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]

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

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Rubi [A]  time = 0.34, antiderivative size = 287, 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 = {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 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])

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]

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*}

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Mathematica [A]  time = 4.71, 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)

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fricas [A]  time = 0.85, size = 161, normalized size = 0.56 \[ \frac {{\left (840 \, {\left (4 \, A + i \, B\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-6 i \, A - 6 \, B\right )} e^{\left (16 i \, f x + 16 i \, e\right )} + {\left (-60 i \, A - 45 \, B\right )} e^{\left (14 i \, f x + 14 i \, e\right )} + {\left (-280 i \, A - 140 \, B\right )} e^{\left (12 i \, f x + 12 i \, e\right )} + {\left (-840 i \, A - 210 \, B\right )} e^{\left (10 i \, f x + 10 i \, e\right )} - 2100 i \, A e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (840 i \, A - 420 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (120 i \, A - 90 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 10 i \, A - 10 \, B\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{15360 \, a^{3} c^{5} f} \]

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)

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giac [A]  time = 5.28, size = 291, normalized size = 1.01 \[ -\frac {\frac {60 \, {\left (-28 i \, A + 7 \, B\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{3} c^{5}} + \frac {60 \, {\left (28 i \, A - 7 \, B\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c^{5}} + \frac {10 \, {\left (-308 i \, A \tan \left (f x + e\right )^{3} + 77 \, B \tan \left (f x + e\right )^{3} - 1050 \, A \tan \left (f x + e\right )^{2} - 285 i \, B \tan \left (f x + e\right )^{2} + 1212 i \, A \tan \left (f x + e\right ) - 363 \, B \tan \left (f x + e\right ) + 478 \, A + 163 i \, B\right )}}{a^{3} c^{5} {\left (\tan \left (f x + e\right ) - i\right )}^{3}} + \frac {3836 i \, A \tan \left (f x + e\right )^{5} - 959 \, B \tan \left (f x + e\right )^{5} - 21280 \, A \tan \left (f x + e\right )^{4} - 5095 i \, B \tan \left (f x + e\right )^{4} - 47960 i \, A \tan \left (f x + e\right )^{3} + 10790 \, B \tan \left (f x + e\right )^{3} + 55360 \, A \tan \left (f x + e\right )^{2} + 11230 i \, B \tan \left (f x + e\right )^{2} + 33260 i \, A \tan \left (f x + e\right ) - 5435 \, B \tan \left (f x + e\right ) - 8608 \, A - 667 i \, B}{a^{3} c^{5} {\left (\tan \left (f x + e\right ) + i\right )}^{5}}}{15360 \, f} \]

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

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maple [A]  time = 0.47, size = 445, normalized size = 1.55 \[ -\frac {i B}{80 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )+i\right )^{5}}+\frac {A}{80 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )+i\right )^{5}}+\frac {i B}{96 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )+i\right )^{3}}-\frac {5 A}{96 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )+i\right )^{3}}+\frac {9 i B}{256 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )-i\right )}+\frac {35 A}{256 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )+i\right )}-\frac {7 i \ln \left (\tan \left (f x +e \right )-i\right ) A}{64 f \,a^{3} c^{5}}-\frac {B}{64 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )+i\right )^{4}}-\frac {3 i A}{128 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {7 \ln \left (\tan \left (f x +e \right )+i\right ) B}{256 f \,a^{3} c^{5}}+\frac {5 i A}{64 a^{3} c^{5} f \left (\tan \left (f x +e \right )+i\right )^{2}}-\frac {A}{192 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {7 i \ln \left (\tan \left (f x +e \right )+i\right ) A}{64 f \,a^{3} c^{5}}+\frac {7 \ln \left (\tan \left (f x +e \right )-i\right ) B}{256 f \,a^{3} c^{5}}-\frac {i A}{32 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )+i\right )^{4}}+\frac {21 A}{256 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )-i\right )}+\frac {5 i B}{256 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )+i\right )}+\frac {B}{64 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i B}{192 f \,a^{3} c^{5} \left (\tan \left (f x +e \right )-i\right )^{3}} \]

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]

-1/80*I/f/a^3/c^5/(tan(f*x+e)+I)^5*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-5/9
6/f/a^3/c^5/(tan(f*x+e)+I)^3*A+9/256*I/f/a^3/c^5/(tan(f*x+e)-I)*B+35/256/f/a^3/c^5/(tan(f*x+e)+I)*A-7/64*I/f/a
^3/c^5*ln(tan(f*x+e)-I)*A-1/64/f/a^3/c^5/(tan(f*x+e)+I)^4*B-3/128*I/f/a^3/c^5/(tan(f*x+e)-I)^2*A-7/256/f/a^3/c
^5*ln(tan(f*x+e)+I)*B+5/64*I*A/a^3/c^5/f/(tan(f*x+e)+I)^2-1/192/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-1/32*I/f/a^3/c^5/(tan(f*x+e)+I)^4*A+21/256/f/a^3/c^5/(ta
n(f*x+e)-I)*A+5/256*I/f/a^3/c^5/(tan(f*x+e)+I)*B+1/64/f/a^3/c^5/(tan(f*x+e)-I)^2*B-1/192*I/f/a^3/c^5/(tan(f*x+
e)-I)^3*B

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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

