\(\int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx\) [739]

Optimal result

Integrand size = 41, antiderivative size = 319 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=\frac {7 (3 A+i B) x}{128 a^3 c^6}+\frac {A+i B}{384 a^3 c^6 f (i-\tan (e+f x))^3}-\frac {7 i A-5 B}{512 a^3 c^6 f (i-\tan (e+f x))^2}-\frac {7 (2 A+i B)}{256 a^3 c^6 f (i-\tan (e+f x))}+\frac {i A+B}{96 a^3 c^6 f (i+\tan (e+f x))^6}+\frac {2 A-i B}{80 a^3 c^6 f (i+\tan (e+f x))^5}-\frac {5 i A+B}{128 a^3 c^6 f (i+\tan (e+f x))^4}-\frac {5 A}{96 a^3 c^6 f (i+\tan (e+f x))^3}+\frac {5 (7 i A-B)}{512 a^3 c^6 f (i+\tan (e+f x))^2}+\frac {7 (4 A+i B)}{256 a^3 c^6 f (i+\tan (e+f x))} \]

[Out]

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+tan(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))

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {3669, 78, 209} \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=-\frac {7 (2 A+i B)}{256 a^3 c^6 f (-\tan (e+f x)+i)}+\frac {7 (4 A+i B)}{256 a^3 c^6 f (\tan (e+f x)+i)}-\frac {-5 B+7 i A}{512 a^3 c^6 f (-\tan (e+f x)+i)^2}+\frac {5 (-B+7 i A)}{512 a^3 c^6 f (\tan (e+f x)+i)^2}+\frac {A+i B}{384 a^3 c^6 f (-\tan (e+f x)+i)^3}-\frac {B+5 i A}{128 a^3 c^6 f (\tan (e+f x)+i)^4}+\frac {2 A-i B}{80 a^3 c^6 f (\tan (e+f x)+i)^5}+\frac {B+i A}{96 a^3 c^6 f (\tan (e+f x)+i)^6}+\frac {7 x (3 A+i B)}{128 a^3 c^6}-\frac {5 A}{96 a^3 c^6 f (\tan (e+f x)+i)^3} \]

[In]

Int[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^6),x]

[Out]

(7*(3*A + I*B)*x)/(128*a^3*c^6) + (A + I*B)/(384*a^3*c^6*f*(I - Tan[e + f*x])^3) - ((7*I)*A - 5*B)/(512*a^3*c^
6*f*(I - Tan[e + f*x])^2) - (7*(2*A + I*B))/(256*a^3*c^6*f*(I - Tan[e + f*x])) + (I*A + B)/(96*a^3*c^6*f*(I +
Tan[e + f*x])^6) + (2*A - I*B)/(80*a^3*c^6*f*(I + Tan[e + f*x])^5) - ((5*I)*A + B)/(128*a^3*c^6*f*(I + Tan[e +
f*x])^4) - (5*A)/(96*a^3*c^6*f*(I + Tan[e + f*x])^3) + (5*((7*I)*A - B))/(512*a^3*c^6*f*(I + Tan[e + f*x])^2)
+ (7*(4*A + I*B))/(256*a^3*c^6*f*(I + Tan[e + f*x]))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((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]))))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 3669

