Integrand size = 41, antiderivative size = 319 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=\frac {7 (3 A+i B) x}{128 a^3 c^6}+\frac {A+i B}{384 a^3 c^6 f (i-\tan (e+f x))^3}-\frac {7 i A-5 B}{512 a^3 c^6 f (i-\tan (e+f x))^2}-\frac {7 (2 A+i B)}{256 a^3 c^6 f (i-\tan (e+f x))}+\frac {i A+B}{96 a^3 c^6 f (i+\tan (e+f x))^6}+\frac {2 A-i B}{80 a^3 c^6 f (i+\tan (e+f x))^5}-\frac {5 i A+B}{128 a^3 c^6 f (i+\tan (e+f x))^4}-\frac {5 A}{96 a^3 c^6 f (i+\tan (e+f x))^3}+\frac {5 (7 i A-B)}{512 a^3 c^6 f (i+\tan (e+f x))^2}+\frac {7 (4 A+i B)}{256 a^3 c^6 f (i+\tan (e+f x))} \] Output:
7/128*(3*A+I*B)*x/a^3/c^6+1/384*(A+I*B)/a^3/c^6/f/(I-tan(f*x+e))^3-1/512*( 7*I*A-5*B)/a^3/c^6/f/(I-tan(f*x+e))^2-7/256*(2*A+I*B)/a^3/c^6/f/(I-tan(f*x +e))+1/96*(I*A+B)/a^3/c^6/f/(I+tan(f*x+e))^6+1/80*(2*A-I*B)/a^3/c^6/f/(I+t an(f*x+e))^5-1/128*(5*I*A+B)/a^3/c^6/f/(I+tan(f*x+e))^4-5/96*A/a^3/c^6/f/( I+tan(f*x+e))^3+5/512*(7*I*A-B)/a^3/c^6/f/(I+tan(f*x+e))^2+7/256*(4*A+I*B) /a^3/c^6/f/(I+tan(f*x+e))
Time = 6.41 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.87 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=\frac {\sec ^8(e+f x) (-1271 A+43 i B-8 (391 A+37 i B) \cos (2 (e+f x))+(734 A+618 i B) \cos (4 (e+f x))+76 A \cos (6 (e+f x))+132 i B \cos (6 (e+f x))+5 A \cos (8 (e+f x))+15 i B \cos (8 (e+f x))+1890 i A \sin (2 (e+f x))-630 B \sin (2 (e+f x))+840 (-3 i A+B) \arctan (\tan (e+f x)) \sec (e+f x) (\cos (3 (e+f x))-i \sin (3 (e+f x)))-1176 i A \sin (4 (e+f x))+392 B \sin (4 (e+f x))-174 i A \sin (6 (e+f x))+58 B \sin (6 (e+f x))-15 i A \sin (8 (e+f x))+5 B \sin (8 (e+f x)))}{15360 a^3 c^6 f (-i+\tan (e+f x))^3 (i+\tan (e+f x))^6} \] Input:
Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^6),x]
Output:
(Sec[e + f*x]^8*(-1271*A + (43*I)*B - 8*(391*A + (37*I)*B)*Cos[2*(e + f*x) ] + (734*A + (618*I)*B)*Cos[4*(e + f*x)] + 76*A*Cos[6*(e + f*x)] + (132*I) *B*Cos[6*(e + f*x)] + 5*A*Cos[8*(e + f*x)] + (15*I)*B*Cos[8*(e + f*x)] + ( 1890*I)*A*Sin[2*(e + f*x)] - 630*B*Sin[2*(e + f*x)] + 840*((-3*I)*A + B)*A rcTan[Tan[e + f*x]]*Sec[e + f*x]*(Cos[3*(e + f*x)] - I*Sin[3*(e + f*x)]) - (1176*I)*A*Sin[4*(e + f*x)] + 392*B*Sin[4*(e + f*x)] - (174*I)*A*Sin[6*(e + f*x)] + 58*B*Sin[6*(e + f*x)] - (15*I)*A*Sin[8*(e + f*x)] + 5*B*Sin[8*( e + f*x)]))/(15360*a^3*c^6*f*(-I + Tan[e + f*x])^3*(I + Tan[e + f*x])^6)
Time = 0.83 (sec) , antiderivative size = 248, 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.