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mupad [B]  time = 10.73, size = 319, normalized size = 1.11 \[ \frac {\frac {3\,B}{40\,a^3\,c^5}+{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (-\frac {7\,B}{24\,a^3\,c^5}+\frac {A\,7{}\mathrm {i}}{6\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (-\frac {7\,B}{64\,a^3\,c^5}+\frac {A\,7{}\mathrm {i}}{16\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (\frac {7\,A}{32\,a^3\,c^5}+\frac {B\,7{}\mathrm {i}}{128\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (\frac {35\,A}{96\,a^3\,c^5}+\frac {B\,35{}\mathrm {i}}{384\,a^3\,c^5}\right )+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (-\frac {77\,B}{320\,a^3\,c^5}+\frac {A\,77{}\mathrm {i}}{80\,a^3\,c^5}\right )-{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {49\,A}{480\,a^3\,c^5}+\frac {B\,49{}\mathrm {i}}{1920\,a^3\,c^5}\right )-\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {61\,A}{160\,a^3\,c^5}+\frac {B\,61{}\mathrm {i}}{640\,a^3\,c^5}\right )+\frac {A\,1{}\mathrm {i}}{5\,a^3\,c^5}}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^8+{\mathrm {tan}\left (e+f\,x\right )}^7\,2{}\mathrm {i}+2\,{\mathrm {tan}\left (e+f\,x\right )}^6+{\mathrm {tan}\left (e+f\,x\right )}^5\,6{}\mathrm {i}+{\mathrm {tan}\left (e+f\,x\right )}^3\,6{}\mathrm {i}-2\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,2{}\mathrm {i}-1\right )}+\frac {7\,x\,\left (4\,A+B\,1{}\mathrm {i}\right )}{128\,a^3\,c^5} \]

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)^5),x)

[Out]

(tan(e + f*x)^4*((A*7i)/(6*a^3*c^5) - (7*B)/(24*a^3*c^5)) - tan(e + f*x)*((61*A)/(160*a^3*c^5) + (B*61i)/(640*
a^3*c^5)) + tan(e + f*x)^6*((A*7i)/(16*a^3*c^5) - (7*B)/(64*a^3*c^5)) + tan(e + f*x)^7*((7*A)/(32*a^3*c^5) + (
B*7i)/(128*a^3*c^5)) + tan(e + f*x)^5*((35*A)/(96*a^3*c^5) + (B*35i)/(384*a^3*c^5)) + tan(e + f*x)^2*((A*77i)/
(80*a^3*c^5) - (77*B)/(320*a^3*c^5)) - tan(e + f*x)^3*((49*A)/(480*a^3*c^5) + (B*49i)/(1920*a^3*c^5)) + (A*1i)
/(5*a^3*c^5) + (3*B)/(40*a^3*c^5))/(f*(tan(e + f*x)*2i - 2*tan(e + f*x)^2 + tan(e + f*x)^3*6i + tan(e + f*x)^5
*6i + 2*tan(e + f*x)^6 + tan(e + f*x)^7*2i + tan(e + f*x)^8 - 1)) + (7*x*(4*A + B*1i))/(128*a^3*c^5)

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sympy [A]  time = 1.97, size = 649, normalized size = 2.26 \[ \begin {cases} \frac {\left (- 7263405479023135948800 i A a^{21} c^{35} f^{7} e^{14 i e} e^{2 i f x} + \left (34587645138205409280 i A a^{21} c^{35} f^{7} e^{6 i e} - 34587645138205409280 B a^{21} c^{35} f^{7} e^{6 i e}\right ) e^{- 6 i f x} + \left (415051741658464911360 i A a^{21} c^{35} f^{7} e^{8 i e} - 311288806243848683520 B a^{21} c^{35} f^{7} e^{8 i e}\right ) e^{- 4 i f x} + \left (2905362191609254379520 i A a^{21} c^{35} f^{7} e^{10 i e} - 1452681095804627189760 B a^{21} c^{35} f^{7} e^{10 i e}\right ) e^{- 2 i f x} + \left (- 2905362191609254379520 i A a^{21} c^{35} f^{7} e^{16 i e} - 726340547902313594880 B a^{21} c^{35} f^{7} e^{16 i e}\right ) e^{4 i f x} + \left (- 968454063869751459840 i A a^{21} c^{35} f^{7} e^{18 i e} - 484227031934875729920 B a^{21} c^{35} f^{7} e^{18 i e}\right ) e^{6 i f x} + \left (- 207525870829232455680 i A a^{21} c^{35} f^{7} e^{20 i e} - 155644403121924341760 B a^{21} c^{35} f^{7} e^{20 i e}\right ) e^{8 i f x} + \left (- 20752587082923245568 i A a^{21} c^{35} f^{7} e^{22 i e} - 20752587082923245568 B a^{21} c^{35} f^{7} e^{22 i e}\right ) e^{10 i f x}\right ) e^{- 12 i e}}{53126622932283508654080 a^{24} c^{40} f^{8}} & \text {for}\: 53126622932283508654080 a^{24} c^{40} f^{8} e^{12 i e} \neq 0 \\x \left (- \frac {28 A + 7 i B}{128 a^{3} c^{5}} + \frac {\left (A e^{16 i e} + 8 A e^{14 i e} + 28 A e^{12 i e} + 56 A e^{10 i e} + 70 A e^{8 i e} + 56 A e^{6 i e} + 28 A e^{4 i e} + 8 A e^{2 i e} + A - i B e^{16 i e} - 6 i B e^{14 i e} - 14 i B e^{12 i e} - 14 i B e^{10 i e} + 14 i B e^{6 i e} + 14 i B e^{4 i e} + 6 i B e^{2 i e} + i B\right ) e^{- 6 i e}}{256 a^{3} c^{5}}\right ) & \text {otherwise} \end {cases} - \frac {x \left (- 28 A - 7 i B\right )}{128 a^{3} c^{5}} \]

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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