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^4 (c-i c x)^7} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (\frac {A+i B}{128 a^4 c^7 (-i+x)^4}+\frac {i (7 A+5 i B)}{256 a^4 c^7 (-i+x)^3}-\frac {7 (2 A+i B)}{256 a^4 c^7 (-i+x)^2}-\frac {i (A-i B)}{16 a^4 c^7 (i+x)^7}+\frac {-2 A+i B}{16 a^4 c^7 (i+x)^6}+\frac {5 i A+B}{32 a^4 c^7 (i+x)^5}+\frac {5 A}{32 a^4 c^7 (i+x)^4}+\frac {5 (-7 i A+B)}{256 a^4 c^7 (i+x)^3}-\frac {7 (4 A+i B)}{256 a^4 c^7 (i+x)^2}+\frac {7 (3 A+i B)}{128 a^4 c^7 \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \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))}+\frac {(7 (3 A+i B)) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{128 a^3 c^6 f} \\ & = \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))} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.20 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.87 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=\frac {\sec ^8(e+f x) (-1271 A+43 i B-8 (391 A+37 i B) \cos (2 (e+f x))+(734 A+618 i B) \cos (4 (e+f x))+76 A \cos (6 (e+f x))+132 i B \cos (6 (e+f x))+5 A \cos (8 (e+f x))+15 i B \cos (8 (e+f x))+1890 i A \sin (2 (e+f x))-630 B \sin (2 (e+f x))+840 (-3 i A+B) \arctan (\tan (e+f x)) \sec (e+f x) (\cos (3 (e+f x))-i \sin (3 (e+f x)))-1176 i A \sin (4 (e+f x))+392 B \sin (4 (e+f x))-174 i A \sin (6 (e+f x))+58 B \sin (6 (e+f x))-15 i A \sin (8 (e+f x))+5 B \sin (8 (e+f x)))}{15360 a^3 c^6 f (-i+\tan (e+f x))^3 (i+\tan (e+f x))^6} \]

[In]

Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^6),x]

[Out]

(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)*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)

Maple [A] (verified)

Time = 0.48 (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	\(-\frac {9 i \sin \left (6 f x +6 e \right ) B}{1024 a^{3} c^{6} f}+\frac {21 x A}{128 a^{3} c^{6}}-\frac {{\mathrm e}^{12 i \left (f x +e \right )} B}{6144 a^{3} c^{6} f}-\frac {9 i {\mathrm e}^{10 i \left (f x +e \right )} A}{5120 a^{3} c^{6} f}-\frac {7 \,{\mathrm e}^{10 i \left (f x +e \right )} B}{5120 a^{3} c^{6} f}-\frac {117 i \cos \left (4 f x +4 e \right ) A}{2048 a^{3} c^{6} f}-\frac {5 \,{\mathrm e}^{8 i \left (f x +e \right )} B}{1024 a^{3} c^{6} f}+\frac {17 i \sin \left (2 f x +2 e \right ) B}{512 a^{3} c^{6} f}-\frac {29 \cos \left (6 f x +6 e \right ) B}{3072 a^{3} c^{6} f}-\frac {i {\mathrm e}^{12 i \left (f x +e \right )} A}{6144 a^{3} c^{6} f}-\frac {9 i {\mathrm e}^{8 i \left (f x +e \right )} A}{1024 a^{3} c^{6} f}+\frac {85 \sin \left (6 f x +6 e \right ) A}{3072 a^{3} c^{6} f}-\frac {21 \cos \left (4 f x +4 e \right ) B}{2048 a^{3} c^{6} f}+\frac {7 i x B}{128 a^{3} c^{6}}-\frac {83 i \cos \left (6 f x +6 e \right ) A}{3072 a^{3} c^{6} f}+\frac {135 \sin \left (4 f x +4 e \right ) A}{2048 a^{3} c^{6} f}-\frac {3 \cos \left (2 f x +2 e \right ) B}{512 a^{3} c^{6} f}-\frac {45 i \cos \left (2 f x +2 e \right ) A}{512 a^{3} c^{6} f}-\frac {7 i \sin \left (4 f x +4 e \right ) B}{2048 a^{3} c^{6} f}+\frac {81 \sin \left (2 f x +2 e \right ) A}{512 a^{3} c^{6} f}\)	\(410\)
derivativedivides	\(-\frac {A}{384 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {7 A}{128 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )}+\frac {5 B}{512 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {7 A}{64 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )}+\frac {B}{96 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{6}}-\frac {5 B}{512 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {B}{128 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {21 A \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{6}}+\frac {A}{40 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {5 A}{96 a^{3} c^{6} f \left (i+\tan \left (f x +e \right )\right )^{3}}-\frac {i B}{384 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {7 i B \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{6}}+\frac {35 i A}{512 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B}{80 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {7 i B}{256 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )}-\frac {7 i A}{512 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {i A}{96 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{6}}-\frac {5 i A}{128 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {7 i B}{256 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )}\)	\(440\)
default	\(-\frac {A}{384 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {7 A}{128 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )}+\frac {5 B}{512 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {7 A}{64 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )}+\frac {B}{96 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{6}}-\frac {5 B}{512 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {B}{128 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {21 A \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{6}}+\frac {A}{40 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {5 A}{96 a^{3} c^{6} f \left (i+\tan \left (f x +e \right )\right )^{3}}-\frac {i B}{384 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{3}}+\frac {7 i B \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{6}}+\frac {35 i A}{512 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B}{80 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {7 i B}{256 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )}-\frac {7 i A}{512 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {i A}{96 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{6}}-\frac {5 i A}{128 f \,a^{3} c^{6} \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {7 i B}{256 f \,a^{3} c^{6} \left (-i+\tan \left (f x +e \right )\right )}\)	\(440\)

