Integrand size = 41, antiderivative size = 287 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {7 (4 A+i B) x}{128 a^3 c^5}+\frac {A+i B}{192 a^3 c^5 f (i-\tan (e+f x))^3}-\frac {3 i A-2 B}{128 a^3 c^5 f (i-\tan (e+f x))^2}-\frac {3 (7 A+3 i B)}{256 a^3 c^5 f (i-\tan (e+f x))}+\frac {A-i B}{80 a^3 c^5 f (i+\tan (e+f x))^5}-\frac {2 i A+B}{64 a^3 c^5 f (i+\tan (e+f x))^4}-\frac {5 A-i B}{96 a^3 c^5 f (i+\tan (e+f x))^3}+\frac {5 i A}{64 a^3 c^5 f (i+\tan (e+f x))^2}+\frac {5 (7 A+i B)}{256 a^3 c^5 f (i+\tan (e+f x))} \] Output:
7/128*(4*A+I*B)*x/a^3/c^5+1/192*(A+I*B)/a^3/c^5/f/(I-tan(f*x+e))^3-1/128*( 3*I*A-2*B)/a^3/c^5/f/(I-tan(f*x+e))^2-3/256*(7*A+3*I*B)/a^3/c^5/f/(I-tan(f *x+e))+1/80*(A-I*B)/a^3/c^5/f/(I+tan(f*x+e))^5-1/64*(2*I*A+B)/a^3/c^5/f/(I +tan(f*x+e))^4-1/96*(5*A-I*B)/a^3/c^5/f/(I+tan(f*x+e))^3+5/64*I*A/a^3/c^5/ f/(I+tan(f*x+e))^2+5/256*(7*A+I*B)/a^3/c^5/f/(I+tan(f*x+e))
Time = 6.13 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.91 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {\sec ^7(e+f x) (2100 i A \cos (e+f x)+63 (-8 i A+7 B) \cos (3 (e+f x))-56 i A \cos (5 (e+f x))+119 B \cos (5 (e+f x))-4 i A \cos (7 (e+f x))+16 B \cos (7 (e+f x))+872 A \sin (e+f x)+218 i B \sin (e+f x)-420 (4 A+i B) \arctan (\tan (e+f x)) \sec (e+f x) (\cos (2 (e+f x))-i \sin (2 (e+f x)))-956 A \sin (3 (e+f x))-239 i B \sin (3 (e+f x))-164 A \sin (5 (e+f x))-41 i B \sin (5 (e+f x))-16 A \sin (7 (e+f x))-4 i B \sin (7 (e+f x)))}{7680 a^3 c^5 f (-i+\tan (e+f x))^3 (i+\tan (e+f x))^5} \] Input:
Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^5),x]
Output:
(Sec[e + f*x]^7*((2100*I)*A*Cos[e + f*x] + 63*((-8*I)*A + 7*B)*Cos[3*(e + f*x)] - (56*I)*A*Cos[5*(e + f*x)] + 119*B*Cos[5*(e + f*x)] - (4*I)*A*Cos[7 *(e + f*x)] + 16*B*Cos[7*(e + f*x)] + 872*A*Sin[e + f*x] + (218*I)*B*Sin[e + f*x] - 420*(4*A + I*B)*ArcTan[Tan[e + f*x]]*Sec[e + f*x]*(Cos[2*(e + f* x)] - I*Sin[2*(e + f*x)]) - 956*A*Sin[3*(e + f*x)] - (239*I)*B*Sin[3*(e + f*x)] - 164*A*Sin[5*(e + f*x)] - (41*I)*B*Sin[5*(e + f*x)] - 16*A*Sin[7*(e + f*x)] - (4*I)*B*Sin[7*(e + f*x)]))/(7680*a^3*c^5*f*(-I + Tan[e + f*x])^ 3*(I + Tan[e + f*x])^5)
Time = 0.80 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.78, 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.
