3.94 \(\int \cot ^4(e+f x) \sqrt{a+a \sin (e+f x)} \, dx\)

Optimal. Leaf size=163 \[ -\frac{2 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}+\frac{11 a \cot (e+f x)}{8 f \sqrt{a \sin (e+f x)+a}}+\frac{11 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{8 f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{a \cot (e+f x) \csc (e+f x)}{12 f \sqrt{a \sin (e+f x)+a}} \]

[Out]

(11*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (2*a*Cos[e + f*x])/(f*Sqrt[a + a
*Sin[e + f*x]]) + (11*a*Cot[e + f*x])/(8*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cot[e + f*x]*Csc[e + f*x])/(12*f*Sqr
t[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(3*f)

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Rubi [A]  time = 0.37672, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2718, 2646, 3044, 2980, 2772, 2773, 206} \[ -\frac{2 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}+\frac{11 a \cot (e+f x)}{8 f \sqrt{a \sin (e+f x)+a}}+\frac{11 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{8 f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{a \cot (e+f x) \csc (e+f x)}{12 f \sqrt{a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4*Sqrt[a + a*Sin[e + f*x]],x]

[Out]

(11*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/(8*f) - (2*a*Cos[e + f*x])/(f*Sqrt[a + a
*Sin[e + f*x]]) + (11*a*Cot[e + f*x])/(8*f*Sqrt[a + a*Sin[e + f*x]]) - (a*Cot[e + f*x]*Csc[e + f*x])/(12*f*Sqr
t[a + a*Sin[e + f*x]]) - (Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(3*f)

Rule 2718

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Int[(a + b*Sin[e + f*x
])^m, x] + Int[((a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2))/Sin[e + f*x]^4, x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] &&  !LtQ[m, -1]

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3044

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[
e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^
m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n +
2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b
*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0
])

Rule 2980

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^2*(B*c - A*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n + 1)
*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A
, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]

Rule 2772

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[((b*c - a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]]), x]
+ Dist[((2*n + 3)*(b*c - a*d))/(2*b*(n + 1)*(c^2 - d^2)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n
+ 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &
& LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]

Rule 2773

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(-2*
b)/f, Subst[Int[1/(b*c + a*d - d*x^2), x], x, (b*Cos[e + f*x])/Sqrt[a + b*Sin[e + f*x]]], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \cot ^4(e+f x) \sqrt{a+a \sin (e+f x)} \, dx &=\int \sqrt{a+a \sin (e+f x)} \, dx+\int \csc ^4(e+f x) \sqrt{a+a \sin (e+f x)} \left (1-2 \sin ^2(e+f x)\right ) \, dx\\ &=-\frac{2 a \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{\int \csc ^3(e+f x) \left (\frac{a}{2}-\frac{9}{2} a \sin (e+f x)\right ) \sqrt{a+a \sin (e+f x)} \, dx}{3 a}\\ &=-\frac{2 a \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}-\frac{a \cot (e+f x) \csc (e+f x)}{12 f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{11}{8} \int \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{2 a \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}+\frac{11 a \cot (e+f x)}{8 f \sqrt{a+a \sin (e+f x)}}-\frac{a \cot (e+f x) \csc (e+f x)}{12 f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{11}{16} \int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{2 a \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}+\frac{11 a \cot (e+f x)}{8 f \sqrt{a+a \sin (e+f x)}}-\frac{a \cot (e+f x) \csc (e+f x)}{12 f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{(11 a) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{8 f}\\ &=\frac{11 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{8 f}-\frac{2 a \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}+\frac{11 a \cot (e+f x)}{8 f \sqrt{a+a \sin (e+f x)}}-\frac{a \cot (e+f x) \csc (e+f x)}{12 f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{3 f}\\ \end{align*}

Mathematica [A]  time = 1.56776, size = 309, normalized size = 1.9 \[ \frac{\csc ^{10}\left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sin (e+f x)+1)} \left (-252 \sin \left (\frac{1}{2} (e+f x)\right )-250 \sin \left (\frac{3}{2} (e+f x)\right )+114 \sin \left (\frac{5}{2} (e+f x)\right )+48 \sin \left (\frac{7}{2} (e+f x)\right )+252 \cos \left (\frac{1}{2} (e+f x)\right )-250 \cos \left (\frac{3}{2} (e+f x)\right )-114 \cos \left (\frac{5}{2} (e+f x)\right )+48 \cos \left (\frac{7}{2} (e+f x)\right )+99 \sin (e+f x) \log \left (-\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )+1\right )-99 \sin (e+f x) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )-\cos \left (\frac{1}{2} (e+f x)\right )+1\right )-33 \sin (3 (e+f x)) \log \left (-\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )+1\right )+33 \sin (3 (e+f x)) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )-\cos \left (\frac{1}{2} (e+f x)\right )+1\right )\right )}{24 f \left (\cot \left (\frac{1}{2} (e+f x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (e+f x)\right )-\sec ^2\left (\frac{1}{4} (e+f x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^4*Sqrt[a + a*Sin[e + f*x]],x]

[Out]

(Csc[(e + f*x)/2]^10*Sqrt[a*(1 + Sin[e + f*x])]*(252*Cos[(e + f*x)/2] - 250*Cos[(3*(e + f*x))/2] - 114*Cos[(5*
(e + f*x))/2] + 48*Cos[(7*(e + f*x))/2] - 252*Sin[(e + f*x)/2] + 99*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2
]]*Sin[e + f*x] - 99*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] - 250*Sin[(3*(e + f*x))/2] + 11
4*Sin[(5*(e + f*x))/2] - 33*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] + 33*Log[1 - Cos[(e
+ f*x)/2] + Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] + 48*Sin[(7*(e + f*x))/2]))/(24*f*(1 + Cot[(e + f*x)/2])*(Csc[(
e + f*x)/4]^2 - Sec[(e + f*x)/4]^2)^3)

