\(\int \frac {\tan (x)}{(a+b \cot ^2(x))^{5/2}} \, dx\) [57]

Optimal result

Integrand size = 15, antiderivative size = 118 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2}}+\frac {b}{3 a (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {(2 a-b) b}{a^2 (a-b)^2 \sqrt {a+b \cot ^2(x)}} \] Output:

arctanh((a+b*cot(x)^2)^(1/2)/a^(1/2))/a^(5/2)-arctanh((a+b*cot(x)^2)^(1/2) 
/(a-b)^(1/2))/(a-b)^(5/2)+1/3*b/a/(a-b)/(a+b*cot(x)^2)^(3/2)+(2*a-b)*b/a^2 
/(a-b)^2/(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 = 78, normalized size of antiderivative = 0.66 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \cot ^2(x)}{a-b}\right )+(-a+b) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+\frac {b \cot ^2(x)}{a}\right )}{3 a (a-b) \left (a+b \cot ^2(x)\right )^{3/2}} \] Input:

Integrate[Tan[x]/(a + b*Cot[x]^2)^(5/2),x]
 

Output:

(a*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Cot[x]^2)/(a - b)] + (-a + b)*H 
ypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Cot[x]^2)/a])/(3*a*(a - b)*(a + b*C 
ot[x]^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.35, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {3042, 25, 4153, 25, 354, 96, 169, 27, 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.

\(\displaystyle \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {1}{\tan \left (x+\frac {\pi }{2}\right ) \left (a+b \tan \left (x+\frac {\pi }{2}\right )^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {1}{\tan \left (x+\frac {\pi }{2}\right ) \left (b \tan \left (x+\frac {\pi }{2}\right )^2+a\right )^{5/2}}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle \int -\frac {\tan (x)}{\left (\cot ^2(x)+1\right ) \left (a+b \cot ^2(x)\right )^{5/2}}d\cot (x)\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\tan (x)}{\left (\cot ^2(x)+1\right ) \left (b \cot ^2(x)+a\right )^{5/2}}d\cot (x)\)

\(\Big \downarrow \) 354

\(\displaystyle -\frac {1}{2} \int \frac {\tan (x)}{\left (\cot ^2(x)+1\right ) \left (b \cot ^2(x)+a\right )^{5/2}}d\cot ^2(x)\)

\(\Big \downarrow \) 96

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{2} \left (\frac {2 b}{3 a (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {\frac {\frac {2 (a-b)^2 \int \frac {1}{\frac {\cot ^4(x)}{b}-\frac {a}{b}}d\sqrt {b \cot ^2(x)+a}}{b}-\frac {2 a^2 \int \frac {1}{\frac {\cot ^4(x)}{b}-\frac {a}{b}+1}d\sqrt {b \cot ^2(x)+a}}{b}}{a (a-b)}-\frac {2 b (2 a-b)}{a (a-b) \sqrt {a+b \cot ^2(x)}}}{a (a-b)}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{2} \left (\frac {2 b}{3 a (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {\frac {\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {2 (a-b)^2 \text {arctanh}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{a (a-b)}-\frac {2 b (2 a-b)}{a (a-b) \sqrt {a+b \cot ^2(x)}}}{a (a-b)}\right )\)

Input:

Int[Tan[x]/(a + b*Cot[x]^2)^(5/2),x]
 

Output:

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

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 [F]

\[\int \frac {\tan \left (x \right )}{\left (a +b \cot \left (x \right )^{2}\right )^{\frac {5}{2}}}d x\]

Input:

int(tan(x)/(a+b*cot(x)^2)^(5/2),x)
 

Output:

int(tan(x)/(a+b*cot(x)^2)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (100) = 200\).

