\(\int \frac {\tan (x)}{(a+b \cot ^2(x))^{3/2}} \, dx\) [52]

Optimal result

Integrand size = 15, antiderivative size = 84 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}} \]

[Out]

arctanh((a+b*cot(x)^2)^(1/2)/a^(1/2))/a^(3/2)-arctanh((a+b*cot(x)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)+b/a/(a-b)/
(a+b*cot(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3751, 457, 87, 162, 65, 214} \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}} \]

[In]

Int[Tan[x]/(a + b*Cot[x]^2)^(3/2),x]

[Out]

ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]]/a^(3/2) - ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(3/2) + b/(a
*(a - b)*Sqrt[a + b*Cot[x]^2])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 87

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[f*((e + f*x)^(p +
1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + Dist[1/((b*e - a*f)*(d*e - c*f)), Int[(b*d*e - b*c*f - a*d*f - b*
d*f*x)*((e + f*x)^(p + 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -1]

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 214

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

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x \left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot (x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (1+x) (a+b x)^{3/2}} \, dx,x,\cot ^2(x)\right )\right ) \\ & = \frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\text {Subst}\left (\int \frac {a-b-b x}{x (1+x) \sqrt {a+b x}} \, dx,x,\cot ^2(x)\right )}{2 a (a-b)} \\ & = \frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\cot ^2(x)\right )}{2 a}+\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\cot ^2(x)\right )}{2 (a-b)} \\ & = \frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cot ^2(x)}\right )}{a b}+\frac {\text {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cot ^2(x)}\right )}{(a-b) b} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \cot ^2(x)}{a-b}\right )+(-a+b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {b \cot ^2(x)}{a}\right )}{a (a-b) \sqrt {a+b \cot ^2(x)}} \]

[In]

Integrate[Tan[x]/(a + b*Cot[x]^2)^(3/2),x]

[Out]

(a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Cot[x]^2)/(a - b)] + (-a + b)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b
*Cot[x]^2)/a])/(a*(a - b)*Sqrt[a + b*Cot[x]^2])

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(516\) vs. \(2(70)=140\).

Time = 1.52 (sec) , antiderivative size = 517, normalized size of antiderivative = 6.15

method	result	size
default	\(\frac {\sqrt {4}\, \left (\arctan \left (\frac {\sqrt {b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+b}\, \sin \left (x \right )}{2 \left (1-\cos \left (x \right )\right ) \sqrt {-a +b}}\right ) a^{\frac {5}{2}} \sqrt {b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+b}+2 b \sqrt {-a +b}\, a^{\frac {3}{2}} \left (\csc \left (x \right )-\cot \left (x \right )\right )+\operatorname {arctanh}\left (\frac {\sqrt {b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+b}\, \sin \left (x \right )}{2 \left (1-\cos \left (x \right )\right ) \sqrt {a}}\right ) a^{2} \sqrt {b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+b}\, \sqrt {-a +b}-\operatorname {arctanh}\left (\frac {\sqrt {b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+b}\, \sin \left (x \right )}{2 \left (1-\cos \left (x \right )\right ) \sqrt {a}}\right ) a \sqrt {b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+4 a \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}+b}\, \sqrt {-a +b}\, b \right ) \left (b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )+4 a \left (1-\cos \left (x \right )\right )^{2} \sin \left (x \right )-2 b \left (1-\cos \left (x \right )\right )^{2} \sin \left (x \right )+b \sin \left (x \right )^{3}\right )}{2 a^{\frac {5}{2}} \left (a -b \right ) \sqrt {-a +b}\, \left (\frac {b \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{2}+4 a \left (1-\cos \left (x \right )\right )^{2}-2 b \left (1-\cos \left (x \right )\right )^{2}+b \sin \left (x \right )^{2}}{\left (1-\cos \left (x \right )\right )^{2}}\right )^{\frac {3}{2}} \left (1-\cos \left (x \right )\right )^{3}}\)	\(517\)

