3.8.37 \(\int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx\) [737]

Optimal. Leaf size=251 \[ \frac {5 (7 A+i B) x}{128 a^3 c^4}+\frac {A+i B}{96 a^3 c^4 f (i-\tan (e+f x))^3}-\frac {5 i A-3 B}{128 a^3 c^4 f (i-\tan (e+f x))^2}-\frac {5 (3 A+i B)}{128 a^3 c^4 f (i-\tan (e+f x))}-\frac {i A+B}{64 a^3 c^4 f (i+\tan (e+f x))^4}-\frac {2 A-i B}{48 a^3 c^4 f (i+\tan (e+f x))^3}+\frac {5 i A+B}{64 a^3 c^4 f (i+\tan (e+f x))^2}+\frac {5 A}{32 a^3 c^4 f (i+\tan (e+f x))} \]

[Out]

5/128*(7*A+I*B)*x/a^3/c^4+1/96*(A+I*B)/a^3/c^4/f/(I-tan(f*x+e))^3+1/128*(-5*I*A+3*B)/a^3/c^4/f/(I-tan(f*x+e))^
2-5/128*(3*A+I*B)/a^3/c^4/f/(I-tan(f*x+e))+1/64*(-I*A-B)/a^3/c^4/f/(I+tan(f*x+e))^4+1/48*(-2*A+I*B)/a^3/c^4/f/
(I+tan(f*x+e))^3+1/64*(5*I*A+B)/a^3/c^4/f/(I+tan(f*x+e))^2+5/32*A/a^3/c^4/f/(I+tan(f*x+e))

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Rubi [A]
time = 0.21, antiderivative size = 251, normalized size of antiderivative = 1.00, 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 = {3669, 78, 209} \begin {gather*} -\frac {5 (3 A+i B)}{128 a^3 c^4 f (-\tan (e+f x)+i)}-\frac {-3 B+5 i A}{128 a^3 c^4 f (-\tan (e+f x)+i)^2}+\frac {B+5 i A}{64 a^3 c^4 f (\tan (e+f x)+i)^2}+\frac {A+i B}{96 a^3 c^4 f (-\tan (e+f x)+i)^3}-\frac {2 A-i B}{48 a^3 c^4 f (\tan (e+f x)+i)^3}-\frac {B+i A}{64 a^3 c^4 f (\tan (e+f x)+i)^4}+\frac {5 x (7 A+i B)}{128 a^3 c^4}+\frac {5 A}{32 a^3 c^4 f (\tan (e+f x)+i)} \end {gather*}

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])^4),x]

[Out]