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Maple [A]  time = 0.701, size = 170, normalized size = 1. \begin{align*}{\frac{1+\sin \left ( fx+e \right ) }{24\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) f}\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( -48\,\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }{a}^{7/2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}+33\, \left ( -a \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{5/2}{a}^{3/2}+33\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }}{\sqrt{a}}} \right ){a}^{4} \left ( \sin \left ( fx+e \right ) \right ) ^{3}-56\, \left ( -a \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{3/2}{a}^{5/2}+15\,\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) }{a}^{7/2} \right ){a}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^4*(a+a*sin(f*x+e))^(1/2),x)

[Out]

1/24*(1+sin(f*x+e))*(-a*(-1+sin(f*x+e)))^(1/2)/a^(7/2)*(-48*(-a*(-1+sin(f*x+e)))^(1/2)*a^(7/2)*sin(f*x+e)^3+33
*(-a*(-1+sin(f*x+e)))^(5/2)*a^(3/2)+33*arctanh((-a*(-1+sin(f*x+e)))^(1/2)/a^(1/2))*a^4*sin(f*x+e)^3-56*(-a*(-1
+sin(f*x+e)))^(3/2)*a^(5/2)+15*(-a*(-1+sin(f*x+e)))^(1/2)*a^(7/2))/sin(f*x+e)^3/cos(f*x+e)/(a+a*sin(f*x+e))^(1
/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (f x + e\right ) + a} \cot \left (f x + e\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(f*x + e) + a)*cot(f*x + e)^4, x)

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Fricas [B]  time = 1.81584, size = 1025, normalized size = 6.29 \begin{align*} \frac{33 \,{\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} -{\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right ) + 1\right )} \sqrt{a} \log \left (\frac{a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} + 4 \,{\left (\cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{a} - 9 \, a \cos \left (f x + e\right ) +{\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + 4 \,{\left (48 \, \cos \left (f x + e\right )^{4} - 33 \, \cos \left (f x + e\right )^{3} - 139 \, \cos \left (f x + e\right )^{2} +{\left (48 \, \cos \left (f x + e\right )^{3} + 81 \, \cos \left (f x + e\right )^{2} - 58 \, \cos \left (f x + e\right ) - 83\right )} \sin \left (f x + e\right ) + 25 \, \cos \left (f x + e\right ) + 83\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{96 \,{\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} -{\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2} - f \cos \left (f x + e\right ) - f\right )} \sin \left (f x + e\right ) + f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

1/96*(33*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 - (cos(f*x + e)^3 + cos(f*x + e)^2 - cos(f*x + e) - 1)*sin(f*x + e
) + 1)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e
) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x
+ e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e
) - 1)) + 4*(48*cos(f*x + e)^4 - 33*cos(f*x + e)^3 - 139*cos(f*x + e)^2 + (48*cos(f*x + e)^3 + 81*cos(f*x + e)
^2 - 58*cos(f*x + e) - 83)*sin(f*x + e) + 25*cos(f*x + e) + 83)*sqrt(a*sin(f*x + e) + a))/(f*cos(f*x + e)^4 -
2*f*cos(f*x + e)^2 - (f*cos(f*x + e)^3 + f*cos(f*x + e)^2 - f*cos(f*x + e) - f)*sin(f*x + e) + f)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**4*(a+a*sin(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.1154, size = 950, normalized size = 5.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

-1/48*(66*a*arctan(-(sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))/sqrt(-a))*sgn(tan(1/2*
f*x + 1/2*e) + 1)/sqrt(-a) - 33*sqrt(a)*log(abs(-sqrt(a)*tan(1/2*f*x + 1/2*e) + sqrt(a*tan(1/2*f*x + 1/2*e)^2
+ a)))*sgn(tan(1/2*f*x + 1/2*e) + 1) - (330*sqrt(2)*a*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a)) - 165*sqrt(
2)*sqrt(-a)*sqrt(a)*log(sqrt(2)*sqrt(a) + sqrt(a)) + 462*a*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a)) - 231*
sqrt(-a)*sqrt(a)*log(sqrt(2)*sqrt(a) + sqrt(a)) + 992*sqrt(2)*sqrt(-a)*sqrt(a) + 1422*sqrt(-a)*sqrt(a))*sgn(ta
n(1/2*f*x + 1/2*e) + 1)/(5*sqrt(2)*sqrt(-a) + 7*sqrt(-a)) + (130*a*sgn(tan(1/2*f*x + 1/2*e) + 1) - (99*a*sgn(t
an(1/2*f*x + 1/2*e) + 1) - (32*a*sgn(tan(1/2*f*x + 1/2*e) + 1) - (2*a*sgn(tan(1/2*f*x + 1/2*e) + 1)*tan(1/2*f*
x + 1/2*e) + 3*a*sgn(tan(1/2*f*x + 1/2*e) + 1))*tan(1/2*f*x + 1/2*e))*tan(1/2*f*x + 1/2*e))*tan(1/2*f*x + 1/2*
e))/sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a) - 2*(3*(sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 +
a))^5*a*sgn(tan(1/2*f*x + 1/2*e) + 1) - 30*(sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))
^4*a^(3/2)*sgn(tan(1/2*f*x + 1/2*e) + 1) + 72*(sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 +
a))^2*a^(5/2)*sgn(tan(1/2*f*x + 1/2*e) + 1) - 3*(sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2
+ a))*a^3*sgn(tan(1/2*f*x + 1/2*e) + 1) - 34*a^(7/2)*sgn(tan(1/2*f*x + 1/2*e) + 1))/((sqrt(a)*tan(1/2*f*x + 1/
2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))^2 - a)^3)/f