\(\int \frac {\cot (x)}{(a+b \cot ^4(x))^{5/2}} \, dx\) [64]

Optimal result

Integrand size = 15, antiderivative size = 117 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}} \] Output:

1/2*arctanh((a-b*cot(x)^2)/(a+b)^(1/2)/(a+b*cot(x)^4)^(1/2))/(a+b)^(5/2)-1 
/6*(a+b*cot(x)^2)/a/(a+b)/(a+b*cot(x)^4)^(3/2)-1/6*(3*a^2+b*(5*a+2*b)*cot( 
x)^2)/a^2/(a+b)^2/(a+b*cot(x)^4)^(1/2)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {a^2 (4 a+b)+3 a b (2 a+b) \cot ^2(x)+3 a^2 b \cot ^4(x)+b^2 (5 a+2 b) \cot ^6(x)}{6 a^2 (a+b)^2 \left (a+b \cot ^4(x)\right )^{3/2}} \] Input:

Integrate[Cot[x]/(a + b*Cot[x]^4)^(5/2),x]
 

Output:

ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*(a + b)^(5 
/2)) - (a^2*(4*a + b) + 3*a*b*(2*a + b)*Cot[x]^2 + 3*a^2*b*Cot[x]^4 + b^2* 
(5*a + 2*b)*Cot[x]^6)/(6*a^2*(a + b)^2*(a + b*Cot[x]^4)^(3/2))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.11, 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, 1577, 496, 25, 686, 27, 488, 219}

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.

\(\displaystyle \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {\tan \left (x+\frac {\pi }{2}\right )}{\left (a+b \tan \left (x+\frac {\pi }{2}\right )^4\right )^{5/2}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\tan \left (x+\frac {\pi }{2}\right )}{\left (b \tan \left (x+\frac {\pi }{2}\right )^4+a\right )^{5/2}}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle \int -\frac {\cot (x)}{\left (\cot ^2(x)+1\right ) \left (a+b \cot ^4(x)\right )^{5/2}}d\cot (x)\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\cot (x)}{\left (\cot ^2(x)+1\right ) \left (b \cot ^4(x)+a\right )^{5/2}}d\cot (x)\)

\(\Big \downarrow \) 1577

\(\displaystyle -\frac {1}{2} \int \frac {1}{\left (\cot ^2(x)+1\right ) \left (b \cot ^4(x)+a\right )^{5/2}}d\cot ^2(x)\)

\(\Big \downarrow \) 496

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 686

\(\displaystyle \frac {1}{2} \left (-\frac {\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{a (a+b) \sqrt {a+b \cot ^4(x)}}-\frac {\int -\frac {3 a^2 b}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^4(x)+a}}d\cot ^2(x)}{a b (a+b)}}{3 a (a+b)}-\frac {a+b \cot ^2(x)}{3 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {\frac {3 a \int \frac {1}{\left (\cot ^2(x)+1\right ) \sqrt {b \cot ^4(x)+a}}d\cot ^2(x)}{a+b}+\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{a (a+b) \sqrt {a+b \cot ^4(x)}}}{3 a (a+b)}-\frac {a+b \cot ^2(x)}{3 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}\right )\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {1}{2} \left (-\frac {\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{a (a+b) \sqrt {a+b \cot ^4(x)}}-\frac {3 a \int \frac {1}{-\cot ^4(x)+a+b}d\frac {a-b \cot ^2(x)}{\sqrt {b \cot ^4(x)+a}}}{a+b}}{3 a (a+b)}-\frac {a+b \cot ^2(x)}{3 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (-\frac {\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{a (a+b) \sqrt {a+b \cot ^4(x)}}-\frac {3 a \text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{(a+b)^{3/2}}}{3 a (a+b)}-\frac {a+b \cot ^2(x)}{3 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}\right )\)

Input:

Int[Cot[x]/(a + b*Cot[x]^4)^(5/2),x]
 

Output:

(-1/3*(a + b*Cot[x]^2)/(a*(a + b)*(a + b*Cot[x]^4)^(3/2)) - ((-3*a*ArcTanh 
[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])])/(a + b)^(3/2) + (3* 
a^2 + b*(5*a + 2*b)*Cot[x]^2)/(a*(a + b)*Sqrt[a + b*Cot[x]^4]))/(3*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 219

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

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 496

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 
 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a 
 + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 
*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad 
raticQ[a, 0, b, c, d, n, p, x]
 

rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1577

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free 
Q[{a, c, d, e, p, q}, x]
 

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. \(585\) vs. \(2(101)=202\).