| \(\displaystyle \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5}dx\) |
| \(\Big \downarrow \) 4071 |
| \(\displaystyle \frac {a c \int \frac {A+B \tan (e+f x)}{a^4 c^6 (1-i \tan (e+f x))^6 (i \tan (e+f x)+1)^4}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {A+B \tan (e+f x)}{(1-i \tan (e+f x))^6 (i \tan (e+f x)+1)^4}d\tan (e+f x)}{a^3 c^5 f}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\int \left (-\frac {5 i A}{32 (\tan (e+f x)+i)^3}+\frac {7 (4 A+i B)}{128 \left (\tan ^2(e+f x)+1\right )}-\frac {3 (7 A+3 i B)}{256 (\tan (e+f x)-i)^2}-\frac {5 (7 A+i B)}{256 (\tan (e+f x)+i)^2}+\frac {i (3 A+2 i B)}{64 (\tan (e+f x)-i)^3}+\frac {A+i B}{64 (\tan (e+f x)-i)^4}+\frac {5 A-i B}{32 (\tan (e+f x)+i)^4}+\frac {2 i A+B}{16 (\tan (e+f x)+i)^5}+\frac {i B-A}{16 (\tan (e+f x)+i)^6}\right )d\tan (e+f x)}{a^3 c^5 f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {7}{128} (4 A+i B) \arctan (\tan (e+f x))-\frac {3 (7 A+3 i B)}{256 (-\tan (e+f x)+i)}+\frac {5 (7 A+i B)}{256 (\tan (e+f x)+i)}-\frac {-2 B+3 i A}{128 (-\tan (e+f x)+i)^2}+\frac {A+i B}{192 (-\tan (e+f x)+i)^3}-\frac {5 A-i B}{96 (\tan (e+f x)+i)^3}-\frac {B+2 i A}{64 (\tan (e+f x)+i)^4}+\frac {A-i B}{80 (\tan (e+f x)+i)^5}+\frac {5 i A}{64 (\tan (e+f x)+i)^2}}{a^3 c^5 f}\) |
Input:
Int[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^ 5),x]
Output:
((7*(4*A + I*B)*ArcTan[Tan[e + f*x]])/128 + (A + I*B)/(192*(I - Tan[e + f* x])^3) - ((3*I)*A - 2*B)/(128*(I - Tan[e + f*x])^2) - (3*(7*A + (3*I)*B))/ (256*(I - Tan[e + f*x])) + (A - I*B)/(80*(I + Tan[e + f*x])^5) - ((2*I)*A + B)/(64*(I + Tan[e + f*x])^4) - (5*A - I*B)/(96*(I + Tan[e + f*x])^3) + ( ((5*I)/64)*A)/(I + Tan[e + f*x])^2 + (5*(7*A + I*B))/(256*(I + Tan[e + f*x ])))/(a^3*c^5*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 = 0.68 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.17
| method | result | size |
| norman | \(\frac {\frac {7 \left (i B +4 A \right ) x}{128 a c}-\frac {8 i A +3 B}{40 a c f}+\frac {B \tan \left (f x +e \right )^{2}}{8 a c f}+\frac {79 \left (i B +4 A \right ) \tan \left (f x +e \right )^{3}}{192 a c f}+\frac {7 \left (i B +4 A \right ) \tan \left (f x +e \right )^{5}}{15 a c f}+\frac {49 \left (i B +4 A \right ) \tan \left (f x +e \right )^{7}}{192 a c f}+\frac {7 \left (i B +4 A \right ) \tan \left (f x +e \right )^{9}}{128 a c f}+\frac {35 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{2}}{128 a c}+\frac {35 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{4}}{64 a c}+\frac {35 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{6}}{64 a c}+\frac {35 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{8}}{128 a c}+\frac {7 \left (i B +4 A \right ) x \tan \left (f x +e \right )^{10}}{128 a c}+\frac {\left (-7 i B +100 A \right ) \tan \left (f x +e \right )}{128 a c f}}{\left (1+\tan \left (f x +e \right )^{2}\right )^{5} a^{2} c^{4}}\) | \(335\) |
