3.738 \(\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^5} \, dx\)

Optimal. Leaf size=287 \[ -\frac{3 (7 A+3 i B)}{256 a^3 c^5 f (-\tan (e+f x)+i)}+\frac{5 (7 A+i B)}{256 a^3 c^5 f (\tan (e+f x)+i)}-\frac{-2 B+3 i A}{128 a^3 c^5 f (-\tan (e+f x)+i)^2}+\frac{A+i B}{192 a^3 c^5 f (-\tan (e+f x)+i)^3}-\frac{5 A-i B}{96 a^3 c^5 f (\tan (e+f x)+i)^3}-\frac{B+2 i A}{64 a^3 c^5 f (\tan (e+f x)+i)^4}+\frac{A-i B}{80 a^3 c^5 f (\tan (e+f x)+i)^5}+\frac{7 x (4 A+i B)}{128 a^3 c^5}+\frac{5 i A}{64 a^3 c^5 f (\tan (e+f x)+i)^2} \]

[Out]

(7*(4*A + I*B)*x)/(128*a^3*c^5) + (A + I*B)/(192*a^3*c^5*f*(I - Tan[e + f*x])^3) - ((3*I)*A - 2*B)/(128*a^3*c^
5*f*(I - Tan[e + f*x])^2) - (3*(7*A + (3*I)*B))/(256*a^3*c^5*f*(I - Tan[e + f*x])) + (A - I*B)/(80*a^3*c^5*f*(
I + Tan[e + f*x])^5) - ((2*I)*A + B)/(64*a^3*c^5*f*(I + Tan[e + f*x])^4) - (5*A - I*B)/(96*a^3*c^5*f*(I + Tan[
e + f*x])^3) + (((5*I)/64)*A)/(a^3*c^5*f*(I + Tan[e + f*x])^2) + (5*(7*A + I*B))/(256*a^3*c^5*f*(I + Tan[e + f
*x]))

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Rubi [A]  time = 0.337937, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.073, Rules used = {3588, 77, 203} \[ -\frac{3 (7 A+3 i B)}{256 a^3 c^5 f (-\tan (e+f x)+i)}+\frac{5 (7 A+i B)}{256 a^3 c^5 f (\tan (e+f x)+i)}-\frac{-2 B+3 i A}{128 a^3 c^5 f (-\tan (e+f x)+i)^2}+\frac{A+i B}{192 a^3 c^5 f (-\tan (e+f x)+i)^3}-\frac{5 A-i B}{96 a^3 c^5 f (\tan (e+f x)+i)^3}-\frac{B+2 i A}{64 a^3 c^5 f (\tan (e+f x)+i)^4}+\frac{A-i B}{80 a^3 c^5 f (\tan (e+f x)+i)^5}+\frac{7 x (4 A+i B)}{128 a^3 c^5}+\frac{5 i A}{64 a^3 c^5 f (\tan (e+f x)+i)^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^5),x]

[Out]

(7*(4*A + I*B)*x)/(128*a^3*c^5) + (A + I*B)/(192*a^3*c^5*f*(I - Tan[e + f*x])^3) - ((3*I)*A - 2*B)/(128*a^3*c^
5*f*(I - Tan[e + f*x])^2) - (3*(7*A + (3*I)*B))/(256*a^3*c^5*f*(I - Tan[e + f*x])) + (A - I*B)/(80*a^3*c^5*f*(
I + Tan[e + f*x])^5) - ((2*I)*A + B)/(64*a^3*c^5*f*(I + Tan[e + f*x])^4) - (5*A - I*B)/(96*a^3*c^5*f*(I + Tan[
e + f*x])^3) + (((5*I)/64)*A)/(a^3*c^5*f*(I + Tan[e + f*x])^2) + (5*(7*A + I*B))/(256*a^3*c^5*f*(I + Tan[e + f
*x]))

Rule 3588

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]

Rule 77

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 203

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

Rubi steps

\begin{align*} \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{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^4 (c-i c x)^6} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{A+i B}{64 a^4 c^6 (-i+x)^4}+\frac{i (3 A+2 i B)}{64 a^4 c^6 (-i+x)^3}-\frac{3 (7 A+3 i B)}{256 a^4 c^6 (-i+x)^2}+\frac{-A+i B}{16 a^4 c^6 (i+x)^6}+\frac{2 i A+B}{16 a^4 c^6 (i+x)^5}+\frac{5 A-i B}{32 a^4 c^6 (i+x)^4}-\frac{5 i A}{32 a^4 c^6 (i+x)^3}-\frac{5 (7 A+i B)}{256 a^4 c^6 (i+x)^2}+\frac{7 (4 A+i B)}{128 a^4 c^6 \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\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))}+\frac{(7 (4 A+i B)) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{128 a^3 c^5 f}\\ &=\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))}\\ \end{align*}

