3.18 \(\int \frac{1}{(a+a \csc (x))^{5/2}} \, dx\)
Optimal. Leaf size=100 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right )}{a^{5/2}}+\frac{43 \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{2} \sqrt{a \csc (x)+a}}\right )}{16 \sqrt{2} a^{5/2}}+\frac{11 \cot (x)}{16 a (a \csc (x)+a)^{3/2}}+\frac{\cot (x)}{4 (a \csc (x)+a)^{5/2}} \]
[Out]
(-2*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/a^(5/2) + (43*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc
[x]])])/(16*Sqrt[2]*a^(5/2)) + Cot[x]/(4*(a + a*Csc[x])^(5/2)) + (11*Cot[x])/(16*a*(a + a*Csc[x])^(3/2))
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Rubi [A] time = 0.148174, antiderivative size = 100, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used =
{3777, 3922, 3920, 3774, 203, 3795} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right )}{a^{5/2}}+\frac{43 \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{2} \sqrt{a \csc (x)+a}}\right )}{16 \sqrt{2} a^{5/2}}+\frac{11 \cot (x)}{16 a (a \csc (x)+a)^{3/2}}+\frac{\cot (x)}{4 (a \csc (x)+a)^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Csc[x])^(-5/2),x]
[Out]
(-2*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/a^(5/2) + (43*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc
[x]])])/(16*Sqrt[2]*a^(5/2)) + Cot[x]/(4*(a + a*Csc[x])^(5/2)) + (11*Cot[x])/(16*a*(a + a*Csc[x])^(3/2))
Rule 3777
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Simp[(Cot[c + d*x]*(a + b*Csc[c + d*x])^n)/(d*(
2*n + 1)), x] + Dist[1/(a^2*(2*n + 1)), Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*x]
), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]
Rule 3922
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> -Simp[((b
*c - a*d)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(b*f*(2*m + 1)), x] + Dist[1/(a^2*(2*m + 1)), Int[(a + b*Csc[e
+ f*x])^(m + 1)*Simp[a*c*(2*m + 1) - (b*c - a*d)*(m + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f},
x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && EqQ[a^2 - b^2, 0] && IntegerQ[2*m]
Rule 3920
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c/a,
Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Rule 3774
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(a + x^2), x], x, (b*C
ot[c + d*x])/Sqrt[a + b*Csc[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
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 3795
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2/f, Subst[Int[1/(2
*a + x^2), x], x, (b*Cot[e + f*x])/Sqrt[a + b*Csc[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0
]
Rubi steps
\begin{align*} \int \frac{1}{(a+a \csc (x))^{5/2}} \, dx &=\frac{\cot (x)}{4 (a+a \csc (x))^{5/2}}-\frac{\int \frac{-4 a+\frac{3}{2} a \csc (x)}{(a+a \csc (x))^{3/2}} \, dx}{4 a^2}\\ &=\frac{\cot (x)}{4 (a+a \csc (x))^{5/2}}+\frac{11 \cot (x)}{16 a (a+a \csc (x))^{3/2}}+\frac{\int \frac{8 a^2-\frac{11}{4} a^2 \csc (x)}{\sqrt{a+a \csc (x)}} \, dx}{8 a^4}\\ &=\frac{\cot (x)}{4 (a+a \csc (x))^{5/2}}+\frac{11 \cot (x)}{16 a (a+a \csc (x))^{3/2}}+\frac{\int \sqrt{a+a \csc (x)} \, dx}{a^3}-\frac{43 \int \frac{\csc (x)}{\sqrt{a+a \csc (x)}} \, dx}{32 a^2}\\ &=\frac{\cot (x)}{4 (a+a \csc (x))^{5/2}}+\frac{11 \cot (x)}{16 a (a+a \csc (x))^{3/2}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,\frac{a \cot (x)}{\sqrt{a+a \csc (x)}}\right )}{a^2}+\frac{43 \operatorname{Subst}\left (\int \frac{1}{2 a+x^2} \, dx,x,\frac{a \cot (x)}{\sqrt{a+a \csc (x)}}\right )}{16 a^2}\\ &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{a+a \csc (x)}}\right )}{a^{5/2}}+\frac{43 \tan ^{-1}\left (\frac{\sqrt{a} \cot (x)}{\sqrt{2} \sqrt{a+a \csc (x)}}\right )}{16 \sqrt{2} a^{5/2}}+\frac{\cot (x)}{4 (a+a \csc (x))^{5/2}}+\frac{11 \cot (x)}{16 a (a+a \csc (x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.503013, size = 139, normalized size = 1.39 \[ \frac{\csc ^2(x) \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \left (8 \sin (x)+15 \cos (2 x)-64 \sqrt{\csc (x)-1} \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^4 \tan ^{-1}\left (\sqrt{\csc (x)-1}\right )+43 \sqrt{2} \sqrt{\csc (x)-1} \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )^4 \tan ^{-1}\left (\frac{\sqrt{\csc (x)-1}}{\sqrt{2}}\right )+7\right )}{32 (a (\csc (x)+1))^{5/2} \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Csc[x])^(-5/2),x]
