\(\int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx\) [756]

Optimal result

Integrand size = 43, antiderivative size = 144 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\frac {8 a^3 (i A+B) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {8 a^3 (i A+2 B) (c-i c \tan (e+f x))^{9/2}}{9 c f}+\frac {2 a^3 (i A+5 B) (c-i c \tan (e+f x))^{11/2}}{11 c^2 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{13/2}}{13 c^3 f} \]

[Out]

8/7*a^3*(I*A+B)*(c-I*c*tan(f*x+e))^(7/2)/f-8/9*a^3*(I*A+2*B)*(c-I*c*tan(f*x+e))^(9/2)/c/f+2/11*a^3*(I*A+5*B)*(
c-I*c*tan(f*x+e))^(11/2)/c^2/f-2/13*a^3*B*(c-I*c*tan(f*x+e))^(13/2)/c^3/f

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {3669, 78} \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\frac {2 a^3 (5 B+i A) (c-i c \tan (e+f x))^{11/2}}{11 c^2 f}-\frac {8 a^3 (2 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c f}+\frac {8 a^3 (B+i A) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{13/2}}{13 c^3 f} \]

[In]

Int[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2),x]

[Out]

(8*a^3*(I*A + B)*(c - I*c*Tan[e + f*x])^(7/2))/(7*f) - (8*a^3*(I*A + 2*B)*(c - I*c*Tan[e + f*x])^(9/2))/(9*c*f
) + (2*a^3*(I*A + 5*B)*(c - I*c*Tan[e + f*x])^(11/2))/(11*c^2*f) - (2*a^3*B*(c - I*c*Tan[e + f*x])^(13/2))/(13
*c^3*f)

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 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*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int (a+i a x)^2 (A+B x) (c-i c x)^{5/2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (4 a^2 (A-i B) (c-i c x)^{5/2}-\frac {4 a^2 (A-2 i B) (c-i c x)^{7/2}}{c}+\frac {a^2 (A-5 i B) (c-i c x)^{9/2}}{c^2}+\frac {i a^2 B (c-i c x)^{11/2}}{c^3}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {8 a^3 (i A+B) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {8 a^3 (i A+2 B) (c-i c \tan (e+f x))^{9/2}}{9 c f}+\frac {2 a^3 (i A+5 B) (c-i c \tan (e+f x))^{11/2}}{11 c^2 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{13/2}}{13 c^3 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.80 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.85 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\frac {a^3 c^4 \sec ^6(e+f x) (i \cos (4 (e+f x))+\sin (4 (e+f x))) (2 (572 A+737 i B+7 (169 i A+86 B) \tan (e+f x))+\cos (2 (e+f x)) (2782 A-2558 i B+14 (169 i A+185 B) \tan (e+f x)))}{9009 f \sqrt {c-i c \tan (e+f x)}} \]

[In]

Integrate[(a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2),x]

[Out]

(a^3*c^4*Sec[e + f*x]^6*(I*Cos[4*(e + f*x)] + Sin[4*(e + f*x)])*(2*(572*A + (737*I)*B + 7*((169*I)*A + 86*B)*T
an[e + f*x]) + Cos[2*(e + f*x)]*(2782*A - (2558*I)*B + 14*((169*I)*A + 185*B)*Tan[e + f*x])))/(9009*f*Sqrt[c -
I*c*Tan[e + f*x]])