(5*(7*A + I*B)*x)/(128*a^3*c^4) + (A + I*B)/(96*a^3*c^4*f*(I - Tan[e + f*x])^3) - ((5*I)*A - 3*B)/(128*a^3*c^4
*f*(I - Tan[e + f*x])^2) - (5*(3*A + I*B))/(128*a^3*c^4*f*(I - Tan[e + f*x])) - (I*A + B)/(64*a^3*c^4*f*(I + T
an[e + f*x])^4) - (2*A - I*B)/(48*a^3*c^4*f*(I + Tan[e + f*x])^3) + ((5*I)*A + B)/(64*a^3*c^4*f*(I + Tan[e + f
*x])^2) + (5*A)/(32*a^3*c^4*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*} \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^4 (c-i c x)^5} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \text {Subst}\left (\int \left (\frac {A+i B}{32 a^4 c^5 (-i+x)^4}+\frac {i (5 A+3 i B)}{64 a^4 c^5 (-i+x)^3}-\frac {5 (3 A+i B)}{128 a^4 c^5 (-i+x)^2}+\frac {i A+B}{16 a^4 c^5 (i+x)^5}+\frac {2 A-i B}{16 a^4 c^5 (i+x)^4}-\frac {i (5 A-i B)}{32 a^4 c^5 (i+x)^3}-\frac {5 A}{32 a^4 c^5 (i+x)^2}+\frac {5 (7 A+i B)}{128 a^4 c^5 \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {A+i B}{96 a^3 c^4 f (i-\tan (e+f x))^3}-\frac {5 i A-3 B}{128 a^3 c^4 f (i-\tan (e+f x))^2}-\frac {5 (3 A+i B)}{128 a^3 c^4 f (i-\tan (e+f x))}-\frac {i A+B}{64 a^3 c^4 f (i+\tan (e+f x))^4}-\frac {2 A-i B}{48 a^3 c^4 f (i+\tan (e+f x))^3}+\frac {5 i A+B}{64 a^3 c^4 f (i+\tan (e+f x))^2}+\frac {5 A}{32 a^3 c^4 f (i+\tan (e+f x))}+\frac {(5 (7 A+i B)) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{128 a^3 c^4 f}\\ &=\frac {5 (7 A+i B) x}{128 a^3 c^4}+\frac {A+i B}{96 a^3 c^4 f (i-\tan (e+f x))^3}-\frac {5 i A-3 B}{128 a^3 c^4 f (i-\tan (e+f x))^2}-\frac {5 (3 A+i B)}{128 a^3 c^4 f (i-\tan (e+f x))}-\frac {i A+B}{64 a^3 c^4 f (i+\tan (e+f x))^4}-\frac {2 A-i B}{48 a^3 c^4 f (i+\tan (e+f x))^3}+\frac {5 i A+B}{64 a^3 c^4 f (i+\tan (e+f x))^2}+\frac {5 A}{32 a^3 c^4 f (i+\tan (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 1.53, size = 267, normalized size = 1.06 \begin {gather*} \frac {\sec ^3(e+f x) (-\cos (4 (e+f x))-i \sin (4 (e+f x))) (60 (A (-7-14 i f x)+B (i+2 f x)) \cos (e+f x)+18 (7 A+9 i B) \cos (3 (e+f x))+14 A \cos (5 (e+f x))+50 i B \cos (5 (e+f x))+A \cos (7 (e+f x))+7 i B \cos (7 (e+f x))-420 i A \sin (e+f x)-60 B \sin (e+f x)-840 A f x \sin (e+f x)-120 i B f x \sin (e+f x)-378 i A \sin (3 (e+f x))+54 B \sin (3 (e+f x))-70 i A \sin (5 (e+f x))+10 B \sin (5 (e+f x))-7 i A \sin (7 (e+f x))+B \sin (7 (e+f x)))}{3072 a^3 c^4 f (-i+\tan (e+f x))^3} \end {gather*}

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])^4),x]

[Out]

(Sec[e + f*x]^3*(-Cos[4*(e + f*x)] - I*Sin[4*(e + f*x)])*(60*(A*(-7 - (14*I)*f*x) + B*(I + 2*f*x))*Cos[e + f*x
] + 18*(7*A + (9*I)*B)*Cos[3*(e + f*x)] + 14*A*Cos[5*(e + f*x)] + (50*I)*B*Cos[5*(e + f*x)] + A*Cos[7*(e + f*x
)] + (7*I)*B*Cos[7*(e + f*x)] - (420*I)*A*Sin[e + f*x] - 60*B*Sin[e + f*x] - 840*A*f*x*Sin[e + f*x] - (120*I)*
B*f*x*Sin[e + f*x] - (378*I)*A*Sin[3*(e + f*x)] + 54*B*Sin[3*(e + f*x)] - (70*I)*A*Sin[5*(e + f*x)] + 10*B*Sin
[5*(e + f*x)] - (7*I)*A*Sin[7*(e + f*x)] + B*Sin[7*(e + f*x)]))/(3072*a^3*c^4*f*(-I + Tan[e + f*x])^3)

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Maple [A]
time = 0.33, size = 190, normalized size = 0.76

