3.725 \(\int \frac{A+B \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^4} \, dx\)

Optimal. Leaf size=221 \[ -\frac{5 A+3 i B}{64 a^2 c^4 f (-\tan (e+f x)+i)}+\frac{5 A+i B}{32 a^2 c^4 f (\tan (e+f x)+i)}-\frac{-B+i A}{64 a^2 c^4 f (-\tan (e+f x)+i)^2}-\frac{3 A-i B}{48 a^2 c^4 f (\tan (e+f x)+i)^3}-\frac{B+i A}{32 a^2 c^4 f (\tan (e+f x)+i)^4}+\frac{5 x (3 A+i B)}{64 a^2 c^4}+\frac{3 i A}{32 a^2 c^4 f (\tan (e+f x)+i)^2} \]

[Out]

(5*(3*A + I*B)*x)/(64*a^2*c^4) - (I*A - B)/(64*a^2*c^4*f*(I - Tan[e + f*x])^2) - (5*A + (3*I)*B)/(64*a^2*c^4*f
*(I - Tan[e + f*x])) - (I*A + B)/(32*a^2*c^4*f*(I + Tan[e + f*x])^4) - (3*A - I*B)/(48*a^2*c^4*f*(I + Tan[e +
f*x])^3) + (((3*I)/32)*A)/(a^2*c^4*f*(I + Tan[e + f*x])^2) + (5*A + I*B)/(32*a^2*c^4*f*(I + Tan[e + f*x]))

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Rubi [A]  time = 0.26789, antiderivative size = 221, 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} \[ -\frac{5 A+3 i B}{64 a^2 c^4 f (-\tan (e+f x)+i)}+\frac{5 A+i B}{32 a^2 c^4 f (\tan (e+f x)+i)}-\frac{-B+i A}{64 a^2 c^4 f (-\tan (e+f x)+i)^2}-\frac{3 A-i B}{48 a^2 c^4 f (\tan (e+f x)+i)^3}-\frac{B+i A}{32 a^2 c^4 f (\tan (e+f x)+i)^4}+\frac{5 x (3 A+i B)}{64 a^2 c^4}+\frac{3 i A}{32 a^2 c^4 f (\tan (e+f x)+i)^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^4),x]

[Out]

(5*(3*A + I*B)*x)/(64*a^2*c^4) - (I*A - B)/(64*a^2*c^4*f*(I - Tan[e + f*x])^2) - (5*A + (3*I)*B)/(64*a^2*c^4*f
*(I - Tan[e + f*x])) - (I*A + B)/(32*a^2*c^4*f*(I + Tan[e + f*x])^4) - (3*A - I*B)/(48*a^2*c^4*f*(I + Tan[e +
f*x])^3) + (((3*I)/32)*A)/(a^2*c^4*f*(I + Tan[e + f*x])^2) + (5*A + I*B)/(32*a^2*c^4*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))^2 (c-i c \tan (e+f x))^4} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^3 (c-i c x)^5} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(a c) \operatorname{Subst}\left (\int \left (\frac{i (A+i B)}{32 a^3 c^5 (-i+x)^3}+\frac{-5 A-3 i B}{64 a^3 c^5 (-i+x)^2}+\frac{i A+B}{8 a^3 c^5 (i+x)^5}+\frac{3 A-i B}{16 a^3 c^5 (i+x)^4}-\frac{3 i A}{16 a^3 c^5 (i+x)^3}+\frac{-5 A-i B}{32 a^3 c^5 (i+x)^2}+\frac{5 (3 A+i B)}{64 a^3 c^5 \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{i A-B}{64 a^2 c^4 f (i-\tan (e+f x))^2}-\frac{5 A+3 i B}{64 a^2 c^4 f (i-\tan (e+f x))}-\frac{i A+B}{32 a^2 c^4 f (i+\tan (e+f x))^4}-\frac{3 A-i B}{48 a^2 c^4 f (i+\tan (e+f x))^3}+\frac{3 i A}{32 a^2 c^4 f (i+\tan (e+f x))^2}+\frac{5 A+i B}{32 a^2 c^4 f (i+\tan (e+f x))}+\frac{(5 (3 A+i B)) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{64 a^2 c^4 f}\\ &=\frac{5 (3 A+i B) x}{64 a^2 c^4}-\frac{i A-B}{64 a^2 c^4 f (i-\tan (e+f x))^2}-\frac{5 A+3 i B}{64 a^2 c^4 f (i-\tan (e+f x))}-\frac{i A+B}{32 a^2 c^4 f (i+\tan (e+f x))^4}-\frac{3 A-i B}{48 a^2 c^4 f (i+\tan (e+f x))^3}+\frac{3 i A}{32 a^2 c^4 f (i+\tan (e+f x))^2}+\frac{5 A+i B}{32 a^2 c^4 f (i+\tan (e+f x))}\\ \end{align*}