[In]

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=_RETURNVERBOSE)

[Out]

(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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.58 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=\frac {{\left (1680 \, {\left (3 \, A + i \, B\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 5 \, {\left (i \, A + B\right )} e^{\left (18 i \, f x + 18 i \, e\right )} - 6 \, {\left (9 i \, A + 7 \, B\right )} e^{\left (16 i \, f x + 16 i \, e\right )} - 30 \, {\left (9 i \, A + 5 \, B\right )} e^{\left (14 i \, f x + 14 i \, e\right )} - 280 \, {\left (3 i \, A + B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} - 210 \, {\left (9 i \, A + B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} - 420 \, {\left (9 i \, A - B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} - 120 \, {\left (-9 i \, A + 5 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 15 \, {\left (-9 i \, A + 7 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 10 i \, A - 10 \, B\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{30720 \, a^{3} c^{6} f} \]

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^6,x, algorithm="fricas")

[Out]

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)

Sympy [A] (verification not implemented)

Time = 0.84 (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=\begin {cases} \frac {\left (\left (6800207735332289107722240 i A a^{24} c^{48} f^{8} e^{6 i e} - 6800207735332289107722240 B a^{24} c^{48} f^{8} e^{6 i e}\right ) e^{- 6 i f x} + \left (91802804426985902954250240 i A a^{24} c^{48} f^{8} e^{8 i e} - 71402181220989035631083520 B a^{24} c^{48} f^{8} e^{8 i e}\right ) e^{- 4 i f x} + \left (734422435415887223634001920 i A a^{24} c^{48} f^{8} e^{10 i e} - 408012464119937346463334400 B a^{24} c^{48} f^{8} e^{10 i e}\right ) e^{- 2 i f x} + \left (- 2570478523955605282719006720 i A a^{24} c^{48} f^{8} e^{14 i e} + 285608724883956142524334080 B a^{24} c^{48} f^{8} e^{14 i e}\right ) e^{2 i f x} + \left (- 1285239261977802641359503360 i A a^{24} c^{48} f^{8} e^{16 i e} - 142804362441978071262167040 B a^{24} c^{48} f^{8} e^{16 i e}\right ) e^{4 i f x} + \left (- 571217449767912285048668160 i A a^{24} c^{48} f^{8} e^{18 i e} - 190405816589304095016222720 B a^{24} c^{48} f^{8} e^{18 i e}\right ) e^{6 i f x} + \left (- 183605608853971805908500480 i A a^{24} c^{48} f^{8} e^{20 i e} - 102003116029984336615833600 B a^{24} c^{48} f^{8} e^{20 i e}\right ) e^{8 i f x} + \left (- 36721121770794361181700096 i A a^{24} c^{48} f^{8} e^{22 i e} - 28560872488395614252433408 B a^{24} c^{48} f^{8} e^{22 i e}\right ) e^{10 i f x} + \left (- 3400103867666144553861120 i A a^{24} c^{48} f^{8} e^{24 i e} - 3400103867666144553861120 B a^{24} c^{48} f^{8} e^{24 i e}\right ) e^{12 i f x}\right ) e^{- 12 i e}}{20890238162940792138922721280 a^{27} c^{54} f^{9}} & \text {for}\: a^{27} c^{54} f^{9} e^{12 i e} \neq 0 \\x \left (- \frac {21 A + 7 i B}{128 a^{3} c^{6}} + \frac {\left (A e^{18 i e} + 9 A e^{16 i e} + 36 A e^{14 i e} + 84 A e^{12 i e} + 126 A e^{10 i e} + 126 A e^{8 i e} + 84 A e^{6 i e} + 36 A e^{4 i e} + 9 A e^{2 i e} + A - i B e^{18 i e} - 7 i B e^{16 i e} - 20 i B e^{14 i e} - 28 i B e^{12 i e} - 14 i B e^{10 i e} + 14 i B e^{8 i e} + 28 i B e^{6 i e} + 20 i B e^{4 i e} + 7 i B e^{2 i e} + i B\right ) e^{- 6 i e}}{512 a^{3} c^{6}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (21 A + 7 i B\right )}{128 a^{3} c^{6}} \]