Time = 1.36 (sec) , antiderivative size = 586, normalized size of antiderivative = 5.01

method	result	size
derivativedivides	\(\frac {b^{2} \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \left (\sqrt {-a b}+b \right )^{2} \left (\sqrt {-a b}-b \right )^{2} \sqrt {a +b}}+\frac {-\frac {\sqrt {b \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right ) a}-\frac {\frac {\sqrt {b \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right ) a}+\frac {\left (2 \sqrt {-a b}-b \right ) \sqrt {b \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right )^{2} a^{2} \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (2 \sqrt {-a b}+b \right ) \sqrt {b \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right )^{2} a^{2} \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}\)	\(586\)
default	\(\frac {b^{2} \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \left (\sqrt {-a b}+b \right )^{2} \left (\sqrt {-a b}-b \right )^{2} \sqrt {a +b}}+\frac {-\frac {\sqrt {b \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right ) a}-\frac {\frac {\sqrt {b \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right ) a}+\frac {\left (2 \sqrt {-a b}-b \right ) \sqrt {b \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right )^{2} a^{2} \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (2 \sqrt {-a b}+b \right ) \sqrt {b \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right )^{2} a^{2} \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}\)	\(586\)

Input:

int(cot(x)/(a+b*cot(x)^4)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/2*b^2/((-a*b)^(1/2)+b)^2/((-a*b)^(1/2)-b)^2/(a+b)^(1/2)*ln((2*a+2*b-2*b* 
(cot(x)^2+1)+2*(a+b)^(1/2)*(b*(cot(x)^2+1)^2-2*b*(cot(x)^2+1)+a+b)^(1/2))/ 
(cot(x)^2+1))+1/8/((-a*b)^(1/2)+b)/a*(-1/3/(-a*b)^(1/2)/(cot(x)^2-(-a*b)^( 
1/2)/b)^2*(b*(cot(x)^2-(-a*b)^(1/2)/b)^2+2*(-a*b)^(1/2)*(cot(x)^2-(-a*b)^( 
1/2)/b))^(1/2)-1/3/a/(cot(x)^2-(-a*b)^(1/2)/b)*(b*(cot(x)^2-(-a*b)^(1/2)/b 
)^2+2*(-a*b)^(1/2)*(cot(x)^2-(-a*b)^(1/2)/b))^(1/2))-1/8/((-a*b)^(1/2)-b)/ 
a*(1/3/(-a*b)^(1/2)/(cot(x)^2+(-a*b)^(1/2)/b)^2*(b*(cot(x)^2+(-a*b)^(1/2)/ 
b)^2-2*(-a*b)^(1/2)*(cot(x)^2+(-a*b)^(1/2)/b))^(1/2)-1/3/a/(cot(x)^2+(-a*b 
)^(1/2)/b)*(b*(cot(x)^2+(-a*b)^(1/2)/b)^2-2*(-a*b)^(1/2)*(cot(x)^2+(-a*b)^ 
(1/2)/b))^(1/2))+1/8*(2*(-a*b)^(1/2)-b)/((-a*b)^(1/2)-b)^2/a^2/(cot(x)^2+( 
-a*b)^(1/2)/b)*(b*(cot(x)^2+(-a*b)^(1/2)/b)^2-2*(-a*b)^(1/2)*(cot(x)^2+(-a 
*b)^(1/2)/b))^(1/2)-1/8*(2*(-a*b)^(1/2)+b)/((-a*b)^(1/2)+b)^2/a^2/(cot(x)^ 
2-(-a*b)^(1/2)/b)*(b*(cot(x)^2-(-a*b)^(1/2)/b)^2+2*(-a*b)^(1/2)*(cot(x)^2- 
(-a*b)^(1/2)/b))^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (103) = 206\).

Time = 0.26 (sec) , antiderivative size = 1365, normalized size of antiderivative = 11.67 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(x)/(a+b*cot(x)^4)^(5/2),x, algorithm="fricas")
 

Output:

[1/12*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(2*x)^4 + a^4 + 2*a^3*b + a^2*b^2 - 
 4*(a^4 - a^2*b^2)*cos(2*x)^3 + 2*(3*a^4 - 2*a^3*b + 3*a^2*b^2)*cos(2*x)^2 
 - 4*(a^4 - a^2*b^2)*cos(2*x))*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos 
(2*x)^2 + 1/2*a^2 + 1/2*b^2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - 
 b)*sqrt(a + b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(co 
s(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - b^2)*cos(2*x)) - 4*((2*a^4 + a^3*b - 
5*a^2*b^2 - 5*a*b^3 - b^4)*cos(2*x)^4 + 2*a^4 + 7*a^3*b + 9*a^2*b^2 + 5*a* 
b^3 + b^4 - 2*(4*a^4 + 2*a^3*b - a^2*b^2 + 2*a*b^3 + b^4)*cos(2*x)^3 + 12* 
(a^4 + a^3*b)*cos(2*x)^2 - 2*(4*a^4 + 8*a^3*b + 3*a^2*b^2 - 2*a*b^3 - b^4) 
*cos(2*x))*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x 
)^2 - 2*cos(2*x) + 1)))/(a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b 
^4 + a^2*b^5 + (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2* 
b^5)*cos(2*x)^4 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a 
^2*b^5)*cos(2*x)^3 + 2*(3*a^7 + 7*a^6*b + 6*a^5*b^2 + 6*a^4*b^3 + 7*a^3*b^ 
4 + 3*a^2*b^5)*cos(2*x)^2 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a 
^3*b^4 - a^2*b^5)*cos(2*x)), -1/6*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(2*x)^4 
 + a^4 + 2*a^3*b + a^2*b^2 - 4*(a^4 - a^2*b^2)*cos(2*x)^3 + 2*(3*a^4 - 2*a 
^3*b + 3*a^2*b^2)*cos(2*x)^2 - 4*(a^4 - a^2*b^2)*cos(2*x))*sqrt(-a - b)*ar 
ctan(((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(-a - b)*sqrt(((a + b 
)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1...
 

Sympy [F]

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

integrate(cot(x)/(a+b*cot(x)**4)**(5/2),x)
 

Output:

Integral(cot(x)/(a + b*cot(x)**4)**(5/2), x)
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate(cot(x)/(a+b*cot(x)^4)^(5/2),x, algorithm="maxima")
 

Output:

integrate(cot(x)/(b*cot(x)^4 + a)^(5/2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (103) = 206\).

Time = 0.15 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.36 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=-\frac {{\left (2 \, {\left (\frac {{\left (2 \, a^{3} b - a^{2} b^{2} - 4 \, a b^{3} - b^{4}\right )} \sin \left (x\right )^{2}}{a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}} + \frac {3 \, {\left (3 \, a b^{3} + b^{4}\right )}}{a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}}\right )} \sin \left (x\right )^{2} + \frac {3 \, {\left (a^{2} b^{2} - 5 \, a b^{3} - 2 \, b^{4}\right )}}{a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}}\right )} \sin \left (x\right )^{2} + \frac {5 \, a b^{3} + 2 \, b^{4}}{a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}}}{6 \, {\left (a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b\right )}^{\frac {3}{2}}} - \frac {\log \left ({\left | -{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} \sqrt {a + b} + b \right |}\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a + b}} \] Input:

integrate(cot(x)/(a+b*cot(x)^4)^(5/2),x, algorithm="giac")
 

Output:

-1/6*((2*((2*a^3*b - a^2*b^2 - 4*a*b^3 - b^4)*sin(x)^2/(a^4*b + 2*a^3*b^2 
+ a^2*b^3) + 3*(3*a*b^3 + b^4)/(a^4*b + 2*a^3*b^2 + a^2*b^3))*sin(x)^2 + 3 
*(a^2*b^2 - 5*a*b^3 - 2*b^4)/(a^4*b + 2*a^3*b^2 + a^2*b^3))*sin(x)^2 + (5* 
a*b^3 + 2*b^4)/(a^4*b + 2*a^3*b^2 + a^2*b^3))/(a*sin(x)^4 + b*sin(x)^4 - 2 
*b*sin(x)^2 + b)^(3/2) - 1/2*log(abs(-(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x 
)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))*sqrt(a + b) + b))/((a^2 + 2*a*b + b^ 
2)*sqrt(a + b))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=\int \frac {\mathrm {cot}\left (x\right )}{{\left (b\,{\mathrm {cot}\left (x\right )}^4+a\right )}^{5/2}} \,d x \] Input:

int(cot(x)/(a + b*cot(x)^4)^(5/2),x)
 

Output:

int(cot(x)/(a + b*cot(x)^4)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx=\int \frac {\sqrt {\cot \left (x \right )^{4} b +a}\, \cot \left (x \right )}{\cot \left (x \right )^{12} b^{3}+3 \cot \left (x \right )^{8} a \,b^{2}+3 \cot \left (x \right )^{4} a^{2} b +a^{3}}d x \] Input:

int(cot(x)/(a+b*cot(x)^4)^(5/2),x)
 

Output:

int((sqrt(cot(x)**4*b + a)*cot(x))/(cot(x)**12*b**3 + 3*cot(x)**8*a*b**2 + 
 3*cot(x)**4*a**2*b + a**3),x)