Mathematica [A]  time = 2.57616, size = 232, normalized size = 1.05 \[ \frac{\sec ^2(e+f x) (\sin (4 (e+f x))-i \cos (4 (e+f x))) (30 (A (-3-12 i f x)+B (4 f x+i)) \cos (2 (e+f x))+16 (3 A+4 i B) \cos (4 (e+f x))-360 A f x \sin (2 (e+f x))-90 i A \sin (2 (e+f x))-96 i A \sin (4 (e+f x))-9 i A \sin (6 (e+f x))+3 A \cos (6 (e+f x))-240 A-30 B \sin (2 (e+f x))-120 i B f x \sin (2 (e+f x))+32 B \sin (4 (e+f x))+3 B \sin (6 (e+f x))+9 i B \cos (6 (e+f x)))}{1536 a^2 c^4 f (\tan (e+f x)-i)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Tan[e + f*x])/((a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^4),x]

[Out]

(Sec[e + f*x]^2*((-I)*Cos[4*(e + f*x)] + Sin[4*(e + f*x)])*(-240*A + 30*(A*(-3 - (12*I)*f*x) + B*(I + 4*f*x))*
Cos[2*(e + f*x)] + 16*(3*A + (4*I)*B)*Cos[4*(e + f*x)] + 3*A*Cos[6*(e + f*x)] + (9*I)*B*Cos[6*(e + f*x)] - (90
*I)*A*Sin[2*(e + f*x)] - 30*B*Sin[2*(e + f*x)] - 360*A*f*x*Sin[2*(e + f*x)] - (120*I)*B*f*x*Sin[2*(e + f*x)] -
(96*I)*A*Sin[4*(e + f*x)] + 32*B*Sin[4*(e + f*x)] - (9*I)*A*Sin[6*(e + f*x)] + 3*B*Sin[6*(e + f*x)]))/(1536*a
^2*c^4*f*(-I + Tan[e + f*x])^2)

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Maple [A]  time = 0.073, size = 351, normalized size = 1.6 \begin{align*}{\frac{5\,A}{64\,f{a}^{2}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{{\frac{3\,i}{64}}B}{f{a}^{2}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) }}-{\frac{{\frac{i}{64}}A}{f{a}^{2}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{B}{64\,f{a}^{2}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}+{\frac{5\,\ln \left ( \tan \left ( fx+e \right ) -i \right ) B}{128\,f{a}^{2}{c}^{4}}}-{\frac{{\frac{15\,i}{128}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) A}{f{a}^{2}{c}^{4}}}-{\frac{{\frac{i}{32}}A}{f{a}^{2}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}-{\frac{B}{32\,f{a}^{2}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}+{\frac{5\,A}{32\,f{a}^{2}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) }}+{\frac{{\frac{i}{32}}B}{f{a}^{2}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) }}-{\frac{5\,\ln \left ( \tan \left ( fx+e \right ) +i \right ) B}{128\,f{a}^{2}{c}^{4}}}+{\frac{{\frac{15\,i}{128}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) A}{f{a}^{2}{c}^{4}}}-{\frac{A}{16\,f{a}^{2}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}}+{\frac{{\frac{i}{48}}B}{f{a}^{2}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}}+{\frac{{\frac{3\,i}{32}}A}{f{a}^{2}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}} \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))^2/(c-I*c*tan(f*x+e))^4,x)

[Out]