| \(\displaystyle \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6}dx\) |
| \(\Big \downarrow \) 4071 |
| \(\displaystyle \frac {a c \int \frac {A+B \tan (e+f x)}{a^4 c^7 (1-i \tan (e+f x))^7 (i \tan (e+f x)+1)^4}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {A+B \tan (e+f x)}{(1-i \tan (e+f x))^7 (i \tan (e+f x)+1)^4}d\tan (e+f x)}{a^3 c^6 f}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\int \left (\frac {5 A}{32 (\tan (e+f x)+i)^4}+\frac {7 (3 A+i B)}{128 \left (\tan ^2(e+f x)+1\right )}-\frac {7 (2 A+i B)}{256 (\tan (e+f x)-i)^2}-\frac {7 (4 A+i B)}{256 (\tan (e+f x)+i)^2}+\frac {i (7 A+5 i B)}{256 (\tan (e+f x)-i)^3}+\frac {5 (B-7 i A)}{256 (\tan (e+f x)+i)^3}+\frac {A+i B}{128 (\tan (e+f x)-i)^4}+\frac {5 i A+B}{32 (\tan (e+f x)+i)^5}+\frac {i B-2 A}{16 (\tan (e+f x)+i)^6}-\frac {i (A-i B)}{16 (\tan (e+f x)+i)^7}\right )d\tan (e+f x)}{a^3 c^6 f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {7}{128} (3 A+i B) \arctan (\tan (e+f x))-\frac {7 (2 A+i B)}{256 (-\tan (e+f x)+i)}+\frac {7 (4 A+i B)}{256 (\tan (e+f x)+i)}-\frac {-5 B+7 i A}{512 (-\tan (e+f x)+i)^2}+\frac {5 (-B+7 i A)}{512 (\tan (e+f x)+i)^2}+\frac {A+i B}{384 (-\tan (e+f x)+i)^3}-\frac {B+5 i A}{128 (\tan (e+f x)+i)^4}+\frac {2 A-i B}{80 (\tan (e+f x)+i)^5}+\frac {B+i A}{96 (\tan (e+f x)+i)^6}-\frac {5 A}{96 (\tan (e+f x)+i)^3}}{a^3 c^6 f}\) |
Input:
Int[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^ 6),x]
Output:
((7*(3*A + I*B)*ArcTan[Tan[e + f*x]])/128 + (A + I*B)/(384*(I - Tan[e + f* x])^3) - ((7*I)*A - 5*B)/(512*(I - Tan[e + f*x])^2) - (7*(2*A + I*B))/(256 *(I - Tan[e + f*x])) + (I*A + B)/(96*(I + Tan[e + f*x])^6) + (2*A - I*B)/( 80*(I + Tan[e + f*x])^5) - ((5*I)*A + B)/(128*(I + Tan[e + f*x])^4) - (5*A )/(96*(I + Tan[e + f*x])^3) + (5*((7*I)*A - B))/(512*(I + Tan[e + f*x])^2) + (7*(4*A + I*B))/(256*(I + Tan[e + f*x])))/(a^3*c^6*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 = 0.75 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.23
| method | result | size |
| norman | \(\frac {\frac {7 \left (i B +3 A \right ) x}{128 a c}-\frac {7 i A +B}{30 a c f}+\frac {281 \left (i B +3 A \right ) \tan \left (f x +e \right )^{5}}{320 a c f}+\frac {231 \left (i B +3 A \right ) \tan \left (f x +e \right )^{7}}{320 a c f}+\frac {119 \left (i B +3 A \right ) \tan \left (f x +e \right )^{9}}{384 a c f}+\frac {7 \left (i B +3 A \right ) \tan \left (f x +e \right )^{11}}{128 a c f}+\frac {21 \left (i B +3 A \right ) x \tan \left (f x +e \right )^{2}}{64 a c}+\frac {105 \left (i B +3 A \right ) x \tan \left (f x +e \right )^{4}}{128 a c}+\frac {35 \left (i B +3 A \right ) x \tan \left (f x +e \right )^{6}}{32 a c}+\frac {105 \left (i B +3 A \right ) x \tan \left (f x +e \right )^{8}}{128 a c}+\frac {21 \left (i B +3 A \right ) x \tan \left (f x +e \right )^{10}}{64 a c}+\frac {7 \left (i B +3 A \right ) x \tan \left (f x +e \right )^{12}}{128 a c}+\frac {\left (-7 i B +107 A \right ) \tan \left (f x +e \right )}{128 a c f}+\frac {\left (265 i B +667 A \right ) \tan \left (f x +e \right )^{3}}{384 a c f}+\frac {\left (i A +3 B \right ) \tan \left (f x +e \right )^{2}}{10 a c f}}{\left (1+\tan \left (f x +e \right )^{2}\right )^{6} a^{2} c^{5}}\) | \(392\) |