[In]

int(tan(x)/(a+b*cot(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/2*4^(1/2)/a^(5/2)/(a-b)/(-a+b)^(1/2)*(arctan(1/2*(b*(1-cos(x))^4*csc(x)^4+4*a*(1-cos(x))^2*csc(x)^2-2*b*(1-c
os(x))^2*csc(x)^2+b)^(1/2)/(1-cos(x))*sin(x)/(-a+b)^(1/2))*a^(5/2)*(b*(1-cos(x))^4*csc(x)^4+4*a*(1-cos(x))^2*c
sc(x)^2-2*b*(1-cos(x))^2*csc(x)^2+b)^(1/2)+2*b*(-a+b)^(1/2)*a^(3/2)*(csc(x)-cot(x))+arctanh(1/2*(b*(1-cos(x))^
4*csc(x)^4+4*a*(1-cos(x))^2*csc(x)^2-2*b*(1-cos(x))^2*csc(x)^2+b)^(1/2)/(1-cos(x))*sin(x)/a^(1/2))*a^2*(b*(1-c
os(x))^4*csc(x)^4+4*a*(1-cos(x))^2*csc(x)^2-2*b*(1-cos(x))^2*csc(x)^2+b)^(1/2)*(-a+b)^(1/2)-arctanh(1/2*(b*(1-
cos(x))^4*csc(x)^4+4*a*(1-cos(x))^2*csc(x)^2-2*b*(1-cos(x))^2*csc(x)^2+b)^(1/2)/(1-cos(x))*sin(x)/a^(1/2))*a*(
b*(1-cos(x))^4*csc(x)^4+4*a*(1-cos(x))^2*csc(x)^2-2*b*(1-cos(x))^2*csc(x)^2+b)^(1/2)*(-a+b)^(1/2)*b)/(1/(1-cos
(x))^2*(b*(1-cos(x))^4*csc(x)^2+4*a*(1-cos(x))^2-2*b*(1-cos(x))^2+b*sin(x)^2))^(3/2)/(1-cos(x))^3*(b*(1-cos(x)
)^4*csc(x)+4*a*(1-cos(x))^2*sin(x)-2*b*(1-cos(x))^2*sin(x)+b*sin(x)^3)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (70) = 140\).

Time = 0.37 (sec) , antiderivative size = 863, normalized size of antiderivative = 10.27 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate(tan(x)/(a+b*cot(x)^2)^(3/2),x, algorithm="fricas")

[Out]

[1/2*(2*(a^2*b - a*b^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + (a^2*b - 2*a*b^2 + b^3 + (a^3 - 2*a^2*b + a
*b^2)*tan(x)^2)*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) - (a^3*tan(
x)^2 + a^2*b)*sqrt(a - b)*log(((2*a - b)*tan(x)^2 + 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b
)/(tan(x)^2 + 1)))/(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2), 1/2*(2*(a^2*b - a*b^2)*
sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 - 2*(a^3*tan(x)^2 + a^2*b)*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt((a*
tan(x)^2 + b)/tan(x)^2)/(a - b)) + (a^2*b - 2*a*b^2 + b^3 + (a^3 - 2*a^2*b + a*b^2)*tan(x)^2)*sqrt(a)*log(2*a*
tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b))/(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^
4*b + a^3*b^2)*tan(x)^2), 1/2*(2*(a^2*b - a*b^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 - 2*(a^2*b - 2*a*b^2
+ b^3 + (a^3 - 2*a^2*b + a*b^2)*tan(x)^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) - (a^3*
tan(x)^2 + a^2*b)*sqrt(a - b)*log(((2*a - b)*tan(x)^2 + 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2
+ b)/(tan(x)^2 + 1)))/(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2), ((a^2*b - a*b^2)*sq
rt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 - (a^2*b - 2*a*b^2 + b^3 + (a^3 - 2*a^2*b + a*b^2)*tan(x)^2)*sqrt(-a)*a
rctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) - (a^3*tan(x)^2 + a^2*b)*sqrt(-a + b)*arctan(-sqrt(-a + b)*s
qrt((a*tan(x)^2 + b)/tan(x)^2)/(a - b)))/(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2)]

