\(\int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx\) [467]

Optimal result

Integrand size = 26, antiderivative size = 124 \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=-\frac {3 \sqrt {a \cos ^2(e+f x)} \csc (e+f x) \sec (e+f x)}{f}+\frac {\sqrt {a \cos ^2(e+f x)} \csc ^3(e+f x) \sec (e+f x)}{f}-\frac {\sqrt {a \cos ^2(e+f x)} \csc ^5(e+f x) \sec (e+f x)}{5 f}-\frac {\sqrt {a \cos ^2(e+f x)} \tan (e+f x)}{f} \]

[Out]

-3*csc(f*x+e)*sec(f*x+e)*(a*cos(f*x+e)^2)^(1/2)/f+csc(f*x+e)^3*sec(f*x+e)*(a*cos(f*x+e)^2)^(1/2)/f-1/5*csc(f*x
+e)^5*sec(f*x+e)*(a*cos(f*x+e)^2)^(1/2)/f-(a*cos(f*x+e)^2)^(1/2)*tan(f*x+e)/f

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 124, 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, 2670, 276} \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=-\frac {\tan (e+f x) \sqrt {a \cos ^2(e+f x)}}{f}-\frac {\csc ^5(e+f x) \sec (e+f x) \sqrt {a \cos ^2(e+f x)}}{5 f}+\frac {\csc ^3(e+f x) \sec (e+f x) \sqrt {a \cos ^2(e+f x)}}{f}-\frac {3 \csc (e+f x) \sec (e+f x) \sqrt {a \cos ^2(e+f x)}}{f} \]

[In]

Int[Cot[e + f*x]^6*Sqrt[a - a*Sin[e + f*x]^2],x]

[Out]

(-3*Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]*Sec[e + f*x])/f + (Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]^3*Sec[e + f*x])
/f - (Sqrt[a*Cos[e + f*x]^2]*Csc[e + f*x]^5*Sec[e + f*x])/(5*f) - (Sqrt[a*Cos[e + f*x]^2]*Tan[e + f*x])/f

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 2670

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

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 \sqrt {a \cos ^2(e+f x)} \cot ^6(e+f x) \, dx \\ & = \left (\sqrt {a \cos ^2(e+f x)} \sec (e+f x)\right ) \int \cos (e+f x) \cot ^6(e+f x) \, dx \\ & = -\frac {\left (\sqrt {a \cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6} \, dx,x,-\sin (e+f x)\right )}{f} \\ & = -\frac {\left (\sqrt {a \cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \left (-1+\frac {1}{x^6}-\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,-\sin (e+f x)\right )}{f} \\ & = -\frac {3 \sqrt {a \cos ^2(e+f x)} \csc (e+f x) \sec (e+f x)}{f}+\frac {\sqrt {a \cos ^2(e+f x)} \csc ^3(e+f x) \sec (e+f x)}{f}-\frac {\sqrt {a \cos ^2(e+f x)} \csc ^5(e+f x) \sec (e+f x)}{5 f}-\frac {\sqrt {a \cos ^2(e+f x)} \tan (e+f x)}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.54 \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\frac {\sqrt {a \cos ^2(e+f x)} (-182+235 \cos (2 (e+f x))-90 \cos (4 (e+f x))+5 \cos (6 (e+f x))) \csc ^5(e+f x) \sec (e+f x)}{160 f} \]

[In]

Integrate[Cot[e + f*x]^6*Sqrt[a - a*Sin[e + f*x]^2],x]

[Out]

(Sqrt[a*Cos[e + f*x]^2]*(-182 + 235*Cos[2*(e + f*x)] - 90*Cos[4*(e + f*x)] + 5*Cos[6*(e + f*x)])*Csc[e + f*x]^
5*Sec[e + f*x])/(160*f)