method	result	size
derivativedivides	\(\frac {-\frac {\frac {i A}{16}+\frac {B}{16}}{4 \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {5 A}{32 \left (i+\tan \left (f x +e \right )\right )}+\left (\frac {35 i A}{256}-\frac {5 B}{256}\right ) \ln \left (i+\tan \left (f x +e \right )\right )-\frac {-\frac {i B}{16}+\frac {A}{8}}{3 \left (i+\tan \left (f x +e \right )\right )^{3}}-\frac {-\frac {5 i A}{32}-\frac {B}{32}}{2 \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {\frac {A}{32}+\frac {i B}{32}}{3 \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {-\frac {15 A}{128}-\frac {5 i B}{128}}{-i+\tan \left (f x +e \right )}-\frac {\frac {5 i A}{64}-\frac {3 B}{64}}{2 \left (-i+\tan \left (f x +e \right )\right )^{2}}+\left (-\frac {35 i A}{256}+\frac {5 B}{256}\right ) \ln \left (-i+\tan \left (f x +e \right )\right )}{f \,a^{3} c^{4}}\)	\(190\)
default	\(\frac {-\frac {\frac {i A}{16}+\frac {B}{16}}{4 \left (i+\tan \left (f x +e \right )\right )^{4}}+\frac {5 A}{32 \left (i+\tan \left (f x +e \right )\right )}+\left (\frac {35 i A}{256}-\frac {5 B}{256}\right ) \ln \left (i+\tan \left (f x +e \right )\right )-\frac {-\frac {i B}{16}+\frac {A}{8}}{3 \left (i+\tan \left (f x +e \right )\right )^{3}}-\frac {-\frac {5 i A}{32}-\frac {B}{32}}{2 \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {\frac {A}{32}+\frac {i B}{32}}{3 \left (-i+\tan \left (f x +e \right )\right )^{3}}-\frac {-\frac {15 A}{128}-\frac {5 i B}{128}}{-i+\tan \left (f x +e \right )}-\frac {\frac {5 i A}{64}-\frac {3 B}{64}}{2 \left (-i+\tan \left (f x +e \right )\right )^{2}}+\left (-\frac {35 i A}{256}+\frac {5 B}{256}\right ) \ln \left (-i+\tan \left (f x +e \right )\right )}{f \,a^{3} c^{4}}\)	\(190\)
norman	\(\frac {\frac {5 \left (i B +7 A \right ) x}{128 a c}-\frac {i A +B}{8 a c f}+\frac {\left (-5 i B +93 A \right ) \tan \left (f x +e \right )}{128 a c f}+\frac {73 \left (i B +7 A \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{384 a c f}+\frac {55 \left (i B +7 A \right ) \left (\tan ^{5}\left (f x +e \right )\right )}{384 a c f}+\frac {5 \left (i B +7 A \right ) \left (\tan ^{7}\left (f x +e \right )\right )}{128 a c f}+\frac {5 \left (i B +7 A \right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{32 a c}+\frac {15 \left (i B +7 A \right ) x \left (\tan ^{4}\left (f x +e \right )\right )}{64 a c}+\frac {5 \left (i B +7 A \right ) x \left (\tan ^{6}\left (f x +e \right )\right )}{32 a c}+\frac {5 \left (i B +7 A \right ) x \left (\tan ^{8}\left (f x +e \right )\right )}{128 a c}}{a^{2} c^{3} \left (1+\tan ^{2}\left (f x +e \right )\right )^{4}}\)	\(261\)
risch	\(\frac {5 i x B}{128 a^{3} c^{4}}+\frac {35 x A}{128 a^{3} c^{4}}-\frac {{\mathrm e}^{8 i \left (f x +e \right )} B}{1024 a^{3} c^{4} f}-\frac {i \sin \left (4 f x +4 e \right ) B}{128 a^{3} c^{4} f}-\frac {\cos \left (6 f x +6 e \right ) B}{128 a^{3} c^{4} f}-\frac {i {\mathrm e}^{8 i \left (f x +e \right )} A}{1024 a^{3} c^{4} f}-\frac {i \sin \left (6 f x +6 e \right ) B}{192 a^{3} c^{4} f}+\frac {\sin \left (6 f x +6 e \right ) A}{96 a^{3} c^{4} f}-\frac {7 \cos \left (4 f x +4 e \right ) B}{256 a^{3} c^{4} f}-\frac {i \cos \left (6 f x +6 e \right ) A}{128 a^{3} c^{4} f}+\frac {i \sin \left (2 f x +2 e \right ) B}{64 a^{3} c^{4} f}+\frac {7 \sin \left (4 f x +4 e \right ) A}{128 a^{3} c^{4} f}-\frac {7 \cos \left (2 f x +2 e \right ) B}{128 a^{3} c^{4} f}-\frac {7 i \cos \left (4 f x +4 e \right ) A}{256 a^{3} c^{4} f}-\frac {7 i \cos \left (2 f x +2 e \right ) A}{128 a^{3} c^{4} f}+\frac {7 \sin \left (2 f x +2 e \right ) A}{32 a^{3} c^{4} f}\)	\(324\)

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))^4,x,method=_RETURNVERBOSE)

[Out]