Time = 0.28 (sec) , antiderivative size = 1565, normalized size of antiderivative = 13.26 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(tan(x)/(a+b*cot(x)^2)^(5/2),x, algorithm="fricas")
 

Output:

[1/6*(3*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5 + (a^5 - 3*a^4*b + 3*a^3*b^2 
- a^2*b^3)*tan(x)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*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) + 3*(a^5*tan(x)^4 + 2*a^4*b*tan(x)^2 + a^3*b^2)*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)) + 2*((7*a^4*b - 11*a^3*b^2 + 4*a^2*b^3)*tan(x)^4 + 
3*(2*a^3*b^2 - 3*a^2*b^3 + a*b^4)*tan(x)^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2 
))/(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5 + (a^8 - 3*a^7*b + 3*a^6*b^2 
 - a^5*b^3)*tan(x)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*tan(x)^ 
2), 1/6*(6*(a^5*tan(x)^4 + 2*a^4*b*tan(x)^2 + a^3*b^2)*sqrt(-a + b)*arctan 
(sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2/(a*tan(x)^2 + b)) + 
 3*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2 
*b^3)*tan(x)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*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) + 2*((7*a^4*b - 11*a^3*b^2 + 4*a^2*b^3)*tan(x)^4 + 3*(2*a^3*b^2 - 3*a^ 
2*b^3 + a*b^4)*tan(x)^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(a^6*b^2 - 3*a^5 
*b^3 + 3*a^4*b^4 - a^3*b^5 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*tan(x)^ 
4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*tan(x)^2), -1/6*(6*(a^3*b^ 
2 - 3*a^2*b^3 + 3*a*b^4 - b^5 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*tan( 
x)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(x)^2)*sqrt(-a)*arc...
 

Sympy [F]

\[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\int \frac {\tan {\left (x \right )}}{\left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(tan(x)/(a+b*cot(x)**2)**(5/2),x)
 

Output:

Integral(tan(x)/(a + b*cot(x)**2)**(5/2), x)
 

Maxima [F]

\[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\int { \frac {\tan \left (x\right )}{{\left (b \cot \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(tan(x)/(a+b*cot(x)^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate(tan(x)/(b*cot(x)^2 + a)^(5/2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (100) = 200\).

Time = 0.17 (sec) , antiderivative size = 483, normalized size of antiderivative = 4.09 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=-\frac {{\left (2 \, a^{3} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) - 6 \, a^{2} b \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) + 6 \, a b^{2} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) - 2 \, b^{3} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) + \sqrt {-a^{2} + a b} a^{2} \log \left (b\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (\sqrt {-a^{2} + a b} \sqrt {a - b} a^{4} - 2 \, \sqrt {-a^{2} + a b} \sqrt {a - b} a^{3} b + \sqrt {-a^{2} + a b} \sqrt {a - b} a^{2} b^{2}\right )}} + \frac {\frac {2 \, {\left (\frac {{\left (7 \, a^{5} b^{2} - 17 \, a^{4} b^{3} + 13 \, a^{3} b^{4} - 3 \, a^{2} b^{5}\right )} \sin \left (x\right )^{2}}{a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}} + \frac {3 \, {\left (2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} + a^{2} b^{5}\right )}}{a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}}\right )} \sin \left (x\right )}{{\left (a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b\right )}^{\frac {3}{2}}} + \frac {3 \, \log \left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a - b}} + \frac {6 \, \sqrt {a - b} \arctan \left (\frac {{\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - 2 \, a + b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} a^{2}}}{6 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \] Input:

integrate(tan(x)/(a+b*cot(x)^2)^(5/2),x, algorithm="giac")
 

Output:

-1/2*(2*a^3*arctan(-(a - b)/sqrt(-a^2 + a*b)) - 6*a^2*b*arctan(-(a - b)/sq 
rt(-a^2 + a*b)) + 6*a*b^2*arctan(-(a - b)/sqrt(-a^2 + a*b)) - 2*b^3*arctan 
(-(a - b)/sqrt(-a^2 + a*b)) + sqrt(-a^2 + a*b)*a^2*log(b))*sgn(sin(x))/(sq 
rt(-a^2 + a*b)*sqrt(a - b)*a^4 - 2*sqrt(-a^2 + a*b)*sqrt(a - b)*a^3*b + sq 
rt(-a^2 + a*b)*sqrt(a - b)*a^2*b^2) + 1/6*(2*((7*a^5*b^2 - 17*a^4*b^3 + 13 
*a^3*b^4 - 3*a^2*b^5)*sin(x)^2/(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4) + 
 3*(2*a^4*b^3 - 3*a^3*b^4 + a^2*b^5)/(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4* 
b^4))*sin(x)/(a*sin(x)^2 - b*sin(x)^2 + b)^(3/2) + 3*log((sqrt(a - b)*sin( 
x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2)/((a^2 - 2*a*b + b^2)*sqrt(a - b 
)) + 6*sqrt(a - b)*arctan(1/2*((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*s 
in(x)^2 + b))^2 - 2*a + b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)*a^2))/sgn(s 
in(x))
 