[Out]
(Csc[x]^2*(Cos[x/2] + Sin[x/2])*(7 + 15*Cos[2*x] - 64*ArcTan[Sqrt[-1 + Csc[x]]]*Sqrt[-1 + Csc[x]]*(Cos[x/2] +
Sin[x/2])^4 + 43*Sqrt[2]*ArcTan[Sqrt[-1 + Csc[x]]/Sqrt[2]]*Sqrt[-1 + Csc[x]]*(Cos[x/2] + Sin[x/2])^4 + 8*Sin[x
]))/(32*(a*(1 + Csc[x]))^(5/2)*(Cos[x/2] - Sin[x/2]))
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Maple [B] time = 0.148, size = 1961, normalized size = 19.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+a*csc(x))^(5/2),x)
[Out]
1/128*2^(1/2)*(-1+cos(x))^2*(11*2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+38*sin(x)*cos(x)*(-(-1+cos(x))/sin(
x))^(3/2)*2^(1/2)+22*sin(x)*cos(x)*(-(-1+cos(x))/sin(x))^(1/2)*2^(1/2)+344*sin(x)*cos(x)*arctan((-(-1+cos(x))/
sin(x))^(1/2))*2^(1/2)+128*sin(x)*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-
(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1))+128*sin(x)*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+
sin(x)-cos(x)+1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1))+512*sin(x)*arctan(2^(1/2)*(-(-1
+cos(x))/sin(x))^(1/2)+1)-256*cos(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)-64*cos(x)*ln(-(2^(1/2)*(-(-
1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1))-
64*cos(x)*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2
)*sin(x)-sin(x)+cos(x)-1))-256*cos(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)+19*(-(-1+cos(x))/sin(x))^(
3/2)*2^(1/2)+11*cos(x)^2*sin(x)*(-(-1+cos(x))/sin(x))^(1/2)*2^(1/2)+172*cos(x)^2*sin(x)*2^(1/2)*arctan((-(-1+c
os(x))/sin(x))^(1/2))-11*cos(x)^2*sin(x)*(-(-1+cos(x))/sin(x))^(7/2)*2^(1/2)-19*cos(x)^2*sin(x)*(-(-1+cos(x))/
sin(x))^(5/2)*2^(1/2)+19*cos(x)^2*sin(x)*(-(-1+cos(x))/sin(x))^(3/2)*2^(1/2)-38*cos(x)*sin(x)*(-(-1+cos(x))/si
n(x))^(5/2)*2^(1/2)-22*cos(x)*sin(x)*(-(-1+cos(x))/sin(x))^(7/2)*2^(1/2)+128*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x)
)^(1/2)*sin(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1))+128*ln(-(2^(1/2)
*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)
-1))+512*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)+512*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)-384*c
os(x)^2*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)-96*cos(x)^2*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin
(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1))-96*cos(x)^2*ln(-(2^(1/2)*(-
(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1)
)-384*cos(x)^2*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)+512*sin(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(
1/2)-1)+32*cos(x)^3*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-(-1+cos(x))/si
n(x))^(1/2)*sin(x)+sin(x)-cos(x)+1))+32*cos(x)^3*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)
+1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1))+128*cos(x)^3*arctan(2^(1/2)*(-(-1+cos(x))/si
n(x))^(1/2)+1)+128*cos(x)^3*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)-19*(-(-1+cos(x))/sin(x))^(5/2)*2^(1/
2)-11*(-(-1+cos(x))/sin(x))^(7/2)*2^(1/2)+516*cos(x)^2*arctan((-(-1+cos(x))/sin(x))^(1/2))*2^(1/2)-688*sin(x)*
arctan((-(-1+cos(x))/sin(x))^(1/2))*2^(1/2)+344*cos(x)*arctan((-(-1+cos(x))/sin(x))^(1/2))*2^(1/2)-172*cos(x)^
3*2^(1/2)*arctan((-(-1+cos(x))/sin(x))^(1/2))-11*cos(x)^3*(-(-1+cos(x))/sin(x))^(1/2)*2^(1/2)+19*cos(x)*(-(-1+
cos(x))/sin(x))^(3/2)*2^(1/2)+11*cos(x)*(-(-1+cos(x))/sin(x))^(1/2)*2^(1/2)-32*cos(x)^2*sin(x)*ln(-(2^(1/2)*(-
(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1)
)-32*cos(x)^2*sin(x)*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1)/(2^(1/2)*(-(-1+cos(x))/s
in(x))^(1/2)*sin(x)-sin(x)+cos(x)-1))-128*cos(x)^2*sin(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)+1)-128*co
s(x)^2*sin(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-1)+11*cos(x)^3*(-(-1+cos(x))/sin(x))^(7/2)*2^(1/2)+11