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.69

method	result	size
default	\(\frac {\cos \left (f x +e \right ) a \left (-5 \left (\sin ^{6}\left (f x +e \right )\right )-15 \left (\sin ^{4}\left (f x +e \right )\right )+5 \left (\sin ^{2}\left (f x +e \right )\right )-1\right )}{5 \left (\cos \left (f x +e \right )-1\right )^{2} \left (1+\cos \left (f x +e \right )\right )^{2} \sin \left (f x +e \right ) \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\)	\(85\)
risch	\(\frac {i \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, \left (5 \,{\mathrm e}^{12 i \left (f x +e \right )}-90 \,{\mathrm e}^{10 i \left (f x +e \right )}+235 \,{\mathrm e}^{8 i \left (f x +e \right )}-364 \,{\mathrm e}^{6 i \left (f x +e \right )}+235 \,{\mathrm e}^{4 i \left (f x +e \right )}-90 \,{\mathrm e}^{2 i \left (f x +e \right )}+5\right )}{10 \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{5} f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\)	\(127\)

[In]

int(cot(f*x+e)^6*(a-a*sin(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/5*cos(f*x+e)*a*(-5*sin(f*x+e)^6-15*sin(f*x+e)^4+5*sin(f*x+e)^2-1)/(cos(f*x+e)-1)^2/(1+cos(f*x+e))^2/sin(f*x+
e)/(a*cos(f*x+e)^2)^(1/2)/f

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.69 \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\frac {{\left (5 \, \cos \left (f x + e\right )^{6} - 30 \, \cos \left (f x + e\right )^{4} + 40 \, \cos \left (f x + e\right )^{2} - 16\right )} \sqrt {a \cos \left (f x + e\right )^{2}}}{5 \, {\left (f \cos \left (f x + e\right )^{5} - 2 \, f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]

[In]

integrate(cot(f*x+e)^6*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

1/5*(5*cos(f*x + e)^6 - 30*cos(f*x + e)^4 + 40*cos(f*x + e)^2 - 16)*sqrt(a*cos(f*x + e)^2)/((f*cos(f*x + e)^5
- 2*f*cos(f*x + e)^3 + f*cos(f*x + e))*sin(f*x + e))

Sympy [F]

\[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\int \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )} \cot ^{6}{\left (e + f x \right )}\, dx \]

[In]

integrate(cot(f*x+e)**6*(a-a*sin(f*x+e)**2)**(1/2),x)

[Out]

Integral(sqrt(-a*(sin(e + f*x) - 1)*(sin(e + f*x) + 1))*cot(e + f*x)**6, x)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.55 \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=-\frac {16 \, \sqrt {a} \tan \left (f x + e\right )^{6} + 8 \, \sqrt {a} \tan \left (f x + e\right )^{4} - 2 \, \sqrt {a} \tan \left (f x + e\right )^{2} + \sqrt {a}}{5 \, \sqrt {\tan \left (f x + e\right )^{2} + 1} f \tan \left (f x + e\right )^{5}} \]

[In]

integrate(cot(f*x+e)^6*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/5*(16*sqrt(a)*tan(f*x + e)^6 + 8*sqrt(a)*tan(f*x + e)^4 - 2*sqrt(a)*tan(f*x + e)^2 + sqrt(a))/(sqrt(tan(f*x
+ e)^2 + 1)*f*tan(f*x + e)^5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8626 vs. \(2 (114) = 228\).

Time = 4.03 (sec) , antiderivative size = 8626, normalized size of antiderivative = 69.56 \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\text {Too large to display} \]

[In]