Sympy [F]

\[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\int \frac {\tan {\left (x \right )}}{\left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(tan(x)/(a+b*cot(x)**2)**(3/2),x)

[Out]

Integral(tan(x)/(a + b*cot(x)**2)**(3/2), x)

Maxima [F]

\[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\int { \frac {\tan \left (x\right )}{{\left (b \cot \left (x\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(tan(x)/(a+b*cot(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(tan(x)/(b*cot(x)^2 + a)^(3/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (70) = 140\).

Time = 0.35 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.51 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=-\frac {{\left (2 \, a^{2} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) - 4 \, a b \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) + 2 \, b^{2} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) + \sqrt {-a^{2} + a b} a \log \left (b\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (\sqrt {-a^{2} + a b} \sqrt {a - b} a^{2} - \sqrt {-a^{2} + a b} \sqrt {a - b} a b\right )}} + \frac {\frac {2 \, b \sin \left (x\right )}{\sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b} {\left (a^{2} - a b\right )}} + \frac {2 \, \sqrt {a - b} \arctan \left (\frac {{\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - 2 \, a + b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} a} + \frac {\log \left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right )}{{\left (a - b\right )}^{\frac {3}{2}}}}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \]

[In]

integrate(tan(x)/(a+b*cot(x)^2)^(3/2),x, algorithm="giac")

[Out]

-1/2*(2*a^2*arctan(-(a - b)/sqrt(-a^2 + a*b)) - 4*a*b*arctan(-(a - b)/sqrt(-a^2 + a*b)) + 2*b^2*arctan(-(a - b
)/sqrt(-a^2 + a*b)) + sqrt(-a^2 + a*b)*a*log(b))*sgn(sin(x))/(sqrt(-a^2 + a*b)*sqrt(a - b)*a^2 - sqrt(-a^2 + a
*b)*sqrt(a - b)*a*b) + 1/2*(2*b*sin(x)/(sqrt(a*sin(x)^2 - b*sin(x)^2 + b)*(a^2 - a*b)) + 2*sqrt(a - b)*arctan(
1/2*((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2 - 2*a + b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)
*a) + log((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2)/(a - b)^(3/2))/sgn(sin(x))

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 1451, normalized size of antiderivative = 17.27 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

int(tan(x)/(a + b*cot(x)^2)^(3/2),x)

[Out]