Mathematica [A]  time = 4.36775, size = 280, normalized size = 0.98 \[ \frac{\sec ^3(e+f x) (\cos (5 (e+f x))+i \sin (5 (e+f x))) (210 (4 A (1+4 i f x)-B (4 f x+i)) \cos (2 (e+f x))-560 (A+i B) \cos (4 (e+f x))+3360 A f x \sin (2 (e+f x))+840 i A \sin (2 (e+f x))+1120 i A \sin (4 (e+f x))+180 i A \sin (6 (e+f x))+16 i A \sin (8 (e+f x))-60 A \cos (6 (e+f x))-4 A \cos (8 (e+f x))+2100 A+210 B \sin (2 (e+f x))+840 i B f x \sin (2 (e+f x))-280 B \sin (4 (e+f x))-45 B \sin (6 (e+f x))-4 B \sin (8 (e+f x))-135 i B \cos (6 (e+f x))-16 i B \cos (8 (e+f x)))}{15360 a^3 c^5 f (\tan (e+f x)-i)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^5),x]

[Out]

(Sec[e + f*x]^3*(Cos[5*(e + f*x)] + I*Sin[5*(e + f*x)])*(2100*A + 210*(4*A*(1 + (4*I)*f*x) - B*(I + 4*f*x))*Co
s[2*(e + f*x)] - 560*(A + I*B)*Cos[4*(e + f*x)] - 60*A*Cos[6*(e + f*x)] - (135*I)*B*Cos[6*(e + f*x)] - 4*A*Cos
[8*(e + f*x)] - (16*I)*B*Cos[8*(e + f*x)] + (840*I)*A*Sin[2*(e + f*x)] + 210*B*Sin[2*(e + f*x)] + 3360*A*f*x*S
in[2*(e + f*x)] + (840*I)*B*f*x*Sin[2*(e + f*x)] + (1120*I)*A*Sin[4*(e + f*x)] - 280*B*Sin[4*(e + f*x)] + (180
*I)*A*Sin[6*(e + f*x)] - 45*B*Sin[6*(e + f*x)] + (16*I)*A*Sin[8*(e + f*x)] - 4*B*Sin[8*(e + f*x)]))/(15360*a^3
*c^5*f*(-I + Tan[e + f*x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.078, size = 445, normalized size = 1.6 \begin{align*} -{\frac{A}{192\,f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}-{\frac{{\frac{7\,i}{64}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) A}{f{a}^{3}{c}^{5}}}+{\frac{7\,\ln \left ( \tan \left ( fx+e \right ) -i \right ) B}{256\,f{a}^{3}{c}^{5}}}-{\frac{{\frac{i}{32}}A}{f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}+{\frac{21\,A}{256\,f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{{\frac{i}{192}}B}{f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}+{\frac{B}{64\,f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{{\frac{3\,i}{128}}A}{f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{{\frac{5\,i}{256}}B}{f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{A}{80\,f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{5}}}+{\frac{{\frac{i}{96}}B}{f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}}-{\frac{5\,A}{96\,f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}}+{\frac{{\frac{7\,i}{64}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) A}{f{a}^{3}{c}^{5}}}+{\frac{35\,A}{256\,f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{{\frac{9\,i}{256}}B}{f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{B}{64\,f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}+{\frac{{\frac{5\,i}{64}}A}{f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}-{\frac{7\,\ln \left ( \tan \left ( fx+e \right ) +i \right ) B}{256\,f{a}^{3}{c}^{5}}}-{\frac{{\frac{i}{80}}B}{f{a}^{3}{c}^{5} \left ( \tan \left ( fx+e \right ) +i \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x)

[Out]

-1/192/f/a^3/c^5/(tan(f*x+e)-I)^3*A-7/64*I/f/a^3/c^5*ln(tan(f*x+e)-I)*A+7/256/f/a^3/c^5*ln(tan(f*x+e)-I)*B-1/3
2*I/f/a^3/c^5/(tan(f*x+e)+I)^4*A+21/256/f/a^3/c^5/(tan(f*x+e)-I)*A-1/192*I/f/a^3/c^5/(tan(f*x+e)-I)^3*B+1/64/f
/a^3/c^5/(tan(f*x+e)-I)^2*B-3/128*I/f/a^3/c^5/(tan(f*x+e)-I)^2*A+5/256*I/f/a^3/c^5/(tan(f*x+e)+I)*B+1/80/f/a^3
/c^5/(tan(f*x+e)+I)^5*A+1/96*I/f/a^3/c^5/(tan(f*x+e)+I)^3*B-5/96/f/a^3/c^5/(tan(f*x+e)+I)^3*A+7/64*I/f/a^3/c^5
*ln(tan(f*x+e)+I)*A+35/256/f/a^3/c^5/(tan(f*x+e)+I)*A+9/256*I/f/a^3/c^5/(tan(f*x+e)-I)*B-1/64/f/a^3/c^5/(tan(f
*x+e)+I)^4*B+5/64*I*A/a^3/c^5/f/(tan(f*x+e)+I)^2-7/256/f/a^3/c^5*ln(tan(f*x+e)+I)*B-1/80*I/f/a^3/c^5/(tan(f*x+
e)+I)^5*B