| risch | \(-\frac {7 i \sin \left (4 f x +4 e \right ) B}{2048 a^{3} c^{6} f}+\frac {21 x A}{128 a^{3} c^{6}}-\frac {{\mathrm e}^{12 i \left (f x +e \right )} B}{6144 a^{3} c^{6} f}-\frac {9 i {\mathrm e}^{8 i \left (f x +e \right )} A}{1024 a^{3} c^{6} f}-\frac {7 \,{\mathrm e}^{10 i \left (f x +e \right )} B}{5120 a^{3} c^{6} f}-\frac {9 i \sin \left (6 f x +6 e \right ) B}{1024 a^{3} c^{6} f}-\frac {5 \,{\mathrm e}^{8 i \left (f x +e \right )} B}{1024 a^{3} c^{6} f}+\frac {7 i x B}{128 a^{3} c^{6}}-\frac {29 \cos \left (6 f x +6 e \right ) B}{3072 a^{3} c^{6} f}-\frac {9 i {\mathrm e}^{10 i \left (f x +e \right )} A}{5120 a^{3} c^{6} f}+\frac {17 i \sin \left (2 f x +2 e \right ) B}{512 a^{3} c^{6} f}+\frac {85 \sin \left (6 f x +6 e \right ) A}{3072 a^{3} c^{6} f}-\frac {21 \cos \left (4 f x +4 e \right ) B}{2048 a^{3} c^{6} f}-\frac {i {\mathrm e}^{12 i \left (f x +e \right )} A}{6144 a^{3} c^{6} f}-\frac {45 i \cos \left (2 f x +2 e \right ) A}{512 a^{3} c^{6} f}+\frac {135 \sin \left (4 f x +4 e \right ) A}{2048 a^{3} c^{6} f}-\frac {3 \cos \left (2 f x +2 e \right ) B}{512 a^{3} c^{6} f}-\frac {83 i \cos \left (6 f x +6 e \right ) A}{3072 a^{3} c^{6} f}-\frac {117 i \cos \left (4 f x +4 e \right ) A}{2048 a^{3} c^{6} f}+\frac {81 \sin \left (2 f x +2 e \right ) A}{512 a^{3} c^{6} f}\) | \(410\) |
| derivativedivides | \(\frac {7 A}{128 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )}+\frac {A}{40 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {A}{384 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {21 A \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{6}}+\frac {7 A}{64 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )}+\frac {5 B}{512 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {B}{128 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{4}}-\frac {5 A}{96 a^{3} c^{6} f \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {B}{96 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{6}}-\frac {5 B}{512 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {5 i A}{128 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{4}}-\frac {7 i A}{512 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B}{80 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {i A}{96 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{6}}+\frac {7 i B}{256 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )}+\frac {7 i B \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{6}}+\frac {35 i A}{512 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{2}}+\frac {7 i B}{256 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )}-\frac {i B}{384 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{3}}\) | \(440\) |
| default | \(\frac {7 A}{128 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )}+\frac {A}{40 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {A}{384 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {21 A \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{6}}+\frac {7 A}{64 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )}+\frac {5 B}{512 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {B}{128 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{4}}-\frac {5 A}{96 a^{3} c^{6} f \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {B}{96 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{6}}-\frac {5 B}{512 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {5 i A}{128 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{4}}-\frac {7 i A}{512 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B}{80 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {i A}{96 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{6}}+\frac {7 i B}{256 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )}+\frac {7 i B \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{6}}+\frac {35 i A}{512 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{2}}+\frac {7 i B}{256 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )}-\frac {i B}{384 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{3}}\) | \(440\) |
Input:
int((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^6,x,method=_R ETURNVERBOSE)
Output:
(7/128*(3*A+I*B)/a/c*x-1/30*(7*I*A+B)/a/c/f+281/320*(3*A+I*B)/a/c/f*tan(f* x+e)^5+231/320*(3*A+I*B)/a/c/f*tan(f*x+e)^7+119/384*(3*A+I*B)/a/c/f*tan(f* x+e)^9+7/128*(3*A+I*B)/a/c/f*tan(f*x+e)^11+21/64*(3*A+I*B)/a/c*x*tan(f*x+e )^2+105/128*(3*A+I*B)/a/c*x*tan(f*x+e)^4+35/32*(3*A+I*B)/a/c*x*tan(f*x+e)^ 6+105/128*(3*A+I*B)/a/c*x*tan(f*x+e)^8+21/64*(3*A+I*B)/a/c*x*tan(f*x+e)^10 +7/128*(3*A+I*B)/a/c*x*tan(f*x+e)^12+1/128*(107*A-7*I*B)/a/c/f*tan(f*x+e)+ 1/384*(667*A+265*I*B)/a/c/f*tan(f*x+e)^3+1/10/a/c/f*(I*A+3*B)*tan(f*x+e)^2 )/(1+tan(f*x+e)^2)^6/a^2/c^5
Time = 0.08 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.58 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=\frac {{\left (1680 \, {\left (3 \, A + i \, B\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 5 \, {\left (i \, A + B\right )} e^{\left (18 i \, f x + 18 i \, e\right )} - 6 \, {\left (9 i \, A + 7 \, B\right )} e^{\left (16 i \, f x + 16 i \, e\right )} - 30 \, {\left (9 i \, A + 5 \, B\right )} e^{\left (14 i \, f x + 14 i \, e\right )} - 280 \, {\left (3 i \, A + B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} - 210 \, {\left (9 i \, A + B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} - 420 \, {\left (9 i \, A - B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} - 120 \, {\left (-9 i \, A + 5 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 15 \, {\left (-9 i \, A + 7 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 10 i \, A - 10 \, B\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{30720 \, a^{3} c^{6} f} \] Input:
integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^6,x, al gorithm="fricas")
Output:
1/30720*(1680*(3*A + I*B)*f*x*e^(6*I*f*x + 6*I*e) - 5*(I*A + B)*e^(18*I*f* x + 18*I*e) - 6*(9*I*A + 7*B)*e^(16*I*f*x + 16*I*e) - 30*(9*I*A + 5*B)*e^( 14*I*f*x + 14*I*e) - 280*(3*I*A + B)*e^(12*I*f*x + 12*I*e) - 210*(9*I*A + B)*e^(10*I*f*x + 10*I*e) - 420*(9*I*A - B)*e^(8*I*f*x + 8*I*e) - 120*(-9*I *A + 5*B)*e^(4*I*f*x + 4*I*e) - 15*(-9*I*A + 7*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^6*f)
Time = 0.79 (sec) , antiderivative size = 753, normalized size of antiderivative = 2.36 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx =\text {Too large to display} \] Input:
integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))**3/(c-I*c*tan(f*x+e))**6,x)
Output:
Piecewise((((6800207735332289107722240*I*A*a**24*c**48*f**8*exp(6*I*e) - 6 800207735332289107722240*B*a**24*c**48*f**8*exp(6*I*e))*exp(-6*I*f*x) + (9 1802804426985902954250240*I*A*a**24*c**48*f**8*exp(8*I*e) - 71402181220989 035631083520*B*a**24*c**48*f**8*exp(8*I*e))*exp(-4*I*f*x) + (7344224354158 87223634001920*I*A*a**24*c**48*f**8*exp(10*I*e) - 408012464119937346463334 400*B*a**24*c**48*f**8*exp(10*I*e))*exp(-2*I*f*x) + (-25704785239556052827 19006720*I*A*a**24*c**48*f**8*exp(14*I*e) + 285608724883956142524334080*B* a**24*c**48*f**8*exp(14*I*e))*exp(2*I*f*x) + (-128523926197780264135950336 0*I*A*a**24*c**48*f**8*exp(16*I*e) - 142804362441978071262167040*B*a**24*c **48*f**8*exp(16*I*e))*exp(4*I*f*x) + (-571217449767912285048668160*I*A*a* *24*c**48*f**8*exp(18*I*e) - 190405816589304095016222720*B*a**24*c**48*f** 8*exp(18*I*e))*exp(6*I*f*x) + (-183605608853971805908500480*I*A*a**24*c**4 8*f**8*exp(20*I*e) - 102003116029984336615833600*B*a**24*c**48*f**8*exp(20 *I*e))*exp(8*I*f*x) + (-36721121770794361181700096*I*A*a**24*c**48*f**8*ex p(22*I*e) - 28560872488395614252433408*B*a**24*c**48*f**8*exp(22*I*e))*exp (10*I*f*x) + (-3400103867666144553861120*I*A*a**24*c**48*f**8*exp(24*I*e) - 3400103867666144553861120*B*a**24*c**48*f**8*exp(24*I*e))*exp(12*I*f*x)) *exp(-12*I*e)/(20890238162940792138922721280*a**27*c**54*f**9), Ne(a**27*c **54*f**9*exp(12*I*e), 0)), (x*(-(21*A + 7*I*B)/(128*a**3*c**6) + (A*exp(1 8*I*e) + 9*A*exp(16*I*e) + 36*A*exp(14*I*e) + 84*A*exp(12*I*e) + 126*A*...