integrate(cot(f*x+e)^6*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

1/160*(320*(sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1
/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x
)*tan(1/2*e)^2 - sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*
tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/
2*f*x) - 2*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/
2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*e))/
((tan(1/2*f*x)^2 + 1)*(tan(1/2*e)^2 + 1)) - (5*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(
1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1
/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^9*tan(1/2*e)^16 + 10*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*
f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^
4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^8*tan(1/2*e)^17 + 10*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 -
4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 -
tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^7*tan(1/2*e)^18 + 5*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(
1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1
/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^6*tan(1/2*e)^19 + sqrt(a)*sgn(tan(1/2*f*x
)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*
x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^5*tan(1/2*e)^20 - 40*sqrt(a)*sgn(
tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4
*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^9*tan(1/2*e)^14 - 160*
sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*ta
n(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^8*tan(1/2*
e)^15 - 200*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1
/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x
)^7*tan(1/2*e)^16 - 125*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)
^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)
*tan(1/2*f*x)^6*tan(1/2*e)^17 - 40*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - t
an(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(
1/2*e) + 1)*tan(1/2*f*x)^5*tan(1/2*e)^18 - 5*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/
2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2
*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^4*tan(1/2*e)^19 + 145*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f
*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4
- 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^9*tan(1/2*e)^12 + 860*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 -
4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 -
tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^8*tan(1/2*e)^13 + 1820*sqrt(a)*sgn(tan(1/2*f*x)^4*t
an(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*ta
n(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^7*tan(1/2*e)^14 + 1610*sqrt(a)*sgn(tan
(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*ta
n(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^6*tan(1/2*e)^15 + 665*sqr
t(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1
/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^5*tan(1/2*e)^
16 + 125*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*
f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^4
*tan(1/2*e)^17 + 10*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 -
4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan
(1/2*f*x)^3*tan(1/2*e)^18 - 870*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(
1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2
*e) + 1)*tan(1/2*f*x)^8*tan(1/2*e)^11 - 4360*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/
2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2
*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^7*tan(1/2*e)^12 - 7970*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*
f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^
4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^6*tan(1/2*e)^13 - 6160*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4
- 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3
- tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^5*tan(1/2*e)^14 - 1990*sqrt(a)*sgn(tan(1/2*f*x)^4
*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*
tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^4*tan(1/2*e)^15 - 200*sqrt(a)*sgn(ta
n(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*t
an(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^3*tan(1/2*e)^16 - 10*sqr
t(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1
/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^2*tan(1/2*e)^
17 - 145*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*
f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^9
*tan(1/2*e)^8 - 870*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 -
4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan
(1/2*f*x)^8*tan(1/2*e)^9 + 7440*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(
1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2
*e) + 1)*tan(1/2*f*x)^6*tan(1/2*e)^11 + 15760*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1
/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/
2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^5*tan(1/2*e)^12 + 12230*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/
2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e
)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^4*tan(1/2*e)^13 + 3580*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)
^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^
3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^3*tan(1/2*e)^14 + 190*sqrt(a)*sgn(tan(1/2*f*x)^
4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)
*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^2*tan(1/2*e)^15 + 5*sqrt(a)*sgn(tan
(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*ta
n(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)*tan(1/2*e)^16 + 40*sqrt(a
)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*
e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^9*tan(1/2*e)^6 +
860*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)
^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^8*tan
(1/2*e)^7 + 4360*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*
tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/
2*f*x)^7*tan(1/2*e)^8 + 7440*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2
*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e)
+ 1)*tan(1/2*f*x)^6*tan(1/2*e)^9 - 12920*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e
)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*
x)*tan(1/2*e) + 1)*tan(1/2*f*x)^4*tan(1/2*e)^11 - 12360*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*
x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4
- 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^3*tan(1/2*e)^12 - 3710*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 -
4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 -
tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^2*tan(1/2*e)^13 - 120*sqrt(a)*sgn(tan(1/2*f*x)^4*ta
n(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan
(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)*tan(1/2*e)^14 - 2*sqrt(a)*sgn(tan(1/2*f
*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*
f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*e)^15 - 5*sqrt(a)*sgn(tan(1/2*f*x)^4
*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*
tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^9*tan(1/2*e)^4 - 160*sqrt(a)*sgn(tan
(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*ta
n(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^8*tan(1/2*e)^5 - 1820*sqr