Mupad [B] (verification not implemented)

Time = 10.09 (sec) , antiderivative size = 2817, normalized size of antiderivative = 23.87 \[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

int(tan(x)/(a + b*cot(x)^2)^(5/2),x)
 

Output:

atanh((2*a^5*b^13*(a + b/tan(x)^2)^(1/2))/((a^5)^(1/2)*(2*a^3*b^13 - 22*a^ 
4*b^12 + 110*a^5*b^11 - 330*a^6*b^10 + 660*a^7*b^9 - 922*a^8*b^8 + 912*a^9 
*b^7 - 630*a^10*b^6 + 290*a^11*b^5 - 80*a^12*b^4 + 10*a^13*b^3)) - (22*a^6 
*b^12*(a + b/tan(x)^2)^(1/2))/((a^5)^(1/2)*(2*a^3*b^13 - 22*a^4*b^12 + 110 
*a^5*b^11 - 330*a^6*b^10 + 660*a^7*b^9 - 922*a^8*b^8 + 912*a^9*b^7 - 630*a 
^10*b^6 + 290*a^11*b^5 - 80*a^12*b^4 + 10*a^13*b^3)) + (110*a^7*b^11*(a + 
b/tan(x)^2)^(1/2))/((a^5)^(1/2)*(2*a^3*b^13 - 22*a^4*b^12 + 110*a^5*b^11 - 
 330*a^6*b^10 + 660*a^7*b^9 - 922*a^8*b^8 + 912*a^9*b^7 - 630*a^10*b^6 + 2 
90*a^11*b^5 - 80*a^12*b^4 + 10*a^13*b^3)) - (330*a^8*b^10*(a + b/tan(x)^2) 
^(1/2))/((a^5)^(1/2)*(2*a^3*b^13 - 22*a^4*b^12 + 110*a^5*b^11 - 330*a^6*b^ 
10 + 660*a^7*b^9 - 922*a^8*b^8 + 912*a^9*b^7 - 630*a^10*b^6 + 290*a^11*b^5 
 - 80*a^12*b^4 + 10*a^13*b^3)) + (660*a^9*b^9*(a + b/tan(x)^2)^(1/2))/((a^ 
5)^(1/2)*(2*a^3*b^13 - 22*a^4*b^12 + 110*a^5*b^11 - 330*a^6*b^10 + 660*a^7 
*b^9 - 922*a^8*b^8 + 912*a^9*b^7 - 630*a^10*b^6 + 290*a^11*b^5 - 80*a^12*b 
^4 + 10*a^13*b^3)) - (922*a^10*b^8*(a + b/tan(x)^2)^(1/2))/((a^5)^(1/2)*(2 
*a^3*b^13 - 22*a^4*b^12 + 110*a^5*b^11 - 330*a^6*b^10 + 660*a^7*b^9 - 922* 
a^8*b^8 + 912*a^9*b^7 - 630*a^10*b^6 + 290*a^11*b^5 - 80*a^12*b^4 + 10*a^1 
3*b^3)) + (912*a^11*b^7*(a + b/tan(x)^2)^(1/2))/((a^5)^(1/2)*(2*a^3*b^13 - 
 22*a^4*b^12 + 110*a^5*b^11 - 330*a^6*b^10 + 660*a^7*b^9 - 922*a^8*b^8 + 9 
12*a^9*b^7 - 630*a^10*b^6 + 290*a^11*b^5 - 80*a^12*b^4 + 10*a^13*b^3)) ...
 

Reduce [F]

\[ \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx=\int \frac {\sqrt {\cot \left (x \right )^{2} b +a}\, \tan \left (x \right )}{\cot \left (x \right )^{6} b^{3}+3 \cot \left (x \right )^{4} a \,b^{2}+3 \cot \left (x \right )^{2} a^{2} b +a^{3}}d x \] Input:

int(tan(x)/(a+b*cot(x)^2)^(5/2),x)
 

Output:

int((sqrt(cot(x)**2*b + a)*tan(x))/(cot(x)**6*b**3 + 3*cot(x)**4*a*b**2 + 
3*cot(x)**2*a**2*b + a**3),x)