atanh((2*a^2*b^8*(a + b/tan(x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8 - 12*a^2*b^7 + 30*a^3*b^6 - 38*a^4*b^5 + 24*a^5
*b^4 - 6*a^6*b^3)) - (12*a^3*b^7*(a + b/tan(x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8 - 12*a^2*b^7 + 30*a^3*b^6 - 38*
a^4*b^5 + 24*a^5*b^4 - 6*a^6*b^3)) + (30*a^4*b^6*(a + b/tan(x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8 - 12*a^2*b^7 +
30*a^3*b^6 - 38*a^4*b^5 + 24*a^5*b^4 - 6*a^6*b^3)) - (38*a^5*b^5*(a + b/tan(x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8
- 12*a^2*b^7 + 30*a^3*b^6 - 38*a^4*b^5 + 24*a^5*b^4 - 6*a^6*b^3)) + (24*a^6*b^4*(a + b/tan(x)^2)^(1/2))/((a^3
)^(1/2)*(2*a*b^8 - 12*a^2*b^7 + 30*a^3*b^6 - 38*a^4*b^5 + 24*a^5*b^4 - 6*a^6*b^3)) - (6*a^7*b^3*(a + b/tan(x)^
2)^(1/2))/((a^3)^(1/2)*(2*a*b^8 - 12*a^2*b^7 + 30*a^3*b^6 - 38*a^4*b^5 + 24*a^5*b^4 - 6*a^6*b^3)))/(a^3)^(1/2)
- (atan(((((a - b)^3)^(1/2)*(((a + b/tan(x)^2)^(1/2)*(2*a^3*b^7 - 10*a^4*b^6 + 22*a^5*b^5 - 26*a^6*b^4 + 16*a
^7*b^3 - 4*a^8*b^2))/2 + (((a - b)^3)^(1/2)*(12*a^5*b^7 - 2*a^4*b^8 - 28*a^6*b^6 + 32*a^7*b^5 - 18*a^8*b^4 + 4
*a^9*b^3 + ((a + b/tan(x)^2)^(1/2)*((a - b)^3)^(1/2)*(8*a^5*b^8 - 56*a^6*b^7 + 160*a^7*b^6 - 240*a^8*b^5 + 200
*a^9*b^4 - 88*a^10*b^3 + 16*a^11*b^2))/(4*(a - b)^3)))/(2*(a - b)^3))*1i)/(a - b)^3 + (((a - b)^3)^(1/2)*(((a
+ b/tan(x)^2)^(1/2)*(2*a^3*b^7 - 10*a^4*b^6 + 22*a^5*b^5 - 26*a^6*b^4 + 16*a^7*b^3 - 4*a^8*b^2))/2 + (((a - b)
^3)^(1/2)*(2*a^4*b^8 - 12*a^5*b^7 + 28*a^6*b^6 - 32*a^7*b^5 + 18*a^8*b^4 - 4*a^9*b^3 + ((a + b/tan(x)^2)^(1/2)
*((a - b)^3)^(1/2)*(8*a^5*b^8 - 56*a^6*b^7 + 160*a^7*b^6 - 240*a^8*b^5 + 200*a^9*b^4 - 88*a^10*b^3 + 16*a^11*b
^2))/(4*(a - b)^3)))/(2*(a - b)^3))*1i)/(a - b)^3)/(2*a^3*b^6 - 6*a^4*b^5 + 6*a^5*b^4 - 2*a^6*b^3 - (((a - b)^
3)^(1/2)*(((a + b/tan(x)^2)^(1/2)*(2*a^3*b^7 - 10*a^4*b^6 + 22*a^5*b^5 - 26*a^6*b^4 + 16*a^7*b^3 - 4*a^8*b^2))
/2 + (((a - b)^3)^(1/2)*(12*a^5*b^7 - 2*a^4*b^8 - 28*a^6*b^6 + 32*a^7*b^5 - 18*a^8*b^4 + 4*a^9*b^3 + ((a + b/t
an(x)^2)^(1/2)*((a - b)^3)^(1/2)*(8*a^5*b^8 - 56*a^6*b^7 + 160*a^7*b^6 - 240*a^8*b^5 + 200*a^9*b^4 - 88*a^10*b
^3 + 16*a^11*b^2))/(4*(a - b)^3)))/(2*(a - b)^3)))/(a - b)^3 + (((a - b)^3)^(1/2)*(((a + b/tan(x)^2)^(1/2)*(2*
a^3*b^7 - 10*a^4*b^6 + 22*a^5*b^5 - 26*a^6*b^4 + 16*a^7*b^3 - 4*a^8*b^2))/2 + (((a - b)^3)^(1/2)*(2*a^4*b^8 -
12*a^5*b^7 + 28*a^6*b^6 - 32*a^7*b^5 + 18*a^8*b^4 - 4*a^9*b^3 + ((a + b/tan(x)^2)^(1/2)*((a - b)^3)^(1/2)*(8*a
^5*b^8 - 56*a^6*b^7 + 160*a^7*b^6 - 240*a^8*b^5 + 200*a^9*b^4 - 88*a^10*b^3 + 16*a^11*b^2))/(4*(a - b)^3)))/(2
*(a - b)^3)))/(a - b)^3))*((a - b)^3)^(1/2)*1i)/(a - b)^3 - b/((a*b - a^2)*(a + b/tan(x)^2)^(1/2))