1/f/a^3/c^4*(-1/4*(1/16*I*A+1/16*B)/(I+tan(f*x+e))^4+5/32*A/(I+tan(f*x+e))+(35/256*I*A-5/256*B)*ln(I+tan(f*x+e
))-1/3*(-1/16*I*B+1/8*A)/(I+tan(f*x+e))^3-1/2*(-5/32*I*A-1/32*B)/(I+tan(f*x+e))^2-1/3*(1/32*A+1/32*I*B)/(-I+ta
n(f*x+e))^3-(-15/128*A-5/128*I*B)/(-I+tan(f*x+e))-1/2*(5/64*I*A-3/64*B)/(-I+tan(f*x+e))^2+(-35/256*I*A+5/256*B
)*ln(-I+tan(f*x+e)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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))^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [A]
time = 4.46, size = 159, normalized size = 0.63 \begin {gather*} \frac {{\left (120 \, {\left (7 \, A + i \, B\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 3 \, {\left (i \, A + B\right )} e^{\left (14 i \, f x + 14 i \, e\right )} - 4 \, {\left (7 i \, A + 5 \, B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} - 18 \, {\left (7 i \, A + 3 \, B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} - 60 \, {\left (7 i \, A + B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} - 36 \, {\left (-7 i \, A + 3 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 6 \, {\left (-7 i \, A + 5 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i \, A - 4 \, B\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{3072 \, a^{3} c^{4} f} \end {gather*}

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))^4,x, algorithm="fricas")

[Out]

1/3072*(120*(7*A + I*B)*f*x*e^(6*I*f*x + 6*I*e) - 3*(I*A + B)*e^(14*I*f*x + 14*I*e) - 4*(7*I*A + 5*B)*e^(12*I*
f*x + 12*I*e) - 18*(7*I*A + 3*B)*e^(10*I*f*x + 10*I*e) - 60*(7*I*A + B)*e^(8*I*f*x + 8*I*e) - 36*(-7*I*A + 3*B
)*e^(4*I*f*x + 4*I*e) - 6*(-7*I*A + 5*B)*e^(2*I*f*x + 2*I*e) + 4*I*A - 4*B)*e^(-6*I*f*x - 6*I*e)/(a^3*c^4*f)

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Sympy [A]
time = 0.69, size = 605, normalized size = 2.41 \begin {gather*} \begin {cases} \frac {\left (\left (13510798882111488 i A a^{18} c^{24} f^{6} e^{6 i e} - 13510798882111488 B a^{18} c^{24} f^{6} e^{6 i e}\right ) e^{- 6 i f x} + \left (141863388262170624 i A a^{18} c^{24} f^{6} e^{8 i e} - 101330991615836160 B a^{18} c^{24} f^{6} e^{8 i e}\right ) e^{- 4 i f x} + \left (851180329573023744 i A a^{18} c^{24} f^{6} e^{10 i e} - 364791569817010176 B a^{18} c^{24} f^{6} e^{10 i e}\right ) e^{- 2 i f x} + \left (- 1418633882621706240 i A a^{18} c^{24} f^{6} e^{14 i e} - 202661983231672320 B a^{18} c^{24} f^{6} e^{14 i e}\right ) e^{2 i f x} + \left (- 425590164786511872 i A a^{18} c^{24} f^{6} e^{16 i e} - 182395784908505088 B a^{18} c^{24} f^{6} e^{16 i e}\right ) e^{4 i f x} + \left (- 94575592174780416 i A a^{18} c^{24} f^{6} e^{18 i e} - 67553994410557440 B a^{18} c^{24} f^{6} e^{18 i e}\right ) e^{6 i f x} + \left (- 10133099161583616 i A a^{18} c^{24} f^{6} e^{20 i e} - 10133099161583616 B a^{18} c^{24} f^{6} e^{20 i e}\right ) e^{8 i f x}\right ) e^{- 12 i e}}{10376293541461622784 a^{21} c^{28} f^{7}} & \text {for}\: a^{21} c^{28} f^{7} e^{12 i e} \neq 0 \\x \left (- \frac {35 A + 5 i B}{128 a^{3} c^{4}} + \frac {\left (A e^{14 i e} + 7 A e^{12 i e} + 21 A e^{10 i e} + 35 A e^{8 i e} + 35 A e^{6 i e} + 21 A e^{4 i e} + 7 A e^{2 i e} + A - i B e^{14 i e} - 5 i B e^{12 i e} - 9 i B e^{10 i e} - 5 i B e^{8 i e} + 5 i B e^{6 i e} + 9 i B e^{4 i e} + 5 i B e^{2 i e} + i B\right ) e^{- 6 i e}}{128 a^{3} c^{4}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (35 A + 5 i B\right )}{128 a^{3} c^{4}} \end {gather*}