5/64/f/a^2/c^4/(tan(f*x+e)-I)*A+3/64*I/f/a^2/c^4/(tan(f*x+e)-I)*B-1/64*I/f/a^2/c^4/(tan(f*x+e)-I)^2*A+1/64/f/a
^2/c^4/(tan(f*x+e)-I)^2*B+5/128/f/a^2/c^4*ln(tan(f*x+e)-I)*B-15/128*I/f/a^2/c^4*ln(tan(f*x+e)-I)*A-1/32*I/f/a^
2/c^4/(tan(f*x+e)+I)^4*A-1/32/f/a^2/c^4/(tan(f*x+e)+I)^4*B+5/32/f/a^2/c^4/(tan(f*x+e)+I)*A+1/32*I/f/a^2/c^4/(t
an(f*x+e)+I)*B-5/128/f/a^2/c^4*ln(tan(f*x+e)+I)*B+15/128*I/f/a^2/c^4*ln(tan(f*x+e)+I)*A-1/16/f/a^2/c^4/(tan(f*
x+e)+I)^3*A+1/48*I/f/a^2/c^4/(tan(f*x+e)+I)^3*B+3/32*I*A/a^2/c^4/f/(tan(f*x+e)+I)^2

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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))^2/(c-I*c*tan(f*x+e))^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.08907, size = 383, normalized size = 1.73 \begin{align*} \frac{{\left (120 \,{\left (3 \, A + i \, B\right )} f x e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-3 i \, A - 3 \, B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} +{\left (-24 i \, A - 16 \, B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} +{\left (-90 i \, A - 30 \, B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} - 240 i \, A e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (72 i \, A - 48 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 6 i \, A - 6 \, B\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{1536 \, a^{2} c^{4} 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))^2/(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")

[Out]

1/1536*(120*(3*A + I*B)*f*x*e^(4*I*f*x + 4*I*e) + (-3*I*A - 3*B)*e^(12*I*f*x + 12*I*e) + (-24*I*A - 16*B)*e^(1
0*I*f*x + 10*I*e) + (-90*I*A - 30*B)*e^(8*I*f*x + 8*I*e) - 240*I*A*e^(6*I*f*x + 6*I*e) + (72*I*A - 48*B)*e^(2*
I*f*x + 2*I*e) + 6*I*A - 6*B)*e^(-4*I*f*x - 4*I*e)/(a^2*c^4*f)

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Sympy [A]  time = 5.56188, size = 500, normalized size = 2.26 \begin{align*} \begin{cases} \frac{\left (- 2061584302080 i A a^{10} c^{20} f^{5} e^{8 i e} e^{2 i f x} + \left (51539607552 i A a^{10} c^{20} f^{5} e^{2 i e} - 51539607552 B a^{10} c^{20} f^{5} e^{2 i e}\right ) e^{- 4 i f x} + \left (618475290624 i A a^{10} c^{20} f^{5} e^{4 i e} - 412316860416 B a^{10} c^{20} f^{5} e^{4 i e}\right ) e^{- 2 i f x} + \left (- 773094113280 i A a^{10} c^{20} f^{5} e^{10 i e} - 257698037760 B a^{10} c^{20} f^{5} e^{10 i e}\right ) e^{4 i f x} + \left (- 206158430208 i A a^{10} c^{20} f^{5} e^{12 i e} - 137438953472 B a^{10} c^{20} f^{5} e^{12 i e}\right ) e^{6 i f x} + \left (- 25769803776 i A a^{10} c^{20} f^{5} e^{14 i e} - 25769803776 B a^{10} c^{20} f^{5} e^{14 i e}\right ) e^{8 i f x}\right ) e^{- 6 i e}}{13194139533312 a^{12} c^{24} f^{6}} & \text{for}\: 13194139533312 a^{12} c^{24} f^{6} e^{6 i e} \neq 0 \\x \left (- \frac{15 A + 5 i B}{64 a^{2} c^{4}} + \frac{\left (A e^{12 i e} + 6 A e^{10 i e} + 15 A e^{8 i e} + 20 A e^{6 i e} + 15 A e^{4 i e} + 6 A e^{2 i e} + A - i B e^{12 i e} - 4 i B e^{10 i e} - 5 i B e^{8 i e} + 5 i B e^{4 i e} + 4 i B e^{2 i e} + i B\right ) e^{- 4 i e}}{64 a^{2} c^{4}}\right ) & \text{otherwise} \end{cases} + \frac{x \left (15 A + 5 i B\right )}{64 a^{2} c^{4}} \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))**2/(c-I*c*tan(f*x+e))**4,x)

[Out]