*cos(x)^2*(-(-1+cos(x))/sin(x))^(7/2)*2^(1/2)+19*cos(x)^3*(-(-1+cos(x))/sin(x))^(5/2)*2^(1/2)-11*cos(x)*(-(-1+
cos(x))/sin(x))^(7/2)*2^(1/2)-11*sin(x)*(-(-1+cos(x))/sin(x))^(7/2)*2^(1/2)+19*cos(x)^2*(-(-1+cos(x))/sin(x))^
(5/2)*2^(1/2)-19*cos(x)^3*(-(-1+cos(x))/sin(x))^(3/2)*2^(1/2)-19*cos(x)*(-(-1+cos(x))/sin(x))^(5/2)*2^(1/2)-19
*sin(x)*(-(-1+cos(x))/sin(x))^(5/2)*2^(1/2)+11*2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)-688*2^(1/2)*arctan((-(-1+co
s(x))/sin(x))^(1/2))-19*cos(x)^2*(-(-1+cos(x))/sin(x))^(3/2)*2^(1/2)+19*sin(x)*(-(-1+cos(x))/sin(x))^(3/2)*2^(
1/2)-11*cos(x)^2*(-(-1+cos(x))/sin(x))^(1/2)*2^(1/2)-256*sin(x)*cos(x)*arctan(2^(1/2)*(-(-1+cos(x))/sin(x))^(1
/2)-1)-64*sin(x)*cos(x)*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1)/(2^(1/2)*(-(-1+cos(x)
)/sin(x))^(1/2)*sin(x)+sin(x)-cos(x)+1))-64*sin(x)*cos(x)*ln(-(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)+sin(
x)-cos(x)+1)/(2^(1/2)*(-(-1+cos(x))/sin(x))^(1/2)*sin(x)-sin(x)+cos(x)-1))-256*sin(x)*cos(x)*arctan(2^(1/2)*(-
(-1+cos(x))/sin(x))^(1/2)+1))/(a*(sin(x)+1)/sin(x))^(5/2)/sin(x)^5/(-(-1+cos(x))/sin(x))^(5/2)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \csc \left (x\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+a*csc(x))^(5/2),x, algorithm="maxima")
[Out]
integrate((a*csc(x) + a)^(-5/2), x)
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Fricas [B] time = 0.546279, size = 1677, normalized size = 16.77 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+a*csc(x))^(5/2),x, algorithm="fricas")
[Out]
[-1/32*(43*sqrt(2)*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*sqrt(-a)*log(-(sq
rt(2)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x))*sin(x) - a*cos(x))/(sin(x) + 1)) + 32*(cos(x)^3 + 3*cos(x)^2 + (cos
(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*sqrt(-a)*log((2*a*cos(x)^2 + 2*(cos(x)^2 + (cos(x) + 1)*sin(x) -
1)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x)) + a*cos(x) - (2*a*cos(x) + a)*sin(x) - a)/(cos(x) + sin(x) + 1)) - 2*(
15*cos(x)^3 + 4*cos(x)^2 - (15*cos(x)^2 + 11*cos(x) - 4)*sin(x) - 15*cos(x) - 4)*sqrt((a*sin(x) + a)/sin(x)))/
(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x)), 1/16*(4
3*sqrt(2)*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*sqrt(a)*arctan(sqrt(2)*sqr
t(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) + 1)/(a*cos(x) + a*sin(x) + a)) + 32*(cos(x)^3 + 3*cos(x)^2 + (cos(x)
^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*sqrt(a)*arctan(-sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) - sin(x)
+ 1)/(a*cos(x) + a*sin(x) + a)) + (15*cos(x)^3 + 4*cos(x)^2 - (15*cos(x)^2 + 11*cos(x) - 4)*sin(x) - 15*cos(x
) - 4)*sqrt((a*sin(x) + a)/sin(x)))/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*
a^3*cos(x) - 4*a^3)*sin(x))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \csc{\left (x \right )} + a\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+a*csc(x))**(5/2),x)
[Out]
Integral((a*csc(x) + a)**(-5/2), x)
________________________________________________________________________________________
Giac [B] time = 2.52895, size = 470, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+a*csc(x))^(5/2),x, algorithm="giac")
[Out]
-1/16*(43*sqrt(2)*arctan(sqrt(a*tan(1/2*x))/sqrt(a))/(a^(5/2)*sgn(tan(1/2*x) + 1)) - 16*(a*sqrt(abs(a)) + abs(
a)^(3/2))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) + 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a)))/(a^4*sgn(tan(1/2*x) +
1)) - 16*(a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) - 2*sqrt(a*tan(1/2*x)))/sq
rt(abs(a)))/(a^4*sgn(tan(1/2*x) + 1)) - 8*(a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x) + sqrt(2)*sqrt(a*ta
n(1/2*x))*sqrt(abs(a)) + abs(a))/(a^4*sgn(tan(1/2*x) + 1)) + 8*(a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x
) - sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))/(a^4*sgn(tan(1/2*x) + 1)) + sqrt(2)*(11*sqrt(a*tan(1/2*x
))*a^3*tan(1/2*x)^3 + 19*sqrt(a*tan(1/2*x))*a^3*tan(1/2*x)^2 - 19*sqrt(a*tan(1/2*x))*a^3*tan(1/2*x) - 11*sqrt(
a*tan(1/2*x))*a^3)/((a*tan(1/2*x) + a)^4*a^2*sgn(tan(1/2*x) + 1)))*sgn(tan(1/2*x))