\(\int \frac {\cot ^8(e+f x)}{(a-a \sin ^2(e+f x))^{3/2}} \, dx\) [487]

Optimal result

Integrand size = 26, antiderivative size = 115 \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}}+\frac {2 \cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^6(e+f x)}{7 a f \sqrt {a \cos ^2(e+f x)}} \]

[Out]

-1/3*cot(f*x+e)*csc(f*x+e)^2/a/f/(a*cos(f*x+e)^2)^(1/2)+2/5*cot(f*x+e)*csc(f*x+e)^4/a/f/(a*cos(f*x+e)^2)^(1/2)
-1/7*cot(f*x+e)*csc(f*x+e)^6/a/f/(a*cos(f*x+e)^2)^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3255, 3286, 2686, 276} \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cot (e+f x) \csc ^6(e+f x)}{7 a f \sqrt {a \cos ^2(e+f x)}}+\frac {2 \cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}} \]

[In]

Int[Cot[e + f*x]^8/(a - a*Sin[e + f*x]^2)^(3/2),x]

[Out]

-1/3*(Cot[e + f*x]*Csc[e + f*x]^2)/(a*f*Sqrt[a*Cos[e + f*x]^2]) + (2*Cot[e + f*x]*Csc[e + f*x]^4)/(5*a*f*Sqrt[
a*Cos[e + f*x]^2]) - (Cot[e + f*x]*Csc[e + f*x]^6)/(7*a*f*Sqrt[a*Cos[e + f*x]^2])

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 3255

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3286

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^8(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx \\ & = \frac {\cos (e+f x) \int \cot ^5(e+f x) \csc ^3(e+f x) \, dx}{a \sqrt {a \cos ^2(e+f x)}} \\ & = -\frac {\cos (e+f x) \text {Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a f \sqrt {a \cos ^2(e+f x)}} \\ & = -\frac {\cos (e+f x) \text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (e+f x)\right )}{a f \sqrt {a \cos ^2(e+f x)}} \\ & = -\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}}+\frac {2 \cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^6(e+f x)}{7 a f \sqrt {a \cos ^2(e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.44 \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\cot ^3(e+f x) \left (35-42 \csc ^2(e+f x)+15 \csc ^4(e+f x)\right )}{105 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]

[In]

Integrate[Cot[e + f*x]^8/(a - a*Sin[e + f*x]^2)^(3/2),x]

[Out]

-1/105*(Cot[e + f*x]^3*(35 - 42*Csc[e + f*x]^2 + 15*Csc[e + f*x]^4))/(f*(a*Cos[e + f*x]^2)^(3/2))

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.50

method	result	size
default	\(-\frac {\cos \left (f x +e \right ) \left (35 \left (\cos ^{4}\left (f x +e \right )\right )-28 \left (\cos ^{2}\left (f x +e \right )\right )+8\right )}{105 a \sin \left (f x +e \right )^{7} \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\)	\(57\)
risch	\(\frac {8 i \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left (35 \,{\mathrm e}^{10 i \left (f x +e \right )}+28 \,{\mathrm e}^{8 i \left (f x +e \right )}+114 \,{\mathrm e}^{6 i \left (f x +e \right )}+28 \,{\mathrm e}^{4 i \left (f x +e \right )}+35 \,{\mathrm e}^{2 i \left (f x +e \right )}\right )}{105 \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{7} f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}\)	\(116\)

[In]

int(cot(f*x+e)^8/(a-a*sin(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/105/a*cos(f*x+e)*(35*cos(f*x+e)^4-28*cos(f*x+e)^2+8)/sin(f*x+e)^7/(a*cos(f*x+e)^2)^(1/2)/f

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {{\left (35 \, \cos \left (f x + e\right )^{4} - 28 \, \cos \left (f x + e\right )^{2} + 8\right )} \sqrt {a \cos \left (f x + e\right )^{2}}}{105 \, {\left (a^{2} f \cos \left (f x + e\right )^{7} - 3 \, a^{2} f \cos \left (f x + e\right )^{5} + 3 \, a^{2} f \cos \left (f x + e\right )^{3} - a^{2} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]

[In]

integrate(cot(f*x+e)^8/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

1/105*(35*cos(f*x + e)^4 - 28*cos(f*x + e)^2 + 8)*sqrt(a*cos(f*x + e)^2)/((a^2*f*cos(f*x + e)^7 - 3*a^2*f*cos(
f*x + e)^5 + 3*a^2*f*cos(f*x + e)^3 - a^2*f*cos(f*x + e))*sin(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(cot(f*x+e)**8/(a-a*sin(f*x+e)**2)**(3/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2026 vs. \(2 (103) = 206\).