| risch | \(-\frac {i {\mathrm e}^{8 i \left (f x +e \right )} A}{256 a^{3} c^{5} f}+\frac {7 x A}{32 a^{3} c^{5}}-\frac {{\mathrm e}^{10 i \left (f x +e \right )} B}{2560 a^{3} c^{5} f}-\frac {21 i \cos \left (2 f x +2 e \right ) A}{256 a^{3} c^{5} f}-\frac {3 \,{\mathrm e}^{8 i \left (f x +e \right )} B}{1024 a^{3} c^{5} f}-\frac {13 i \sin \left (6 f x +6 e \right ) B}{1536 a^{3} c^{5} f}-\frac {5 \cos \left (6 f x +6 e \right ) B}{512 a^{3} c^{5} f}-\frac {9 i \cos \left (6 f x +6 e \right ) A}{512 a^{3} c^{5} f}+\frac {7 i \sin \left (2 f x +2 e \right ) B}{256 a^{3} c^{5} f}+\frac {29 \sin \left (6 f x +6 e \right ) A}{1536 a^{3} c^{5} f}-\frac {5 \cos \left (4 f x +4 e \right ) B}{256 a^{3} c^{5} f}-\frac {i {\mathrm e}^{10 i \left (f x +e \right )} A}{2560 a^{3} c^{5} f}-\frac {i \sin \left (4 f x +4 e \right ) B}{128 a^{3} c^{5} f}+\frac {\sin \left (4 f x +4 e \right ) A}{16 a^{3} c^{5} f}-\frac {7 \cos \left (2 f x +2 e \right ) B}{256 a^{3} c^{5} f}+\frac {7 i x B}{128 a^{3} c^{5}}-\frac {3 i \cos \left (4 f x +4 e \right ) A}{64 a^{3} c^{5} f}+\frac {49 \sin \left (2 f x +2 e \right ) A}{256 a^{3} c^{5} f}\) | \(367\) |
| derivativedivides | \(\frac {5 i B}{256 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )}-\frac {A}{192 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {3 i A}{128 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {9 i B}{256 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {7 A \arctan \left (\tan \left (f x +e \right )\right )}{32 f \,a^{3} c^{5}}+\frac {7 i B \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{5}}+\frac {21 A}{256 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {B}{64 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B}{80 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {i B}{192 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {A}{80 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {i A}{32 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{4}}-\frac {5 A}{96 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {i B}{96 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {35 A}{256 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )}+\frac {5 i A}{64 a^{3} c^{5} f \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {B}{64 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{4}}\) | \(394\) |
| default | \(\frac {5 i B}{256 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )}-\frac {A}{192 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {3 i A}{128 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {9 i B}{256 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {7 A \arctan \left (\tan \left (f x +e \right )\right )}{32 f \,a^{3} c^{5}}+\frac {7 i B \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{5}}+\frac {21 A}{256 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )}+\frac {B}{64 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B}{80 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {i B}{192 f \,a^{3} c^{5} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {A}{80 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {i A}{32 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{4}}-\frac {5 A}{96 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {i B}{96 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {35 A}{256 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )}+\frac {5 i A}{64 a^{3} c^{5} f \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {B}{64 f \,a^{3} c^{5} \left (i+\tan \left (f x +e \right )\right )^{4}}\) | \(394\) |
Input:
int((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x,method=_R ETURNVERBOSE)
Output:
(7/128*(4*A+I*B)/a/c*x-1/40*(3*B+8*I*A)/a/c/f+1/8/a/c/f*B*tan(f*x+e)^2+79/ 192*(4*A+I*B)/a/c/f*tan(f*x+e)^3+7/15*(4*A+I*B)/a/c/f*tan(f*x+e)^5+49/192* (4*A+I*B)/a/c/f*tan(f*x+e)^7+7/128*(4*A+I*B)/a/c/f*tan(f*x+e)^9+35/128*(4* A+I*B)/a/c*x*tan(f*x+e)^2+35/64*(4*A+I*B)/a/c*x*tan(f*x+e)^4+35/64*(4*A+I* B)/a/c*x*tan(f*x+e)^6+35/128*(4*A+I*B)/a/c*x*tan(f*x+e)^8+7/128*(4*A+I*B)/ a/c*x*tan(f*x+e)^10+1/128*(100*A-7*I*B)/a/c/f*tan(f*x+e))/(1+tan(f*x+e)^2) ^5/a^2/c^4
Time = 0.07 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.55 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {{\left (840 \, {\left (4 \, A + i \, B\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 6 \, {\left (i \, A + B\right )} e^{\left (16 i \, f x + 16 i \, e\right )} - 15 \, {\left (4 i \, A + 3 \, B\right )} e^{\left (14 i \, f x + 14 i \, e\right )} - 140 \, {\left (2 i \, A + B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} - 210 \, {\left (4 i \, A + B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} - 2100 i \, A e^{\left (8 i \, f x + 8 i \, e\right )} - 420 \, {\left (-2 i \, A + B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 30 \, {\left (-4 i \, A + 3 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 10 i \, A - 10 \, B\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{15360 \, a^{3} c^{5} f} \] Input:
integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x, al gorithm="fricas")
Output:
1/15360*(840*(4*A + I*B)*f*x*e^(6*I*f*x + 6*I*e) - 6*(I*A + B)*e^(16*I*f*x + 16*I*e) - 15*(4*I*A + 3*B)*e^(14*I*f*x + 14*I*e) - 140*(2*I*A + B)*e^(1 2*I*f*x + 12*I*e) - 210*(4*I*A + B)*e^(10*I*f*x + 10*I*e) - 2100*I*A*e^(8* I*f*x + 8*I*e) - 420*(-2*I*A + B)*e^(4*I*f*x + 4*I*e) - 30*(-4*I*A + 3*B)* e^(2*I*f*x + 2*I*e) + 10*I*A - 10*B)*e^(-6*I*f*x - 6*I*e)/(a^3*c^5*f)
Time = 0.79 (sec) , antiderivative size = 646, normalized size of antiderivative = 2.25 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\begin {cases} \frac {\left (- 7263405479023135948800 i A a^{21} c^{35} f^{7} e^{14 i e} e^{2 i f x} + \left (34587645138205409280 i A a^{21} c^{35} f^{7} e^{6 i e} - 34587645138205409280 B a^{21} c^{35} f^{7} e^{6 i e}\right ) e^{- 6 i f x} + \left (415051741658464911360 i A a^{21} c^{35} f^{7} e^{8 i e} - 311288806243848683520 B a^{21} c^{35} f^{7} e^{8 i e}\right ) e^{- 4 i f x} + \left (2905362191609254379520 i A a^{21} c^{35} f^{7} e^{10 i e} - 1452681095804627189760 B a^{21} c^{35} f^{7} e^{10 i e}\right ) e^{- 2 i f x} + \left (- 2905362191609254379520 i A a^{21} c^{35} f^{7} e^{16 i e} - 726340547902313594880 B a^{21} c^{35} f^{7} e^{16 i e}\right ) e^{4 i f x} + \left (- 968454063869751459840 i A a^{21} c^{35} f^{7} e^{18 i e} - 484227031934875729920 B a^{21} c^{35} f^{7} e^{18 i e}\right ) e^{6 i f x} + \left (- 207525870829232455680 i A a^{21} c^{35} f^{7} e^{20 i e} - 155644403121924341760 B a^{21} c^{35} f^{7} e^{20 i e}\right ) e^{8 i f x} + \left (- 20752587082923245568 i A a^{21} c^{35} f^{7} e^{22 i e} - 20752587082923245568 B a^{21} c^{35} f^{7} e^{22 i e}\right ) e^{10 i f x}\right ) e^{- 12 i e}}{53126622932283508654080 a^{24} c^{40} f^{8}} & \text {for}\: a^{24} c^{40} f^{8} e^{12 i e} \neq 0 \\x \left (- \frac {28 A + 7 i B}{128 a^{3} c^{5}} + \frac {\left (A e^{16 i e} + 8 A e^{14 i e} + 28 A e^{12 i e} + 56 A e^{10 i e} + 70 A e^{8 i e} + 56 A e^{6 i e} + 28 A e^{4 i e} + 8 A e^{2 i e} + A - i B e^{16 i e} - 6 i B e^{14 i e} - 14 i B e^{12 i e} - 14 i B e^{10 i e} + 14 i B e^{6 i e} + 14 i B e^{4 i e} + 6 i B e^{2 i e} + i B\right ) e^{- 6 i e}}{256 a^{3} c^{5}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (28 A + 7 i B\right )}{128 a^{3} c^{5}} \] Input:
integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))**3/(c-I*c*tan(f*x+e))**5,x)
Output:
Piecewise(((-7263405479023135948800*I*A*a**21*c**35*f**7*exp(14*I*e)*exp(2 *I*f*x) + (34587645138205409280*I*A*a**21*c**35*f**7*exp(6*I*e) - 34587645 138205409280*B*a**21*c**35*f**7*exp(6*I*e))*exp(-6*I*f*x) + (4150517416584 64911360*I*A*a**21*c**35*f**7*exp(8*I*e) - 311288806243848683520*B*a**21*c **35*f**7*exp(8*I*e))*exp(-4*I*f*x) + (2905362191609254379520*I*A*a**21*c* *35*f**7*exp(10*I*e) - 1452681095804627189760*B*a**21*c**35*f**7*exp(10*I* e))*exp(-2*I*f*x) + (-2905362191609254379520*I*A*a**21*c**35*f**7*exp(16*I *e) - 726340547902313594880*B*a**21*c**35*f**7*exp(16*I*e))*exp(4*I*f*x) + (-968454063869751459840*I*A*a**21*c**35*f**7*exp(18*I*e) - 48422703193487 5729920*B*a**21*c**35*f**7*exp(18*I*e))*exp(6*I*f*x) + (-20752587082923245 5680*I*A*a**21*c**35*f**7*exp(20*I*e) - 155644403121924341760*B*a**21*c**3 5*f**7*exp(20*I*e))*exp(8*I*f*x) + (-20752587082923245568*I*A*a**21*c**35* f**7*exp(22*I*e) - 20752587082923245568*B*a**21*c**35*f**7*exp(22*I*e))*ex p(10*I*f*x))*exp(-12*I*e)/(53126622932283508654080*a**24*c**40*f**8), Ne(a **24*c**40*f**8*exp(12*I*e), 0)), (x*(-(28*A + 7*I*B)/(128*a**3*c**5) + (A *exp(16*I*e) + 8*A*exp(14*I*e) + 28*A*exp(12*I*e) + 56*A*exp(10*I*e) + 70* A*exp(8*I*e) + 56*A*exp(6*I*e) + 28*A*exp(4*I*e) + 8*A*exp(2*I*e) + A - I* B*exp(16*I*e) - 6*I*B*exp(14*I*e) - 14*I*B*exp(12*I*e) - 14*I*B*exp(10*I*e ) + 14*I*B*exp(6*I*e) + 14*I*B*exp(4*I*e) + 6*I*B*exp(2*I*e) + I*B)*exp(-6 *I*e)/(256*a**3*c**5)), True)) + x*(28*A + 7*I*B)/(128*a**3*c**5)
Exception generated. \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x, al gorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.76 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.70 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=-\frac {7 \, {\left (-4 i \, A + B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{256 \, a^{3} c^{5} f} - \frac {7 \, {\left (4 i \, A - B\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{256 \, a^{3} c^{5} f} + \frac {105 \, {\left (4 \, A + i \, B\right )} \tan \left (f x + e\right )^{7} - 210 \, {\left (-4 i \, A + B\right )} \tan \left (f x + e\right )^{6} + 175 \, {\left (4 \, A + i \, B\right )} \tan \left (f x + e\right )^{5} - 560 \, {\left (-4 i \, A + B\right )} \tan \left (f x + e\right )^{4} - 49 \, {\left (4 \, A + i \, B\right )} \tan \left (f x + e\right )^{3} - 462 \, {\left (-4 i \, A + B\right )} \tan \left (f x + e\right )^{2} - 183 \, {\left (4 \, A + i \, B\right )} \tan \left (f x + e\right ) + 384 i \, A + 144 \, B}{1920 \, a^{3} c^{5} f {\left (\tan \left (f x + e\right ) + i\right )}^{5} {\left (\tan \left (f x + e\right ) - i\right )}^{3}} \] Input:
integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x, al gorithm="giac")
Output:
-7/256*(-4*I*A + B)*log(tan(f*x + e) + I)/(a^3*c^5*f) - 7/256*(4*I*A - B)* log(tan(f*x + e) - I)/(a^3*c^5*f) + 1/1920*(105*(4*A + I*B)*tan(f*x + e)^7 - 210*(-4*I*A + B)*tan(f*x + e)^6 + 175*(4*A + I*B)*tan(f*x + e)^5 - 560* (-4*I*A + B)*tan(f*x + e)^4 - 49*(4*A + I*B)*tan(f*x + e)^3 - 462*(-4*I*A + B)*tan(f*x + e)^2 - 183*(4*A + I*B)*tan(f*x + e) + 384*I*A + 144*B)/(a^3 *c^5*f*(tan(f*x + e) + I)^5*(tan(f*x + e) - I)^3)