Exception generated. \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^6,x, al gorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.79 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.69 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=-\frac {7 \, {\left (-3 i \, A + B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{256 \, a^{3} c^{6} f} - \frac {7 \, {\left (3 i \, A - B\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{256 \, a^{3} c^{6} f} + \frac {105 \, {\left (3 \, A + i \, B\right )} \tan \left (f x + e\right )^{8} - 315 \, {\left (-3 i \, A + B\right )} \tan \left (f x + e\right )^{7} - 35 \, {\left (3 \, A + i \, B\right )} \tan \left (f x + e\right )^{6} - 735 \, {\left (-3 i \, A + B\right )} \tan \left (f x + e\right )^{5} - 609 \, {\left (3 \, A + i \, B\right )} \tan \left (f x + e\right )^{4} - 413 \, {\left (-3 i \, A + B\right )} \tan \left (f x + e\right )^{3} - 645 \, {\left (3 \, A + i \, B\right )} \tan \left (f x + e\right )^{2} - 87 \, {\left (3 i \, A - B\right )} \tan \left (f x + e\right ) - 448 \, A + 64 i \, B}{1920 \, a^{3} c^{6} f {\left (\tan \left (f x + e\right ) + i\right )}^{6} {\left (\tan \left (f x + e\right ) - i\right )}^{3}} \] Input:
integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^6,x, al gorithm="giac")
Output:
-7/256*(-3*I*A + B)*log(tan(f*x + e) + I)/(a^3*c^6*f) - 7/256*(3*I*A - B)* log(tan(f*x + e) - I)/(a^3*c^6*f) + 1/1920*(105*(3*A + I*B)*tan(f*x + e)^8 - 315*(-3*I*A + B)*tan(f*x + e)^7 - 35*(3*A + I*B)*tan(f*x + e)^6 - 735*( -3*I*A + B)*tan(f*x + e)^5 - 609*(3*A + I*B)*tan(f*x + e)^4 - 413*(-3*I*A + B)*tan(f*x + e)^3 - 645*(3*A + I*B)*tan(f*x + e)^2 - 87*(3*I*A - B)*tan( f*x + e) - 448*A + 64*I*B)/(a^3*c^6*f*(tan(f*x + e) + I)^6*(tan(f*x + e) - I)^3)
Time = 7.69 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.10 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (-\frac {29\,B}{640\,a^3\,c^6}+\frac {A\,87{}\mathrm {i}}{640\,a^3\,c^6}\right )-{\mathrm {tan}\left (e+f\,x\right )}^8\,\left (\frac {21\,A}{128\,a^3\,c^6}+\frac {B\,7{}\mathrm {i}}{128\,a^3\,c^6}\right )-{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (-\frac {21\,B}{128\,a^3\,c^6}+\frac {A\,63{}\mathrm {i}}{128\,a^3\,c^6}\right )+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {129\,A}{128\,a^3\,c^6}+\frac {B\,43{}\mathrm {i}}{128\,a^3\,c^6}\right )-{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (-\frac {49\,B}{128\,a^3\,c^6}+\frac {A\,147{}\mathrm {i}}{128\,a^3\,c^6}\right )+{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (\frac {7\,A}{128\,a^3\,c^6}+\frac {B\,7{}\mathrm {i}}{384\,a^3\,c^6}\right )+{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (\frac {609\,A}{640\,a^3\,c^6}+\frac {B\,203{}\mathrm {i}}{640\,a^3\,c^6}\right )-{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (-\frac {413\,B}{1920\,a^3\,c^6}+\frac {A\,413{}\mathrm {i}}{640\,a^3\,c^6}\right )+\frac {7\,A}{30\,a^3\,c^6}-\frac {B\,1{}\mathrm {i}}{30\,a^3\,c^6}}{f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^9-{\mathrm {tan}\left (e+f\,x\right )}^8\,3{}\mathrm {i}-{\mathrm {tan}\left (e+f\,x\right )}^6\,8{}\mathrm {i}+6\,{\mathrm {tan}\left (e+f\,x\right )}^5-{\mathrm {tan}\left (e+f\,x\right )}^4\,6{}\mathrm {i}+8\,{\mathrm {tan}\left (e+f\,x\right )}^3+3\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}+\frac {7\,x\,\left (3\,A+B\,1{}\mathrm {i}\right )}{128\,a^3\,c^6} \] Input:
int((A + B*tan(e + f*x))/((a + a*tan(e + f*x)*1i)^3*(c - c*tan(e + f*x)*1i )^6),x)
Output:
(tan(e + f*x)*((A*87i)/(640*a^3*c^6) - (29*B)/(640*a^3*c^6)) - tan(e + f*x )^8*((21*A)/(128*a^3*c^6) + (B*7i)/(128*a^3*c^6)) - tan(e + f*x)^7*((A*63i )/(128*a^3*c^6) - (21*B)/(128*a^3*c^6)) + tan(e + f*x)^2*((129*A)/(128*a^3 *c^6) + (B*43i)/(128*a^3*c^6)) - tan(e + f*x)^5*((A*147i)/(128*a^3*c^6) - (49*B)/(128*a^3*c^6)) + tan(e + f*x)^6*((7*A)/(128*a^3*c^6) + (B*7i)/(384* a^3*c^6)) + tan(e + f*x)^4*((609*A)/(640*a^3*c^6) + (B*203i)/(640*a^3*c^6) ) - tan(e + f*x)^3*((A*413i)/(640*a^3*c^6) - (413*B)/(1920*a^3*c^6)) + (7* A)/(30*a^3*c^6) - (B*1i)/(30*a^3*c^6))/(f*(3*tan(e + f*x) + 8*tan(e + f*x) ^3 - tan(e + f*x)^4*6i + 6*tan(e + f*x)^5 - tan(e + f*x)^6*8i - tan(e + f* x)^8*3i - tan(e + f*x)^9 + 1i)) + (7*x*(3*A + B*1i))/(128*a^3*c^6)
\[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=\frac {\left (\int \frac {\tan \left (f x +e \right )}{\tan \left (f x +e \right )^{9} i -3 \tan \left (f x +e \right )^{8}-8 \tan \left (f x +e \right )^{6}-6 \tan \left (f x +e \right )^{5} i -6 \tan \left (f x +e \right )^{4}-8 \tan \left (f x +e \right )^{3} i -3 \tan \left (f x +e \right ) i +1}d x \right ) b +\left (\int \frac {1}{\tan \left (f x +e \right )^{9} i -3 \tan \left (f x +e \right )^{8}-8 \tan \left (f x +e \right )^{6}-6 \tan \left (f x +e \right )^{5} i -6 \tan \left (f x +e \right )^{4}-8 \tan \left (f x +e \right )^{3} i -3 \tan \left (f x +e \right ) i +1}d x \right ) a}{a^{3} c^{6}} \] Input:
int((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^6,x)
Output:
(int(tan(e + f*x)/(tan(e + f*x)**9*i - 3*tan(e + f*x)**8 - 8*tan(e + f*x)* *6 - 6*tan(e + f*x)**5*i - 6*tan(e + f*x)**4 - 8*tan(e + f*x)**3*i - 3*tan (e + f*x)*i + 1),x)*b + int(1/(tan(e + f*x)**9*i - 3*tan(e + f*x)**8 - 8*t an(e + f*x)**6 - 6*tan(e + f*x)**5*i - 6*tan(e + f*x)**4 - 8*tan(e + f*x)* *3*i - 3*tan(e + f*x)*i + 1),x)*a)/(a**3*c**6)