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

Optimal. Leaf size=319 \[ -\frac{7 (2 A+i B)}{256 a^3 c^6 f (-\tan (e+f x)+i)}+\frac{7 (4 A+i B)}{256 a^3 c^6 f (\tan (e+f x)+i)}-\frac{-5 B+7 i A}{512 a^3 c^6 f (-\tan (e+f x)+i)^2}+\frac{5 (-B+7 i A)}{512 a^3 c^6 f (\tan (e+f x)+i)^2}+\frac{A+i B}{384 a^3 c^6 f (-\tan (e+f x)+i)^3}-\frac{B+5 i A}{128 a^3 c^6 f (\tan (e+f x)+i)^4}+\frac{2 A-i B}{80 a^3 c^6 f (\tan (e+f x)+i)^5}+\frac{B+i A}{96 a^3 c^6 f (\tan (e+f x)+i)^6}+\frac{7 x (3 A+i B)}{128 a^3 c^6}-\frac{5 A}{96 a^3 c^6 f (\tan (e+f x)+i)^3} \]

[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]))

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Rubi [A]  time = 0.380221, antiderivative size = 319, 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} \[ -\frac{7 (2 A+i B)}{256 a^3 c^6 f (-\tan (e+f x)+i)}+\frac{7 (4 A+i B)}{256 a^3 c^6 f (\tan (e+f x)+i)}-\frac{-5 B+7 i A}{512 a^3 c^6 f (-\tan (e+f x)+i)^2}+\frac{5 (-B+7 i A)}{512 a^3 c^6 f (\tan (e+f x)+i)^2}+\frac{A+i B}{384 a^3 c^6 f (-\tan (e+f x)+i)^3}-\frac{B+5 i A}{128 a^3 c^6 f (\tan (e+f x)+i)^4}+\frac{2 A-i B}{80 a^3 c^6 f (\tan (e+f x)+i)^5}+\frac{B+i A}{96 a^3 c^6 f (\tan (e+f x)+i)^6}+\frac{7 x (3 A+i B)}{128 a^3 c^6}-\frac{5 A}{96 a^3 c^6 f (\tan (e+f x)+i)^3} \]

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])^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 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))^6} \, dx &=\frac{(a c) \operatorname{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) \operatorname{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)) \operatorname{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]  time = 5.02441, size = 321, normalized size = 1.01 \[ \frac{\sec ^3(e+f x) (-\cos (6 (e+f x))-i \sin (6 (e+f x))) (-210 (27 A+i B) \cos (e+f x)+280 (-18 i A f x-3 A+6 B f x+i B) \cos (3 (e+f x))+1890 i A \sin (e+f x)-840 i A \sin (3 (e+f x))-5040 A f x \sin (3 (e+f x))-1350 i A \sin (5 (e+f x))-189 i A \sin (7 (e+f x))-15 i A \sin (9 (e+f x))+810 A \cos (5 (e+f x))+81 A \cos (7 (e+f x))+5 A \cos (9 (e+f x))-630 B \sin (e+f x)-280 B \sin (3 (e+f x))-1680 i B f x \sin (3 (e+f x))+450 B \sin (5 (e+f x))+63 B \sin (7 (e+f x))+5 B \sin (9 (e+f x))+750 i B \cos (5 (e+f x))+147 i B \cos (7 (e+f x))+15 i B \cos (9 (e+f x)))}{30720 a^3 c^6 f (\tan (e+f x)-i)^3} \]

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])^6),x]

[Out]

(Sec[e + f*x]^3*(-Cos[6*(e + f*x)] - I*Sin[6*(e + f*x)])*(-210*(27*A + I*B)*Cos[e + f*x] + 280*(-3*A + I*B - (
18*I)*A*f*x + 6*B*f*x)*Cos[3*(e + f*x)] + 810*A*Cos[5*(e + f*x)] + (750*I)*B*Cos[5*(e + f*x)] + 81*A*Cos[7*(e
+ f*x)] + (147*I)*B*Cos[7*(e + f*x)] + 5*A*Cos[9*(e + f*x)] + (15*I)*B*Cos[9*(e + f*x)] + (1890*I)*A*Sin[e + f
*x] - 630*B*Sin[e + f*x] - (840*I)*A*Sin[3*(e + f*x)] - 280*B*Sin[3*(e + f*x)] - 5040*A*f*x*Sin[3*(e + f*x)] -
(1680*I)*B*f*x*Sin[3*(e + f*x)] - (1350*I)*A*Sin[5*(e + f*x)] + 450*B*Sin[5*(e + f*x)] - (189*I)*A*Sin[7*(e +
f*x)] + 63*B*Sin[7*(e + f*x)] - (15*I)*A*Sin[9*(e + f*x)] + 5*B*Sin[9*(e + f*x)]))/(30720*a^3*c^6*f*(-I + Tan
[e + f*x])^3)