t(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1
/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^7*tan(1/2*e)^
6 - 7970*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*
f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^6
*tan(1/2*e)^7 - 15760*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4
- 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*t
an(1/2*f*x)^5*tan(1/2*e)^8 - 12920*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - t
an(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(
1/2*e) + 1)*tan(1/2*f*x)^4*tan(1/2*e)^9 + 5130*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(
1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1
/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^2*tan(1/2*e)^11 + 1905*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/
2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e
)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)*tan(1/2*e)^12 + 30*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 -
4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 -
tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*e)^13 + 10*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*t
an(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(
1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^8*tan(1/2*e)^3 + 200*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2
*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*
e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^7*tan(1/2*e)^4 + 1610*sqrt(a)*sgn(tan(1/2*f*
x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f
*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^6*tan(1/2*e)^5 + 6160*sqrt(a)*sg
n(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) -
4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^5*tan(1/2*e)^6 + 122
30*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3
*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^4*tan(1
/2*e)^7 + 12360*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*t
an(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2
*f*x)^3*tan(1/2*e)^8 + 5130*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*
f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e)
+ 1)*tan(1/2*f*x)^2*tan(1/2*e)^9 - 380*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3
- tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*
tan(1/2*e) + 1)*tan(1/2*e)^11 - 10*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - t
an(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(
1/2*e) + 1)*tan(1/2*f*x)^7*tan(1/2*e)^2 - 125*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1
/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/
2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^6*tan(1/2*e)^3 - 665*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f
*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4
- 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^5*tan(1/2*e)^4 - 1990*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 -
4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 -
tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^4*tan(1/2*e)^5 - 3580*sqrt(a)*sgn(tan(1/2*f*x)^4*ta
n(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan
(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^3*tan(1/2*e)^6 - 3710*sqrt(a)*sgn(tan(1
/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(
1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^2*tan(1/2*e)^7 - 1905*sqrt(
a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2
*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)*tan(1/2*e)^8 -
380*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^
3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*e)^9 + 5*sq
rt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(
1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^6*tan(1/2*e)
+ 40*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x
)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^5*ta
n(1/2*e)^2 + 125*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*
tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/
2*f*x)^4*tan(1/2*e)^3 + 200*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*
f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e)
+ 1)*tan(1/2*f*x)^3*tan(1/2*e)^4 + 190*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3
- tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*
tan(1/2*e) + 1)*tan(1/2*f*x)^2*tan(1/2*e)^5 + 120*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*t
an(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*ta
n(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)*tan(1/2*e)^6 + 30*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*
f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^
4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*e)^7 - sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*t
an(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*ta
n(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^5 - 5*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1
/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/
2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^4*tan(1/2*e) - 10*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*tan(1/2*f*x)
^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1/2*e)^4 -
4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^3*tan(1/2*e)^2 - 10*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)^4 - 4*ta
n(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^3 - tan(1
/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)^2*tan(1/2*e)^3 - 5*sqrt(a)*sgn(tan(1/2*f*x)^4*tan(1/2*e)
^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(1/2*e)^
3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*f*x)*tan(1/2*e)^4 - 2*sqrt(a)*sgn(tan(1/2*f*x)^4*tan
(1/2*e)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e)^3 - tan(1/2*f*x)^4 - 4*tan(1/2*f*x)^3*tan(1/2*e) - 4*tan(1/2*f*x)*tan(
1/2*e)^3 - tan(1/2*e)^4 - 4*tan(1/2*f*x)*tan(1/2*e) + 1)*tan(1/2*e)^5)/((tan(1/2*f*x)^2*tan(1/2*e) + tan(1/2*f
*x)*tan(1/2*e)^2 - tan(1/2*f*x) - tan(1/2*e))^5*tan(1/2*e)^5))/f

Mupad [B] (verification not implemented)

Time = 24.39 (sec) , antiderivative size = 555, normalized size of antiderivative = 4.48 \[ \int \cot ^6(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=-\frac {\left (\frac {1{}\mathrm {i}}{f}-\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{f}\right )\,\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}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}-\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}\,12{}\mathrm {i}}{f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )\,\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}\,16{}\mathrm {i}}{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}\,144{}\mathrm {i}}{5\,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}\,128{}\mathrm {i}}{5\,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}\,64{}\mathrm {i}}{5\,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 )} \]

[In]

int(cot(e + f*x)^6*(a - a*sin(e + f*x)^2)^(1/2),x)

[Out]

- ((1i/f - (exp(e*2i + f*x*2i)*1i)/f)*(a - a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)^2)^(1/2
))/(exp(e*2i + f*x*2i) + 1) - (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)*12i)/(f*(exp(e*2i + f*x*2i) - 1)*(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)*16i)/(f*(exp(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)*144i)/(5*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)*128i)/(5*f*(e
xp(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)*64i)/(5*f*(exp(e*2i + f*x*2i) - 1)^5*(exp(e*1i + f*x*1i)
+ exp(e*3i + f*x*3i)))