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))**4,x)

[Out]

Piecewise((((13510798882111488*I*A*a**18*c**24*f**6*exp(6*I*e) - 13510798882111488*B*a**18*c**24*f**6*exp(6*I*
e))*exp(-6*I*f*x) + (141863388262170624*I*A*a**18*c**24*f**6*exp(8*I*e) - 101330991615836160*B*a**18*c**24*f**
6*exp(8*I*e))*exp(-4*I*f*x) + (851180329573023744*I*A*a**18*c**24*f**6*exp(10*I*e) - 364791569817010176*B*a**1
8*c**24*f**6*exp(10*I*e))*exp(-2*I*f*x) + (-1418633882621706240*I*A*a**18*c**24*f**6*exp(14*I*e) - 20266198323
1672320*B*a**18*c**24*f**6*exp(14*I*e))*exp(2*I*f*x) + (-425590164786511872*I*A*a**18*c**24*f**6*exp(16*I*e) -
182395784908505088*B*a**18*c**24*f**6*exp(16*I*e))*exp(4*I*f*x) + (-94575592174780416*I*A*a**18*c**24*f**6*ex
p(18*I*e) - 67553994410557440*B*a**18*c**24*f**6*exp(18*I*e))*exp(6*I*f*x) + (-10133099161583616*I*A*a**18*c**
24*f**6*exp(20*I*e) - 10133099161583616*B*a**18*c**24*f**6*exp(20*I*e))*exp(8*I*f*x))*exp(-12*I*e)/(1037629354
1461622784*a**21*c**28*f**7), Ne(a**21*c**28*f**7*exp(12*I*e), 0)), (x*(-(35*A + 5*I*B)/(128*a**3*c**4) + (A*e
xp(14*I*e) + 7*A*exp(12*I*e) + 21*A*exp(10*I*e) + 35*A*exp(8*I*e) + 35*A*exp(6*I*e) + 21*A*exp(4*I*e) + 7*A*ex
p(2*I*e) + A - I*B*exp(14*I*e) - 5*I*B*exp(12*I*e) - 9*I*B*exp(10*I*e) - 5*I*B*exp(8*I*e) + 5*I*B*exp(6*I*e) +
9*I*B*exp(4*I*e) + 5*I*B*exp(2*I*e) + I*B)*exp(-6*I*e)/(128*a**3*c**4)), True)) + x*(35*A + 5*I*B)/(128*a**3*
c**4)

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Giac [A]
time = 0.91, size = 271, normalized size = 1.08 \begin {gather*} \frac {\frac {60 \, {\left (7 i \, A - B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{3} c^{4}} - \frac {60 \, {\left (7 i \, A - B\right )} \log \left (-i \, \tan \left (f x + e\right ) - 1\right )}{a^{3} c^{4}} + \frac {2 \, {\left (385 \, A \tan \left (f x + e\right )^{3} + 55 i \, B \tan \left (f x + e\right )^{3} - 1335 i \, A \tan \left (f x + e\right )^{2} + 225 \, B \tan \left (f x + e\right )^{2} - 1575 \, A \tan \left (f x + e\right ) - 321 i \, B \tan \left (f x + e\right ) + 641 i \, A - 167 \, B\right )}}{a^{3} c^{4} {\left (i \, \tan \left (f x + e\right ) + 1\right )}^{3}} + \frac {-875 i \, A \tan \left (f x + e\right )^{4} + 125 \, B \tan \left (f x + e\right )^{4} + 3980 \, A \tan \left (f x + e\right )^{3} + 500 i \, B \tan \left (f x + e\right )^{3} + 6930 i \, A \tan \left (f x + e\right )^{2} - 702 \, B \tan \left (f x + e\right )^{2} - 5548 \, A \tan \left (f x + e\right ) - 340 i \, B \tan \left (f x + e\right ) - 1771 i \, A - 35 \, B}{a^{3} c^{4} {\left (\tan \left (f x + e\right ) + i\right )}^{4}}}{3072 \, f} \end {gather*}

