e + f*x)**3 - 2*tan(e + f*x)**2*i + tan(e + f* x) - i),x)*tan(e + f*x)**2*a*f*i + 16*int(1/(tan(e + f*x)**5 - tan(e + f*x )**4*i + 2*tan(e + f*x)**3 - 2*tan(e + f*x)**2*i + tan(e + f*x) - i),x)*ta n(e + f*x)**2*b*f - 8*int(1/(tan(e + f*x)**5 - tan(e + f*x)**4*i + 2*tan(e + f*x)**3 - 2*tan(e + f*x)**2*i + tan(e + f*x) - i),x)*a*f*i + 8*int(1/(t an(e + f*x)**5 - tan(e + f*x)**4*i + 2*tan(e + f*x)**3 - 2*tan(e + f*x)**2 *i + tan(e + f*x) - i),x)*b*f - 3*tan(e + f*x)**4*b*e*i - 3*tan(e + f*x)** 4*b*f*i*x - 3*tan(e + f*x)**3*b*i - 6*tan(e + f*x)**2*b*e*i - 6*tan(e + f* x)**2*b*f*i*x - 5*tan(e + f*x)*b*i - 3*b*e*i - 3*b*f*i*x)/(8*a**3*c**2*f*( tan(e + f*x)**4 + 2*tan(e + f*x)**2 + 1))