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))**3/(c-I*c*tan(f*x+e))**6,x)

[Out]

Piecewise((((6800207735332289107722240*I*A*a**24*c**48*f**8*exp(6*I*e) - 6800207735332289107722240*B*a**24*c**
48*f**8*exp(6*I*e))*exp(-6*I*f*x) + (91802804426985902954250240*I*A*a**24*c**48*f**8*exp(8*I*e) - 714021812209
89035631083520*B*a**24*c**48*f**8*exp(8*I*e))*exp(-4*I*f*x) + (734422435415887223634001920*I*A*a**24*c**48*f**
8*exp(10*I*e) - 408012464119937346463334400*B*a**24*c**48*f**8*exp(10*I*e))*exp(-2*I*f*x) + (-2570478523955605
282719006720*I*A*a**24*c**48*f**8*exp(14*I*e) + 285608724883956142524334080*B*a**24*c**48*f**8*exp(14*I*e))*ex
p(2*I*f*x) + (-1285239261977802641359503360*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) - 1
90405816589304095016222720*B*a**24*c**48*f**8*exp(18*I*e))*exp(6*I*f*x) + (-183605608853971805908500480*I*A*a*
*24*c**48*f**8*exp(20*I*e) - 102003116029984336615833600*B*a**24*c**48*f**8*exp(20*I*e))*exp(8*I*f*x) + (-3672
1121770794361181700096*I*A*a**24*c**48*f**8*exp(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) - 340010386766614455386112
0*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(18*I*e) + 9*A*exp(16*I*e)
+ 36*A*exp(14*I*e) + 84*A*exp(12*I*e) + 126*A*exp(10*I*e) + 126*A*exp(8*I*e) + 84*A*exp(6*I*e) + 36*A*exp(4*I
*e) + 9*A*exp(2*I*e) + A - I*B*exp(18*I*e) - 7*I*B*exp(16*I*e) - 20*I*B*exp(14*I*e) - 28*I*B*exp(12*I*e) - 14*
I*B*exp(10*I*e) + 14*I*B*exp(8*I*e) + 28*I*B*exp(6*I*e) + 20*I*B*exp(4*I*e) + 7*I*B*exp(2*I*e) + I*B)*exp(-6*I
*e)/(512*a**3*c**6)), True)) + x*(21*A + 7*I*B)/(128*a**3*c**6)

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^6,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.95 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.92 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=\frac {\frac {420 \, {\left (3 i \, A - B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{3} c^{6}} - \frac {420 \, {\left (3 i \, A - B\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c^{6}} - \frac {10 \, {\left (231 \, A \tan \left (f x + e\right )^{3} + 77 i \, B \tan \left (f x + e\right )^{3} - 777 i \, A \tan \left (f x + e\right )^{2} + 273 \, B \tan \left (f x + e\right )^{2} - 882 \, A \tan \left (f x + e\right ) - 330 i \, B \tan \left (f x + e\right ) + 340 i \, A - 138 \, B\right )}}{a^{3} c^{6} {\left (-i \, \tan \left (f x + e\right ) - 1\right )}^{3}} + \frac {-3087 i \, A \tan \left (f x + e\right )^{6} + 1029 \, B \tan \left (f x + e\right )^{6} + 20202 \, A \tan \left (f x + e\right )^{5} + 6594 i \, B \tan \left (f x + e\right )^{5} + 55755 i \, A \tan \left (f x + e\right )^{4} - 17685 \, B \tan \left (f x + e\right )^{4} - 83540 \, A \tan \left (f x + e\right )^{3} - 25380 i \, B \tan \left (f x + e\right )^{3} - 72405 i \, A \tan \left (f x + e\right )^{2} + 20415 \, B \tan \left (f x + e\right )^{2} + 35106 \, A \tan \left (f x + e\right ) + 8442 i \, B \tan \left (f x + e\right ) + 7761 i \, A - 1127 \, B}{a^{3} c^{6} {\left (\tan \left (f x + e\right ) + i\right )}^{6}}}{15360 \, f} \]