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Maple [A]  time = 0.078, size = 491, normalized size = 1.5 \begin{align*}{\frac{-{\frac{21\,i}{256}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) A}{f{a}^{3}{c}^{6}}}+{\frac{7\,A}{128\,f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{{\frac{i}{80}}B}{f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) +i \right ) ^{5}}}+{\frac{7\,\ln \left ( \tan \left ( fx+e \right ) -i \right ) B}{256\,f{a}^{3}{c}^{6}}}+{\frac{5\,B}{512\,f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{{\frac{5\,i}{128}}A}{f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}-{\frac{A}{384\,f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}-{\frac{{\frac{i}{384}}B}{f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}+{\frac{{\frac{35\,i}{512}}A}{f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}-{\frac{5\,B}{512\,f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}-{\frac{{\frac{7\,i}{512}}A}{f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{A}{40\,f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) +i \right ) ^{5}}}+{\frac{{\frac{21\,i}{256}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) A}{f{a}^{3}{c}^{6}}}-{\frac{B}{128\,f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}+{\frac{7\,A}{64\,f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{{\frac{7\,i}{256}}B}{f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{5\,A}{96\,f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}}+{\frac{{\frac{i}{96}}A}{f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) +i \right ) ^{6}}}-{\frac{7\,\ln \left ( \tan \left ( fx+e \right ) +i \right ) B}{256\,f{a}^{3}{c}^{6}}}+{\frac{{\frac{7\,i}{256}}B}{f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{B}{96\,f{a}^{3}{c}^{6} \left ( \tan \left ( fx+e \right ) +i \right ) ^{6}}} \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))^6,x)

[Out]

-21/256*I/f/a^3/c^6*ln(tan(f*x+e)-I)*A+7/128/f/a^3/c^6/(tan(f*x+e)-I)*A-1/80*I/f/a^3/c^6/(tan(f*x+e)+I)^5*B+7/
256/f/a^3/c^6*ln(tan(f*x+e)-I)*B+5/512/f/a^3/c^6/(tan(f*x+e)-I)^2*B-5/128*I/f/a^3/c^6/(tan(f*x+e)+I)^4*A-1/384
/f/a^3/c^6/(tan(f*x+e)-I)^3*A-1/384*I/f/a^3/c^6/(tan(f*x+e)-I)^3*B+35/512*I/f/a^3/c^6/(tan(f*x+e)+I)^2*A-5/512
/f/a^3/c^6/(tan(f*x+e)+I)^2*B-7/512*I/f/a^3/c^6/(tan(f*x+e)-I)^2*A+1/40/f/a^3/c^6/(tan(f*x+e)+I)^5*A+21/256*I/
f/a^3/c^6*ln(tan(f*x+e)+I)*A-1/128/f/a^3/c^6/(tan(f*x+e)+I)^4*B+7/64/f/a^3/c^6/(tan(f*x+e)+I)*A+7/256*I/f/a^3/
c^6/(tan(f*x+e)-I)*B-5/96*A/a^3/c^6/f/(tan(f*x+e)+I)^3+1/96*I/f/a^3/c^6/(tan(f*x+e)+I)^6*A-7/256/f/a^3/c^6*ln(
tan(f*x+e)+I)*B+7/256*I/f/a^3/c^6/(tan(f*x+e)+I)*B+1/96/f/a^3/c^6/(tan(f*x+e)+I)^6*B

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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))^6,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.07719, size = 586, normalized size = 1.84 \begin{align*} \frac{{\left (1680 \,{\left (3 \, A + i \, B\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-5 i \, A - 5 \, B\right )} e^{\left (18 i \, f x + 18 i \, e\right )} +{\left (-54 i \, A - 42 \, B\right )} e^{\left (16 i \, f x + 16 i \, e\right )} +{\left (-270 i \, A - 150 \, B\right )} e^{\left (14 i \, f x + 14 i \, e\right )} +{\left (-840 i \, A - 280 \, B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} +{\left (-1890 i \, A - 210 \, B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-3780 i \, A + 420 \, B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (1080 i \, A - 600 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (135 i \, A - 105 \, 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} \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))^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 - 5*B)*e^(18*I*f*x + 18*I*e) + (-54*I*A - 42*B)*e^
(16*I*f*x + 16*I*e) + (-270*I*A - 150*B)*e^(14*I*f*x + 14*I*e) + (-840*I*A - 280*B)*e^(12*I*f*x + 12*I*e) + (-
1890*I*A - 210*B)*e^(10*I*f*x + 10*I*e) + (-3780*I*A + 420*B)*e^(8*I*f*x + 8*I*e) + (1080*I*A - 600*B)*e^(4*I*
f*x + 4*I*e) + (135*I*A - 105*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)

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Sympy [A]  time = 10.4687, size = 755, normalized size = 2.37 \begin{align*} \text{result too large to display} \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))**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(20890238162940792138922721280*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)

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Giac [A]  time = 1.425, size = 431, normalized size = 1.35 \begin{align*} \frac{\frac{60 \,{\left (21 i \, A - 7 \, B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{3} c^{6}} - \frac{60 \,{\left (21 i \, A - 7 \, B\right )} \log \left (i \, \tan \left (f x + e\right ) + 1\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} \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))^6,x, algorithm="giac")

[Out]

1/15360*(60*(21*I*A - 7*B)*log(tan(f*x + e) + I)/(a^3*c^6) - 60*(21*I*A - 7*B)*log(I*tan(f*x + e) + 1)/(a^3*c^
6) - 10*(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