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {2 i a^{3} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {13}{2}}}{13}+\frac {\left (-5 i B c +c A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {\left (-4 \left (-i B c +c A \right ) c +4 i B \,c^{2}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {4 \left (-i B c +c A \right ) c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}\right )}{f \,c^{3}}\)	\(121\)
default	\(\frac {2 i a^{3} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {13}{2}}}{13}+\frac {\left (-5 i B c +c A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {\left (-4 \left (-i B c +c A \right ) c +4 i B \,c^{2}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {4 \left (-i B c +c A \right ) c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}\right )}{f \,c^{3}}\)	\(121\)
parts	\(\frac {2 i a^{3} A c \left (-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {2 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-4 \sqrt {c -i c \tan \left (f x +e \right )}\, c^{2}+4 c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f}+\frac {a^{3} \left (3 i A +B \right ) \left (\frac {2 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {2 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {4 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}} c^{2}}{3}+8 \sqrt {c -i c \tan \left (f x +e \right )}\, c^{3}-8 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f}+\frac {2 B \,a^{3} \left (-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {13}{2}}}{13}+\frac {2 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}-\frac {2 c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}-\frac {c^{4} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {2 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}} c^{5}}{3}-4 \sqrt {c -i c \tan \left (f x +e \right )}\, c^{6}+4 c^{\frac {13}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f \,c^{3}}-\frac {6 i a^{3} \left (-i B +A \right ) \left (\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}} c^{3}}{3}+4 \sqrt {c -i c \tan \left (f x +e \right )}\, c^{4}-4 c^{\frac {9}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f c}-\frac {2 a^{3} \left (i A +3 B \right ) \left (-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}-\frac {c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {c^{3} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {2 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}} c^{4}}{3}-4 \sqrt {c -i c \tan \left (f x +e \right )}\, c^{5}+4 c^{\frac {11}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f \,c^{2}}\)	\(649\)

[In]

int((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

2*I/f*a^3/c^3*(1/13*I*B*(c-I*c*tan(f*x+e))^(13/2)+1/11*(-5*I*B*c+c*A)*(c-I*c*tan(f*x+e))^(11/2)+1/9*(-4*(-I*B*
c+c*A)*c+4*I*B*c^2)*(c-I*c*tan(f*x+e))^(9/2)+4/7*(-I*B*c+c*A)*c^2*(c-I*c*tan(f*x+e))^(7/2))

Fricas [A] (verification not implemented)

none

Time = 0.53 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.24 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=-\frac {64 \, \sqrt {2} {\left (1287 \, {\left (-i \, A - B\right )} a^{3} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 143 \, {\left (-13 i \, A + B\right )} a^{3} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 52 \, {\left (-13 i \, A + B\right )} a^{3} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, {\left (-13 i \, A + B\right )} a^{3} c^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{9009 \, {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

[In]

integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

-64/9009*sqrt(2)*(1287*(-I*A - B)*a^3*c^3*e^(6*I*f*x + 6*I*e) + 143*(-13*I*A + B)*a^3*c^3*e^(4*I*f*x + 4*I*e)
+ 52*(-13*I*A + B)*a^3*c^3*e^(2*I*f*x + 2*I*e) + 8*(-13*I*A + B)*a^3*c^3)*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))/(f
*e^(12*I*f*x + 12*I*e) + 6*f*e^(10*I*f*x + 10*I*e) + 15*f*e^(8*I*f*x + 8*I*e) + 20*f*e^(6*I*f*x + 6*I*e) + 15*
f*e^(4*I*f*x + 4*I*e) + 6*f*e^(2*I*f*x + 2*I*e) + f)

Sympy [F]

\[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=- i a^{3} \left (\int i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}\, dx + \int 3 i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\, dx + \int 3 i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\, dx + \int i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{6}{\left (e + f x \right )}\, dx + \int i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\, dx + \int 3 i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\, dx + \int 3 i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\, dx + \int i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{7}{\left (e + f x \right )}\, dx\right ) \]

[In]

integrate((a+I*a*tan(f*x+e))**3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**(7/2),x)

[Out]

