\(\int \sqrt {a \cot ^3(x)} \, dx\) [30]

Optimal result

Integrand size = 10, antiderivative size = 132 \[ \int \sqrt {a \cot ^3(x)} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (x)}}{1+\cot (x)}\right ) \sqrt {a \cot ^3(x)}}{\sqrt {2} \cot ^{\frac {3}{2}}(x)}-2 \sqrt {a \cot ^3(x)} \tan (x) \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \sqrt {a \cot ^3(x)} \, dx=-\frac {\sqrt {a \cot ^3(x)} \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (x)}\right )+8 \sqrt {\cot (x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (x)}+\cot (x)\right )\right )}{4 \cot ^{\frac {3}{2}}(x)} \] Input:

Integrate[Sqrt[a*Cot[x]^3],x]
 

Output:

-1/4*(Sqrt[a*Cot[x]^3]*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]]] - 2*Sqr 
t[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[x]]] + 8*Sqrt[Cot[x]] + Sqrt[2]*Log[1 - S 
qrt[2]*Sqrt[Cot[x]] + Cot[x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot 
[x]]))/Cot[x]^(3/2)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.03, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {3042, 4141, 3042, 3954, 3042, 3957, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \sqrt {a \cot ^3(x)} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4141

\(\displaystyle \frac {\sqrt {a \cot ^3(x)} \int \cot ^{\frac {3}{2}}(x)dx}{\cot ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3954

\(\displaystyle \frac {\sqrt {a \cot ^3(x)} \left (-\int \frac {1}{\sqrt {\cot (x)}}dx-2 \sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3957

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\sqrt {a \cot ^3(x)} \left (2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\cot (x)+\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (x)-\sqrt {2} \sqrt {\cot (x)}+1\right )}{2 \sqrt {2}}\right )\right )-2 \sqrt {\cot (x)}\right )}{\cot ^{\frac {3}{2}}(x)}\)

Input:

Int[Sqrt[a*Cot[x]^3],x]
 

Output:

(Sqrt[a*Cot[x]^3]*(-2*Sqrt[Cot[x]] + 2*((-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[x]] 
]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Cot[x]]]/Sqrt[2])/2 + (-1/2*Log[1 - S 
qrt[2]*Sqrt[Cot[x]] + Cot[x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Cot[x]] + Cot 
[x]]/(2*Sqrt[2]))/2)))/Cot[x]^(3/2)
 

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 217

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

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 3042

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

rule 3954

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d 
*x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2   Int[(b*Tan[c + d*x])^(n - 2), x] 
, x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
 

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

rule 4141

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff 
= FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ 
n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p]))   Int[ActivateTrig[u]*(Ta 
n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] 
 && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / 
; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
 

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.26

method	result	size
derivativedivides	\(-\frac {\sqrt {a \cot \left (x \right )^{3}}\, \left (-\left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}{\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}-a \cot \left (x \right )-\sqrt {a^{2}}}\right )-2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )-2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+8 \sqrt {a \cot \left (x \right )}\right )}{4 \cot \left (x \right ) \sqrt {a \cot \left (x \right )}}\)	\(166\)
default	\(-\frac {\sqrt {a \cot \left (x \right )^{3}}\, \left (-\left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {a \cot \left (x \right )+\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}+\sqrt {a^{2}}}{\left (a^{2}\right )^{\frac {1}{4}} \sqrt {a \cot \left (x \right )}\, \sqrt {2}-a \cot \left (x \right )-\sqrt {a^{2}}}\right )-2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}+\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )-2 \left (a^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a \cot \left (x \right )}-\left (a^{2}\right )^{\frac {1}{4}}}{\left (a^{2}\right )^{\frac {1}{4}}}\right )+8 \sqrt {a \cot \left (x \right )}\right )}{4 \cot \left (x \right ) \sqrt {a \cot \left (x \right )}}\)	\(166\)

Input:

int((a*cot(x)^3)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/4*(a*cot(x)^3)^(1/2)*(-(a^2)^(1/4)*2^(1/2)*ln(-(a*cot(x)+(a^2)^(1/4)*(a 
*cot(x))^(1/2)*2^(1/2)+(a^2)^(1/2))/((a^2)^(1/4)*(a*cot(x))^(1/2)*2^(1/2)- 
a*cot(x)-(a^2)^(1/2)))-2*(a^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(a*cot(x))^(1 
/2)+(a^2)^(1/4))/(a^2)^(1/4))-2*(a^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(a*cot 
(x))^(1/2)-(a^2)^(1/4))/(a^2)^(1/4))+8*(a*cot(x))^(1/2))/cot(x)/(a*cot(x)) 
^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (101) = 202\).