Time = 0.36 (sec) , antiderivative size = 2026, normalized size of antiderivative = 17.62 \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate(cot(f*x+e)^8/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

-8/105*((35*sin(11*f*x + 11*e) + 28*sin(9*f*x + 9*e) + 114*sin(7*f*x + 7*e) + 28*sin(5*f*x + 5*e) + 35*sin(3*f
*x + 3*e))*cos(14*f*x + 14*e) - 7*(35*sin(11*f*x + 11*e) + 28*sin(9*f*x + 9*e) + 114*sin(7*f*x + 7*e) + 28*sin
(5*f*x + 5*e) + 35*sin(3*f*x + 3*e))*cos(12*f*x + 12*e) - 245*(3*sin(10*f*x + 10*e) - 5*sin(8*f*x + 8*e) + 5*s
in(6*f*x + 6*e) - 3*sin(4*f*x + 4*e) + sin(2*f*x + 2*e))*cos(11*f*x + 11*e) + 21*(28*sin(9*f*x + 9*e) + 114*si
n(7*f*x + 7*e) + 28*sin(5*f*x + 5*e) + 35*sin(3*f*x + 3*e))*cos(10*f*x + 10*e) + 196*(5*sin(8*f*x + 8*e) - 5*s
in(6*f*x + 6*e) + 3*sin(4*f*x + 4*e) - sin(2*f*x + 2*e))*cos(9*f*x + 9*e) - 35*(114*sin(7*f*x + 7*e) + 28*sin(
5*f*x + 5*e) + 35*sin(3*f*x + 3*e))*cos(8*f*x + 8*e) - 798*(5*sin(6*f*x + 6*e) - 3*sin(4*f*x + 4*e) + sin(2*f*
x + 2*e))*cos(7*f*x + 7*e) + 245*(4*sin(5*f*x + 5*e) + 5*sin(3*f*x + 3*e))*cos(6*f*x + 6*e) + 196*(3*sin(4*f*x
+ 4*e) - sin(2*f*x + 2*e))*cos(5*f*x + 5*e) - (35*cos(11*f*x + 11*e) + 28*cos(9*f*x + 9*e) + 114*cos(7*f*x +
7*e) + 28*cos(5*f*x + 5*e) + 35*cos(3*f*x + 3*e))*sin(14*f*x + 14*e) + 7*(35*cos(11*f*x + 11*e) + 28*cos(9*f*x
+ 9*e) + 114*cos(7*f*x + 7*e) + 28*cos(5*f*x + 5*e) + 35*cos(3*f*x + 3*e))*sin(12*f*x + 12*e) + 35*(21*cos(10
*f*x + 10*e) - 35*cos(8*f*x + 8*e) + 35*cos(6*f*x + 6*e) - 21*cos(4*f*x + 4*e) + 7*cos(2*f*x + 2*e) - 1)*sin(1
1*f*x + 11*e) - 21*(28*cos(9*f*x + 9*e) + 114*cos(7*f*x + 7*e) + 28*cos(5*f*x + 5*e) + 35*cos(3*f*x + 3*e))*si
n(10*f*x + 10*e) - 28*(35*cos(8*f*x + 8*e) - 35*cos(6*f*x + 6*e) + 21*cos(4*f*x + 4*e) - 7*cos(2*f*x + 2*e) +
1)*sin(9*f*x + 9*e) + 35*(114*cos(7*f*x + 7*e) + 28*cos(5*f*x + 5*e) + 35*cos(3*f*x + 3*e))*sin(8*f*x + 8*e) +
114*(35*cos(6*f*x + 6*e) - 21*cos(4*f*x + 4*e) + 7*cos(2*f*x + 2*e) - 1)*sin(7*f*x + 7*e) - 245*(4*cos(5*f*x
+ 5*e) + 5*cos(3*f*x + 3*e))*sin(6*f*x + 6*e) - 28*(21*cos(4*f*x + 4*e) - 7*cos(2*f*x + 2*e) + 1)*sin(5*f*x +
5*e) + 735*cos(3*f*x + 3*e)*sin(4*f*x + 4*e) + 35*(7*cos(2*f*x + 2*e) - 1)*sin(3*f*x + 3*e) - 735*cos(4*f*x +
4*e)*sin(3*f*x + 3*e) - 245*cos(3*f*x + 3*e)*sin(2*f*x + 2*e))*sqrt(a)/((a^2*cos(14*f*x + 14*e)^2 + 49*a^2*cos
(12*f*x + 12*e)^2 + 441*a^2*cos(10*f*x + 10*e)^2 + 1225*a^2*cos(8*f*x + 8*e)^2 + 1225*a^2*cos(6*f*x + 6*e)^2 +
441*a^2*cos(4*f*x + 4*e)^2 + 49*a^2*cos(2*f*x + 2*e)^2 + a^2*sin(14*f*x + 14*e)^2 + 49*a^2*sin(12*f*x + 12*e)
^2 + 441*a^2*sin(10*f*x + 10*e)^2 + 1225*a^2*sin(8*f*x + 8*e)^2 + 1225*a^2*sin(6*f*x + 6*e)^2 + 441*a^2*sin(4*
f*x + 4*e)^2 - 294*a^2*sin(4*f*x + 4*e)*sin(2*f*x + 2*e) + 49*a^2*sin(2*f*x + 2*e)^2 - 14*a^2*cos(2*f*x + 2*e)
+ a^2 - 2*(7*a^2*cos(12*f*x + 12*e) - 21*a^2*cos(10*f*x + 10*e) + 35*a^2*cos(8*f*x + 8*e) - 35*a^2*cos(6*f*x
+ 6*e) + 21*a^2*cos(4*f*x + 4*e) - 7*a^2*cos(2*f*x + 2*e) + a^2)*cos(14*f*x + 14*e) - 14*(21*a^2*cos(10*f*x +
10*e) - 35*a^2*cos(8*f*x + 8*e) + 35*a^2*cos(6*f*x + 6*e) - 21*a^2*cos(4*f*x + 4*e) + 7*a^2*cos(2*f*x + 2*e) -
a^2)*cos(12*f*x + 12*e) - 42*(35*a^2*cos(8*f*x + 8*e) - 35*a^2*cos(6*f*x + 6*e) + 21*a^2*cos(4*f*x + 4*e) - 7
*a^2*cos(2*f*x + 2*e) + a^2)*cos(10*f*x + 10*e) - 70*(35*a^2*cos(6*f*x + 6*e) - 21*a^2*cos(4*f*x + 4*e) + 7*a^
2*cos(2*f*x + 2*e) - a^2)*cos(8*f*x + 8*e) - 70*(21*a^2*cos(4*f*x + 4*e) - 7*a^2*cos(2*f*x + 2*e) + a^2)*cos(6
*f*x + 6*e) - 42*(7*a^2*cos(2*f*x + 2*e) - a^2)*cos(4*f*x + 4*e) - 14*(a^2*sin(12*f*x + 12*e) - 3*a^2*sin(10*f
*x + 10*e) + 5*a^2*sin(8*f*x + 8*e) - 5*a^2*sin(6*f*x + 6*e) + 3*a^2*sin(4*f*x + 4*e) - a^2*sin(2*f*x + 2*e))*
sin(14*f*x + 14*e) - 98*(3*a^2*sin(10*f*x + 10*e) - 5*a^2*sin(8*f*x + 8*e) + 5*a^2*sin(6*f*x + 6*e) - 3*a^2*si
n(4*f*x + 4*e) + a^2*sin(2*f*x + 2*e))*sin(12*f*x + 12*e) - 294*(5*a^2*sin(8*f*x + 8*e) - 5*a^2*sin(6*f*x + 6*
e) + 3*a^2*sin(4*f*x + 4*e) - a^2*sin(2*f*x + 2*e))*sin(10*f*x + 10*e) - 490*(5*a^2*sin(6*f*x + 6*e) - 3*a^2*s
in(4*f*x + 4*e) + a^2*sin(2*f*x + 2*e))*sin(8*f*x + 8*e) - 490*(3*a^2*sin(4*f*x + 4*e) - a^2*sin(2*f*x + 2*e))
*sin(6*f*x + 6*e))*f)