-I*a**3*(Integral(I*A*c**3*sqrt(-I*c*tan(e + f*x) + c), x) + Integral(3*I*A*c**3*sqrt(-I*c*tan(e + f*x) + c)*t
an(e + f*x)**2, x) + Integral(3*I*A*c**3*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**4, x) + Integral(I*A*c**3*s
qrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**6, x) + Integral(I*B*c**3*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x), x
) + Integral(3*I*B*c**3*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**3, x) + Integral(3*I*B*c**3*sqrt(-I*c*tan(e
+ f*x) + c)*tan(e + f*x)**5, x) + Integral(I*B*c**3*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**7, x))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.72 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\frac {2 i \, {\left (693 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {13}{2}} B a^{3} + 819 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {11}{2}} {\left (A - 5 i \, B\right )} a^{3} c - 4004 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}} {\left (A - 2 i \, B\right )} a^{3} c^{2} + 5148 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} {\left (A - i \, B\right )} a^{3} c^{3}\right )}}{9009 \, c^{3} f} \]

[In]

integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

2/9009*I*(693*I*(-I*c*tan(f*x + e) + c)^(13/2)*B*a^3 + 819*(-I*c*tan(f*x + e) + c)^(11/2)*(A - 5*I*B)*a^3*c -
4004*(-I*c*tan(f*x + e) + c)^(9/2)*(A - 2*I*B)*a^3*c^2 + 5148*(-I*c*tan(f*x + e) + c)^(7/2)*(A - I*B)*a^3*c^3)
/(c^3*f)

Giac [F]

\[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]

[In]

integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2),x, algorithm="giac")

[Out]

integrate((B*tan(f*x + e) + A)*(I*a*tan(f*x + e) + a)^3*(-I*c*tan(f*x + e) + c)^(7/2), x)

Mupad [B] (verification not implemented)

Time = 12.90 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.42 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=-\frac {\left (\frac {a^3\,c^3\,\left (A-B\,1{}\mathrm {i}\right )\,64{}\mathrm {i}}{9\,f}+\frac {a^3\,c^3\,\left (A-B\,3{}\mathrm {i}\right )\,64{}\mathrm {i}}{9\,f}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4}-\frac {\left (\frac {a^3\,c^3\,\left (A-B\,1{}\mathrm {i}\right )\,64{}\mathrm {i}}{13\,f}-\frac {a^3\,c^3\,\left (A+B\,1{}\mathrm {i}\right )\,64{}\mathrm {i}}{13\,f}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^6}+\frac {\left (\frac {256\,B\,a^3\,c^3}{11\,f}+\frac {a^3\,c^3\,\left (A-B\,1{}\mathrm {i}\right )\,64{}\mathrm {i}}{11\,f}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^5}+\frac {a^3\,c^3\,\left (A-B\,1{}\mathrm {i}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,64{}\mathrm {i}}{7\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3} \]

[In]

int((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^3*(c - c*tan(e + f*x)*1i)^(7/2),x)

[Out]

(((a^3*c^3*(A - B*1i)*64i)/(11*f) + (256*B*a^3*c^3)/(11*f))*(c + (c*(exp(e*2i + f*x*2i)*1i - 1i)*1i)/(exp(e*2i
+ f*x*2i) + 1))^(1/2))/(exp(e*2i + f*x*2i) + 1)^5 - (((a^3*c^3*(A - B*1i)*64i)/(13*f) - (a^3*c^3*(A + B*1i)*6
4i)/(13*f))*(c + (c*(exp(e*2i + f*x*2i)*1i - 1i)*1i)/(exp(e*2i + f*x*2i) + 1))^(1/2))/(exp(e*2i + f*x*2i) + 1)
^6 - (((a^3*c^3*(A - B*1i)*64i)/(9*f) + (a^3*c^3*(A - B*3i)*64i)/(9*f))*(c + (c*(exp(e*2i + f*x*2i)*1i - 1i)*1
i)/(exp(e*2i + f*x*2i) + 1))^(1/2))/(exp(e*2i + f*x*2i) + 1)^4 + (a^3*c^3*(A - B*1i)*(c + (c*(exp(e*2i + f*x*2
i)*1i - 1i)*1i)/(exp(e*2i + f*x*2i) + 1))^(1/2)*64i)/(7*f*(exp(e*2i + f*x*2i) + 1)^3)