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.06409, size = 508, normalized size = 1.77 \begin{align*} \frac{{\left (840 \,{\left (4 \, A + i \, B\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-6 i \, A - 6 \, B\right )} e^{\left (16 i \, f x + 16 i \, e\right )} +{\left (-60 i \, A - 45 \, B\right )} e^{\left (14 i \, f x + 14 i \, e\right )} +{\left (-280 i \, A - 140 \, B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} +{\left (-840 i \, A - 210 \, 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 )} +{\left (840 i \, A - 420 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (120 i \, A - 90 \, 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x, algorithm="fricas")

[Out]

1/15360*(840*(4*A + I*B)*f*x*e^(6*I*f*x + 6*I*e) + (-6*I*A - 6*B)*e^(16*I*f*x + 16*I*e) + (-60*I*A - 45*B)*e^(
14*I*f*x + 14*I*e) + (-280*I*A - 140*B)*e^(12*I*f*x + 12*I*e) + (-840*I*A - 210*B)*e^(10*I*f*x + 10*I*e) - 210
0*I*A*e^(8*I*f*x + 8*I*e) + (840*I*A - 420*B)*e^(4*I*f*x + 4*I*e) + (120*I*A - 90*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)

________________________________________________________________________________________

Sympy [A]  time = 10.0078, size = 648, normalized size = 2.26 \begin{align*} \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}\: 53126622932283508654080 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))**3/(c-I*c*tan(f*x+e))**5,x)

[Out]

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) - 34587645138205409280*B*a**21*c**35*f**7*exp(6*I*e))*exp(-6*I*f*x) + (41505174165
8464911360*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(1
0*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) - 48422703
1934875729920*B*a**21*c**35*f**7*exp(18*I*e))*exp(6*I*f*x) + (-207525870829232455680*I*A*a**21*c**35*f**7*exp(
20*I*e) - 155644403121924341760*B*a**21*c**35*f**7*exp(20*I*e))*exp(8*I*f*x) + (-20752587082923245568*I*A*a**2
1*c**35*f**7*exp(22*I*e) - 20752587082923245568*B*a**21*c**35*f**7*exp(22*I*e))*exp(10*I*f*x))*exp(-12*I*e)/(5
3126622932283508654080*a**24*c**40*f**8), Ne(53126622932283508654080*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)

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Giac [A]  time = 1.43331, size = 393, normalized size = 1.37 \begin{align*} -\frac{\frac{60 \,{\left (-28 i \, A + 7 \, B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{3} c^{5}} + \frac{60 \,{\left (28 i \, A - 7 \, B\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c^{5}} + \frac{10 \,{\left (-308 i \, A \tan \left (f x + e\right )^{3} + 77 \, B \tan \left (f x + e\right )^{3} - 1050 \, A \tan \left (f x + e\right )^{2} - 285 i \, B \tan \left (f x + e\right )^{2} + 1212 i \, A \tan \left (f x + e\right ) - 363 \, B \tan \left (f x + e\right ) + 478 \, A + 163 i \, B\right )}}{a^{3} c^{5}{\left (\tan \left (f x + e\right ) - i\right )}^{3}} + \frac{3836 i \, A \tan \left (f x + e\right )^{5} - 959 \, B \tan \left (f x + e\right )^{5} - 21280 \, A \tan \left (f x + e\right )^{4} - 5095 i \, B \tan \left (f x + e\right )^{4} - 47960 i \, A \tan \left (f x + e\right )^{3} + 10790 \, B \tan \left (f x + e\right )^{3} + 55360 \, A \tan \left (f x + e\right )^{2} + 11230 i \, B \tan \left (f x + e\right )^{2} + 33260 i \, A \tan \left (f x + e\right ) - 5435 \, B \tan \left (f x + e\right ) - 8608 \, A - 667 i \, B}{a^{3} c^{5}{\left (\tan \left (f x + e\right ) + i\right )}^{5}}}{15360 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e))/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^5,x, algorithm="giac")

[Out]

-1/15360*(60*(-28*I*A + 7*B)*log(tan(f*x + e) + I)/(a^3*c^5) + 60*(28*I*A - 7*B)*log(tan(f*x + e) - I)/(a^3*c^
5) + 10*(-308*I*A*tan(f*x + e)^3 + 77*B*tan(f*x + e)^3 - 1050*A*tan(f*x + e)^2 - 285*I*B*tan(f*x + e)^2 + 1212
*I*A*tan(f*x + e) - 363*B*tan(f*x + e) + 478*A + 163*I*B)/(a^3*c^5*(tan(f*x + e) - I)^3) + (3836*I*A*tan(f*x +
e)^5 - 959*B*tan(f*x + e)^5 - 21280*A*tan(f*x + e)^4 - 5095*I*B*tan(f*x + e)^4 - 47960*I*A*tan(f*x + e)^3 + 1
0790*B*tan(f*x + e)^3 + 55360*A*tan(f*x + e)^2 + 11230*I*B*tan(f*x + e)^2 + 33260*I*A*tan(f*x + e) - 5435*B*ta
n(f*x + e) - 8608*A - 667*I*B)/(a^3*c^5*(tan(f*x + e) + I)^5))/f