Giac [F(-2)]

Exception generated. \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(cot(f*x+e)^8/(a-a*sin(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m operator + Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 30.49 (sec) , antiderivative size = 589, normalized size of antiderivative = 5.12 \[ \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,16{}\mathrm {i}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^2\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,464{}\mathrm {i}}{15\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^3\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,3072{}\mathrm {i}}{35\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^4\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,4736{}\mathrm {i}}{35\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^5\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,768{}\mathrm {i}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^6\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,256{}\mathrm {i}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^7\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )} \]

[In]

int(cot(e + f*x)^8/(a - a*sin(e + f*x)^2)^(3/2),x)

[Out]

(exp(e*3i + f*x*3i)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*16i)/(3*a^2*f*(e
xp(e*2i + f*x*2i) - 1)^2*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i))) + (exp(e*3i + f*x*3i)*(a - a*((exp(- e*1i
- f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*464i)/(15*a^2*f*(exp(e*2i + f*x*2i) - 1)^3*(exp(e*1i + f
*x*1i) + exp(e*3i + f*x*3i))) + (exp(e*3i + f*x*3i)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*
1i)/2)^2)^(1/2)*3072i)/(35*a^2*f*(exp(e*2i + f*x*2i) - 1)^4*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i))) + (exp(
e*3i + f*x*3i)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*4736i)/(35*a^2*f*(exp
(e*2i + f*x*2i) - 1)^5*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i))) + (exp(e*3i + f*x*3i)*(a - a*((exp(- e*1i -
f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2)*768i)/(7*a^2*f*(exp(e*2i + f*x*2i) - 1)^6*(exp(e*1i + f*x*
1i) + exp(e*3i + f*x*3i))) + (exp(e*3i + f*x*3i)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)
/2)^2)^(1/2)*256i)/(7*a^2*f*(exp(e*2i + f*x*2i) - 1)^7*(exp(e*1i + f*x*1i) + exp(e*3i + f*x*3i)))