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

Optimal result

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

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)
 

Mathematica [C] (verified)

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

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.89 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \cot ^2(x)}{a-b}\right )+(-a+b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {b \cot ^2(x)}{a}\right )}{a (a-b) \sqrt {a+b \cot ^2(x)}} \] Input:

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

Output:

(a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Cot[x]^2)/(a - b)] + (-a + b)*Hy 
pergeometric2F1[-1/2, 1, 1/2, 1 + (b*Cot[x]^2)/a])/(a*(a - b)*Sqrt[a + b*C 
ot[x]^2])
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.30, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 25, 4153, 25, 354, 96, 174, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {1}{\tan \left (x+\frac {\pi }{2}\right ) \left (a+b \tan \left (x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {1}{\tan \left (x+\frac {\pi }{2}\right ) \left (b \tan \left (x+\frac {\pi }{2}\right )^2+a\right )^{3/2}}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle \int -\frac {\tan (x)}{\left (\cot ^2(x)+1\right ) \left (a+b \cot ^2(x)\right )^{3/2}}d\cot (x)\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\tan (x)}{\left (\cot ^2(x)+1\right ) \left (b \cot ^2(x)+a\right )^{3/2}}d\cot (x)\)

\(\Big \downarrow \) 354

\(\displaystyle -\frac {1}{2} \int \frac {\tan (x)}{\left (\cot ^2(x)+1\right ) \left (b \cot ^2(x)+a\right )^{3/2}}d\cot ^2(x)\)

\(\Big \downarrow \) 96

\(\displaystyle \frac {1}{2} \left (\frac {2 b}{a (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\int \frac {\left (-b \cot ^2(x)+a-b\right ) \tan (x)}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot ^2(x)}{a (a-b)}\right )\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {1}{2} \left (\frac {2 b}{a (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {(a-b) \int \frac {\tan (x)}{\sqrt {b \cot ^2(x)+a}}d\cot ^2(x)-a \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^2(x)+a}}d\cot ^2(x)}{a (a-b)}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{2} \left (\frac {2 b}{a (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\frac {2 (a-b) \int \frac {1}{\frac {\cot ^4(x)}{b}-\frac {a}{b}}d\sqrt {b \cot ^2(x)+a}}{b}-\frac {2 a \int \frac {1}{\frac {\cot ^4(x)}{b}-\frac {a}{b}+1}d\sqrt {b \cot ^2(x)+a}}{b}}{a (a-b)}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{2} \left (\frac {2 b}{a (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\frac {2 a \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {2 (a-b) \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{a (a-b)}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 96

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[f*((e + f*x)^(p + 1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + S 
imp[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 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(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 221

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 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4153

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]}, Simp[c*(ff/f)   Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f 
f^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] && Ratio 
nalQ[n]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(306\) vs. \(2(70)=140\).

Time = 3.52 (sec) , antiderivative size = 307, normalized size of antiderivative = 3.65

method	result	size
default	\(\frac {\sqrt {4}\, \left (-\arctan \left (\frac {\sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\sqrt {-a +b}\, \left (-1+\cos \left (x \right )\right )}\right ) a^{\frac {5}{2}} \sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}+b \,a^{\frac {3}{2}} \sqrt {-a +b}\, \left (\csc \left (x \right )-\cot \left (x \right )\right )-\operatorname {arctanh}\left (\frac {\sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\sqrt {a}\, \left (-1+\cos \left (x \right )\right )}\right ) a^{2} \sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {-a +b}+\operatorname {arctanh}\left (\frac {\sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\sqrt {a}\, \left (-1+\cos \left (x \right )\right )}\right ) a \sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {-a +b}\, b \right ) \left (4 a \left (1-\cos \left (x \right )\right )^{2} \sin \left (x \right )+\left (\left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )-2 \left (1-\cos \left (x \right )\right )^{2} \sin \left (x \right )+\sin \left (x \right )^{3}\right ) b \right )}{8 a^{\frac {5}{2}} \left (a -b \right ) \sqrt {-a +b}\, \left (a +b \cot \left (x \right )^{2}\right )^{\frac {3}{2}} \left (1-\cos \left (x \right )\right )^{3}}\)	\(307\)

Input:

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

Output:

1/8*4^(1/2)/a^(5/2)/(a-b)/(-a+b)^(1/2)*(-arctan(1/(-a+b)^(1/2)*((cos(x)^2* 
b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)*sin(x)/(-1+cos(x)))*a^(5/2)*((cos(x)^2*b 
+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)+b*a^(3/2)*(-a+b)^(1/2)*(csc(x)-cot(x))-ar 
ctanh(1/a^(1/2)*((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)*sin(x)/(-1+co 
s(x)))*a^2*((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)*(-a+b)^(1/2)+arcta 
nh(1/a^(1/2)*((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)*sin(x)/(-1+cos(x 
)))*a*((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)*(-a+b)^(1/2)*b)/(a+b*co 
t(x)^2)^(3/2)/(1-cos(x))^3*(4*a*(1-cos(x))^2*sin(x)+((1-cos(x))^4*csc(x)-2 
*(1-cos(x))^2*sin(x)+sin(x)^3)*b)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (70) = 140\).

Time = 0.16 (sec) , antiderivative size = 896, normalized size of antiderivative = 10.67 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[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*ta 
n(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)*tan(x)^2/(a*tan(x)^ 
2 + 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)*tan(x)^2/(a*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*t 
an(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)*sqrt((a*t 
an(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)*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2 
)*tan(x)^2/(a*tan(x)^2 + b)) + (a^3*tan(x)^2 + a^2*b)*sqrt(-a + b)*arct...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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 = 0.16 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.51 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=-\frac {{\left (2 \, a^{2} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) - 4 \, a b \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) + 2 \, b^{2} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) + \sqrt {-a^{2} + a b} a \log \left (b\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (\sqrt {-a^{2} + a b} \sqrt {a - b} a^{2} - \sqrt {-a^{2} + a b} \sqrt {a - b} a b\right )}} + \frac {\frac {2 \, b \sin \left (x\right )}{\sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b} {\left (a^{2} - a b\right )}} + \frac {2 \, \sqrt {a - b} \arctan \left (\frac {{\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - 2 \, a + b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} a} + \frac {\log \left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right )}{{\left (a - b\right )}^{\frac {3}{2}}}}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \] Input:

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

Output:

-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 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

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 - 1 
2*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 + 16...
 

Reduce [F]

\[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx=\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}\, \tan \left (x \right )}{\cot \left (x \right )^{4} b^{2}+2 \cot \left (x \right )^{2} a b +a^{2}}d x \] Input:

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

Output:

int((sqrt(cot(x)**2*b + a)*tan(x))/(cot(x)**4*b**2 + 2*cot(x)**2*a*b + a** 
2),x)