Time = 7.29 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.11 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {\frac {3\,B}{40\,a^3\,c^5}+{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (-\frac {7\,B}{24\,a^3\,c^5}+\frac {A\,7{}\mathrm {i}}{6\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (-\frac {7\,B}{64\,a^3\,c^5}+\frac {A\,7{}\mathrm {i}}{16\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (\frac {7\,A}{32\,a^3\,c^5}+\frac {B\,7{}\mathrm {i}}{128\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (\frac {35\,A}{96\,a^3\,c^5}+\frac {B\,35{}\mathrm {i}}{384\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (-\frac {77\,B}{320\,a^3\,c^5}+\frac {A\,77{}\mathrm {i}}{80\,a^3\,c^5}\right )-{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {49\,A}{480\,a^3\,c^5}+\frac {B\,49{}\mathrm {i}}{1920\,a^3\,c^5}\right )-\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {61\,A}{160\,a^3\,c^5}+\frac {B\,61{}\mathrm {i}}{640\,a^3\,c^5}\right )+\frac {A\,1{}\mathrm {i}}{5\,a^3\,c^5}}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^8+{\mathrm {tan}\left (e+f\,x\right )}^7\,2{}\mathrm {i}+2\,{\mathrm {tan}\left (e+f\,x\right )}^6+{\mathrm {tan}\left (e+f\,x\right )}^5\,6{}\mathrm {i}+{\mathrm {tan}\left (e+f\,x\right )}^3\,6{}\mathrm {i}-2\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,2{}\mathrm {i}-1\right )}+\frac {7\,x\,\left (4\,A+B\,1{}\mathrm {i}\right )}{128\,a^3\,c^5} \] Input:
int((A + B*tan(e + f*x))/((a + a*tan(e + f*x)*1i)^3*(c - c*tan(e + f*x)*1i )^5),x)
Output:
(tan(e + f*x)^4*((A*7i)/(6*a^3*c^5) - (7*B)/(24*a^3*c^5)) - tan(e + f*x)*( (61*A)/(160*a^3*c^5) + (B*61i)/(640*a^3*c^5)) + tan(e + f*x)^6*((A*7i)/(16 *a^3*c^5) - (7*B)/(64*a^3*c^5)) + tan(e + f*x)^7*((7*A)/(32*a^3*c^5) + (B* 7i)/(128*a^3*c^5)) + tan(e + f*x)^5*((35*A)/(96*a^3*c^5) + (B*35i)/(384*a^ 3*c^5)) + tan(e + f*x)^2*((A*77i)/(80*a^3*c^5) - (77*B)/(320*a^3*c^5)) - t an(e + f*x)^3*((49*A)/(480*a^3*c^5) + (B*49i)/(1920*a^3*c^5)) + (A*1i)/(5* a^3*c^5) + (3*B)/(40*a^3*c^5))/(f*(tan(e + f*x)*2i - 2*tan(e + f*x)^2 + ta n(e + f*x)^3*6i + tan(e + f*x)^5*6i + 2*tan(e + f*x)^6 + tan(e + f*x)^7*2i + tan(e + f*x)^8 - 1)) + (7*x*(4*A + B*1i))/(128*a^3*c^5)
\[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx=\frac {-\left (\int \frac {\tan \left (f x +e \right )}{\tan \left (f x +e \right )^{8}+2 \tan \left (f x +e \right )^{7} i +2 \tan \left (f x +e \right )^{6}+6 \tan \left (f x +e \right )^{5} i +6 \tan \left (f x +e \right )^{3} i -2 \tan \left (f x +e \right )^{2}+2 \tan \left (f x +e \right ) i -1}d x \right ) b -\left (\int \frac {1}{\tan \left (f x +e \right )^{8}+2 \tan \left (f x +e \right )^{7} i +2 \tan \left (f x +e \right )^{6}+6 \tan \left (f x +e \right )^{5} i +6 \tan \left (f x +e \right )^{3} i -2 \tan \left (f x +e \right )^{2}+2 \tan \left (f x +e \right ) i -1}d x \right ) a}{a^{3} c^{5}} \] Input:
int((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x)
Output:
( - (int(tan(e + f*x)/(tan(e + f*x)**8 + 2*tan(e + f*x)**7*i + 2*tan(e + f *x)**6 + 6*tan(e + f*x)**5*i + 6*tan(e + f*x)**3*i - 2*tan(e + f*x)**2 + 2 *tan(e + f*x)*i - 1),x)*b + int(1/(tan(e + f*x)**8 + 2*tan(e + f*x)**7*i + 2*tan(e + f*x)**6 + 6*tan(e + f*x)**5*i + 6*tan(e + f*x)**3*i - 2*tan(e + f*x)**2 + 2*tan(e + f*x)*i - 1),x)*a))/(a**3*c**5)