Piecewise(((-2061584302080*I*A*a**10*c**20*f**5*exp(8*I*e)*exp(2*I*f*x) + (51539607552*I*A*a**10*c**20*f**5*ex
p(2*I*e) - 51539607552*B*a**10*c**20*f**5*exp(2*I*e))*exp(-4*I*f*x) + (618475290624*I*A*a**10*c**20*f**5*exp(4
*I*e) - 412316860416*B*a**10*c**20*f**5*exp(4*I*e))*exp(-2*I*f*x) + (-773094113280*I*A*a**10*c**20*f**5*exp(10
*I*e) - 257698037760*B*a**10*c**20*f**5*exp(10*I*e))*exp(4*I*f*x) + (-206158430208*I*A*a**10*c**20*f**5*exp(12
*I*e) - 137438953472*B*a**10*c**20*f**5*exp(12*I*e))*exp(6*I*f*x) + (-25769803776*I*A*a**10*c**20*f**5*exp(14*
I*e) - 25769803776*B*a**10*c**20*f**5*exp(14*I*e))*exp(8*I*f*x))*exp(-6*I*e)/(13194139533312*a**12*c**24*f**6)
, Ne(13194139533312*a**12*c**24*f**6*exp(6*I*e), 0)), (x*(-(15*A + 5*I*B)/(64*a**2*c**4) + (A*exp(12*I*e) + 6*
A*exp(10*I*e) + 15*A*exp(8*I*e) + 20*A*exp(6*I*e) + 15*A*exp(4*I*e) + 6*A*exp(2*I*e) + A - I*B*exp(12*I*e) - 4
*I*B*exp(10*I*e) - 5*I*B*exp(8*I*e) + 5*I*B*exp(4*I*e) + 4*I*B*exp(2*I*e) + I*B)*exp(-4*I*e)/(64*a**2*c**4)),
True)) + x*(15*A + 5*I*B)/(64*a**2*c**4)

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Giac [A]  time = 1.23693, size = 328, normalized size = 1.48 \begin{align*} \frac{\frac{12 \,{\left (15 i \, A - 5 \, B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{2} c^{4}} + \frac{12 \,{\left (-15 i \, A + 5 \, B\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{2} c^{4}} - \frac{6 \,{\left (-45 i \, A \tan \left (f x + e\right )^{2} + 15 \, B \tan \left (f x + e\right )^{2} - 110 \, A \tan \left (f x + e\right ) - 42 i \, B \tan \left (f x + e\right ) + 69 i \, A - 31 \, B\right )}}{a^{2} c^{4}{\left (\tan \left (f x + e\right ) - i\right )}^{2}} + \frac{-375 i \, A \tan \left (f x + e\right )^{4} + 125 \, B \tan \left (f x + e\right )^{4} + 1740 \, A \tan \left (f x + e\right )^{3} + 548 i \, B \tan \left (f x + e\right )^{3} + 3114 i \, A \tan \left (f x + e\right )^{2} - 894 \, B \tan \left (f x + e\right )^{2} - 2604 \, A \tan \left (f x + e\right ) - 612 i \, B \tan \left (f x + e\right ) - 903 i \, A + 93 \, B}{a^{2} c^{4}{\left (\tan \left (f x + e\right ) + i\right )}^{4}}}{1536 \, 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))^2/(c-I*c*tan(f*x+e))^4,x, algorithm="giac")

[Out]

1/1536*(12*(15*I*A - 5*B)*log(tan(f*x + e) + I)/(a^2*c^4) + 12*(-15*I*A + 5*B)*log(tan(f*x + e) - I)/(a^2*c^4)
- 6*(-45*I*A*tan(f*x + e)^2 + 15*B*tan(f*x + e)^2 - 110*A*tan(f*x + e) - 42*I*B*tan(f*x + e) + 69*I*A - 31*B)
/(a^2*c^4*(tan(f*x + e) - I)^2) + (-375*I*A*tan(f*x + e)^4 + 125*B*tan(f*x + e)^4 + 1740*A*tan(f*x + e)^3 + 54
8*I*B*tan(f*x + e)^3 + 3114*I*A*tan(f*x + e)^2 - 894*B*tan(f*x + e)^2 - 2604*A*tan(f*x + e) - 612*I*B*tan(f*x
+ e) - 903*I*A + 93*B)/(a^2*c^4*(tan(f*x + e) + I)^4))/f