Time = 0.08 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.93 \[ \int \sqrt {a \cot ^3(x)} \, dx=\frac {2 \, \sqrt {2} \sqrt {a} {\left (\cos \left (2 \, x\right ) + 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) + a \cos \left (2 \, x\right ) + a}{a \cos \left (2 \, x\right ) + a}\right ) + 2 \, \sqrt {2} \sqrt {a} {\left (\cos \left (2 \, x\right ) + 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} \sin \left (2 \, x\right ) - a \cos \left (2 \, x\right ) - a}{a \cos \left (2 \, x\right ) + a}\right ) - \sqrt {2} \sqrt {a} {\left (\cos \left (2 \, x\right ) + 1\right )} \log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} {\left (\cos \left (2 \, x\right ) - 1\right )} + a \cos \left (2 \, x\right ) + a \sin \left (2 \, x\right ) + a}{\sin \left (2 \, x\right )}\right ) + \sqrt {2} \sqrt {a} {\left (\cos \left (2 \, x\right ) + 1\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} {\left (\cos \left (2 \, x\right ) - 1\right )} - a \cos \left (2 \, x\right ) - a \sin \left (2 \, x\right ) - a}{\sin \left (2 \, x\right )}\right ) - 8 \, \sqrt {-\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )}} \sin \left (2 \, x\right )}{4 \, {\left (\cos \left (2 \, x\right ) + 1\right )}} \] Input:

integrate((a*cot(x)^3)^(1/2),x, algorithm="fricas")
 

Output:

1/4*(2*sqrt(2)*sqrt(a)*(cos(2*x) + 1)*arctan((sqrt(2)*sqrt(a)*sqrt(-(a*cos 
(2*x)^2 + 2*a*cos(2*x) + a)/((cos(2*x) - 1)*sin(2*x)))*sin(2*x) + a*cos(2* 
x) + a)/(a*cos(2*x) + a)) + 2*sqrt(2)*sqrt(a)*(cos(2*x) + 1)*arctan((sqrt( 
2)*sqrt(a)*sqrt(-(a*cos(2*x)^2 + 2*a*cos(2*x) + a)/((cos(2*x) - 1)*sin(2*x 
)))*sin(2*x) - a*cos(2*x) - a)/(a*cos(2*x) + a)) - sqrt(2)*sqrt(a)*(cos(2* 
x) + 1)*log((sqrt(2)*sqrt(a)*sqrt(-(a*cos(2*x)^2 + 2*a*cos(2*x) + a)/((cos 
(2*x) - 1)*sin(2*x)))*(cos(2*x) - 1) + a*cos(2*x) + a*sin(2*x) + a)/sin(2* 
x)) + sqrt(2)*sqrt(a)*(cos(2*x) + 1)*log(-(sqrt(2)*sqrt(a)*sqrt(-(a*cos(2* 
x)^2 + 2*a*cos(2*x) + a)/((cos(2*x) - 1)*sin(2*x)))*(cos(2*x) - 1) - a*cos 
(2*x) - a*sin(2*x) - a)/sin(2*x)) - 8*sqrt(-(a*cos(2*x)^2 + 2*a*cos(2*x) + 
 a)/((cos(2*x) - 1)*sin(2*x)))*sin(2*x))/(cos(2*x) + 1)
 

Sympy [F]

\[ \int \sqrt {a \cot ^3(x)} \, dx=\int \sqrt {a \cot ^{3}{\left (x \right )}}\, dx \] Input:

integrate((a*cot(x)**3)**(1/2),x)
 

Output:

Integral(sqrt(a*cot(x)**3), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.71 \[ \int \sqrt {a \cot ^3(x)} \, dx=-\frac {1}{4} \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (x\right )}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (x\right )} + \tan \left (x\right ) + 1\right )\right )} \sqrt {a} - \frac {2 \, \sqrt {a}}{\sqrt {\tan \left (x\right )}} \] Input:

integrate((a*cot(x)^3)^(1/2),x, algorithm="maxima")
 

Output:

-1/4*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(x)))) + 2*sqrt(2) 
*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(x)))) - sqrt(2)*log(sqrt(2)*sqr 
t(tan(x)) + tan(x) + 1) + sqrt(2)*log(-sqrt(2)*sqrt(tan(x)) + tan(x) + 1)) 
*sqrt(a) - 2*sqrt(a)/sqrt(tan(x))
 

Giac [F]

\[ \int \sqrt {a \cot ^3(x)} \, dx=\int { \sqrt {a \cot \left (x\right )^{3}} \,d x } \] Input:

integrate((a*cot(x)^3)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(a*cot(x)^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a \cot ^3(x)} \, dx=\int \sqrt {a\,{\mathrm {cot}\left (x\right )}^3} \,d x \] Input:

int((a*cot(x)^3)^(1/2),x)
 

Output:

int((a*cot(x)^3)^(1/2), x)
 

Reduce [F]

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

int((a*cot(x)^3)^(1/2),x)
 

Output:

sqrt(a)*( - 2*sqrt(cot(x)) - int(sqrt(cot(x))/cot(x),x))