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))^4,x, algorithm="giac")

[Out]

1/3072*(60*(7*I*A - B)*log(tan(f*x + e) + I)/(a^3*c^4) - 60*(7*I*A - B)*log(-I*tan(f*x + e) - 1)/(a^3*c^4) + 2
*(385*A*tan(f*x + e)^3 + 55*I*B*tan(f*x + e)^3 - 1335*I*A*tan(f*x + e)^2 + 225*B*tan(f*x + e)^2 - 1575*A*tan(f
*x + e) - 321*I*B*tan(f*x + e) + 641*I*A - 167*B)/(a^3*c^4*(I*tan(f*x + e) + 1)^3) + (-875*I*A*tan(f*x + e)^4
+ 125*B*tan(f*x + e)^4 + 3980*A*tan(f*x + e)^3 + 500*I*B*tan(f*x + e)^3 + 6930*I*A*tan(f*x + e)^2 - 702*B*tan(
f*x + e)^2 - 5548*A*tan(f*x + e) - 340*I*B*tan(f*x + e) - 1771*I*A - 35*B)/(a^3*c^4*(tan(f*x + e) + I)^4))/f

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Mupad [B]
time = 10.40, size = 286, normalized size = 1.14 \begin {gather*} \frac {\mathrm {tan}\left (e+f\,x\right )\,\left (-\frac {11\,B}{128\,a^3\,c^4}+\frac {A\,77{}\mathrm {i}}{128\,a^3\,c^4}\right )+{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (-\frac {5\,B}{48\,a^3\,c^4}+\frac {A\,35{}\mathrm {i}}{48\,a^3\,c^4}\right )+{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (\frac {35\,A}{48\,a^3\,c^4}+\frac {B\,5{}\mathrm {i}}{48\,a^3\,c^4}\right )+{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (-\frac {5\,B}{128\,a^3\,c^4}+\frac {A\,35{}\mathrm {i}}{128\,a^3\,c^4}\right )+{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (\frac {35\,A}{128\,a^3\,c^4}+\frac {B\,5{}\mathrm {i}}{128\,a^3\,c^4}\right )+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {77\,A}{128\,a^3\,c^4}+\frac {B\,11{}\mathrm {i}}{128\,a^3\,c^4}\right )+\frac {A}{8\,a^3\,c^4}-\frac {B\,1{}\mathrm {i}}{8\,a^3\,c^4}}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^7+{\mathrm {tan}\left (e+f\,x\right )}^6\,1{}\mathrm {i}+3\,{\mathrm {tan}\left (e+f\,x\right )}^5+{\mathrm {tan}\left (e+f\,x\right )}^4\,3{}\mathrm {i}+3\,{\mathrm {tan}\left (e+f\,x\right )}^3+{\mathrm {tan}\left (e+f\,x\right )}^2\,3{}\mathrm {i}+\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}+\frac {5\,x\,\left (7\,A+B\,1{}\mathrm {i}\right )}{128\,a^3\,c^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*tan(e + f*x))/((a + a*tan(e + f*x)*1i)^3*(c - c*tan(e + f*x)*1i)^4),x)

[Out]

(tan(e + f*x)*((A*77i)/(128*a^3*c^4) - (11*B)/(128*a^3*c^4)) + tan(e + f*x)^3*((A*35i)/(48*a^3*c^4) - (5*B)/(4
8*a^3*c^4)) + tan(e + f*x)^4*((35*A)/(48*a^3*c^4) + (B*5i)/(48*a^3*c^4)) + tan(e + f*x)^5*((A*35i)/(128*a^3*c^
4) - (5*B)/(128*a^3*c^4)) + tan(e + f*x)^6*((35*A)/(128*a^3*c^4) + (B*5i)/(128*a^3*c^4)) + tan(e + f*x)^2*((77
*A)/(128*a^3*c^4) + (B*11i)/(128*a^3*c^4)) + A/(8*a^3*c^4) - (B*1i)/(8*a^3*c^4))/(f*(tan(e + f*x) + tan(e + f*
x)^2*3i + 3*tan(e + f*x)^3 + tan(e + f*x)^4*3i + 3*tan(e + f*x)^5 + tan(e + f*x)^6*1i + tan(e + f*x)^7 + 1i))
+ (5*x*(7*A + B*1i))/(128*a^3*c^4)

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