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^6,x, algorithm="giac")

[Out]

1/15360*(420*(3*I*A - B)*log(tan(f*x + e) + I)/(a^3*c^6) - 420*(3*I*A - B)*log(tan(f*x + e) - I)/(a^3*c^6) - 1
0*(231*A*tan(f*x + e)^3 + 77*I*B*tan(f*x + e)^3 - 777*I*A*tan(f*x + e)^2 + 273*B*tan(f*x + e)^2 - 882*A*tan(f*
x + e) - 330*I*B*tan(f*x + e) + 340*I*A - 138*B)/(a^3*c^6*(-I*tan(f*x + e) - 1)^3) + (-3087*I*A*tan(f*x + e)^6
+ 1029*B*tan(f*x + e)^6 + 20202*A*tan(f*x + e)^5 + 6594*I*B*tan(f*x + e)^5 + 55755*I*A*tan(f*x + e)^4 - 17685
*B*tan(f*x + e)^4 - 83540*A*tan(f*x + e)^3 - 25380*I*B*tan(f*x + e)^3 - 72405*I*A*tan(f*x + e)^2 + 20415*B*tan
(f*x + e)^2 + 35106*A*tan(f*x + e) + 8442*I*B*tan(f*x + e) + 7761*I*A - 1127*B)/(a^3*c^6*(tan(f*x + e) + I)^6)
)/f

Mupad [B] (verification not implemented)

Time = 10.58 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.10 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^6} \, dx=\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (-\frac {29\,B}{640\,a^3\,c^6}+\frac {A\,87{}\mathrm {i}}{640\,a^3\,c^6}\right )-{\mathrm {tan}\left (e+f\,x\right )}^8\,\left (\frac {21\,A}{128\,a^3\,c^6}+\frac {B\,7{}\mathrm {i}}{128\,a^3\,c^6}\right )-{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (-\frac {21\,B}{128\,a^3\,c^6}+\frac {A\,63{}\mathrm {i}}{128\,a^3\,c^6}\right )+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {129\,A}{128\,a^3\,c^6}+\frac {B\,43{}\mathrm {i}}{128\,a^3\,c^6}\right )-{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (-\frac {49\,B}{128\,a^3\,c^6}+\frac {A\,147{}\mathrm {i}}{128\,a^3\,c^6}\right )+{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (\frac {7\,A}{128\,a^3\,c^6}+\frac {B\,7{}\mathrm {i}}{384\,a^3\,c^6}\right )+{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (\frac {609\,A}{640\,a^3\,c^6}+\frac {B\,203{}\mathrm {i}}{640\,a^3\,c^6}\right )-{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (-\frac {413\,B}{1920\,a^3\,c^6}+\frac {A\,413{}\mathrm {i}}{640\,a^3\,c^6}\right )+\frac {7\,A}{30\,a^3\,c^6}-\frac {B\,1{}\mathrm {i}}{30\,a^3\,c^6}}{f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^9-{\mathrm {tan}\left (e+f\,x\right )}^8\,3{}\mathrm {i}-{\mathrm {tan}\left (e+f\,x\right )}^6\,8{}\mathrm {i}+6\,{\mathrm {tan}\left (e+f\,x\right )}^5-{\mathrm {tan}\left (e+f\,x\right )}^4\,6{}\mathrm {i}+8\,{\mathrm {tan}\left (e+f\,x\right )}^3+3\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}+\frac {7\,x\,\left (3\,A+B\,1{}\mathrm {i}\right )}{128\,a^3\,c^6} \]

[In]

int((A + B*tan(e + f*x))/((a + a*tan(e + f*x)*1i)^3*(c - c*tan(e + f*x)*1i)^6),x)

[Out]

(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)