\(\int \cot (x) (a+b \cot ^4(x))^{3/2} \, dx\) [61]

Optimal result

Integrand size = 15, antiderivative size = 137 \[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=\frac {1}{4} \sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+\frac {1}{2} (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\frac {1}{2} (a+b) \sqrt {a+b \cot ^4(x)}+\frac {1}{4} b \cot ^2(x) \sqrt {a+b \cot ^4(x)}-\frac {1}{6} \left (a+b \cot ^4(x)\right )^{3/2} \] Output:

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

Mathematica [A] (verified)

Time = 2.89 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.22 \[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=\frac {1}{12} \left (6 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+6 (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\sqrt {a+b \cot ^4(x)} \left (8 a+6 b-3 b \cot ^2(x)+2 b \cot ^4(x)\right )+\frac {3 \sqrt {a} \sqrt {b} \text {arcsinh}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a}}\right ) \sqrt {a+b \cot ^4(x)}}{\sqrt {1+\frac {b \cot ^4(x)}{a}}}\right ) \] Input:

Integrate[Cot[x]*(a + b*Cot[x]^4)^(3/2),x]
 

Output:

(6*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]] + 6*(a 
 + b)^(3/2)*ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])] - 
 Sqrt[a + b*Cot[x]^4]*(8*a + 6*b - 3*b*Cot[x]^2 + 2*b*Cot[x]^4) + (3*Sqrt[ 
a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Cot[x]^2)/Sqrt[a]]*Sqrt[a + b*Cot[x]^4])/Sqrt[ 
1 + (b*Cot[x]^4)/a])/12
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.95, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.867, Rules used = {3042, 25, 4153, 25, 1577, 493, 682, 27, 719, 224, 219, 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.

\(\displaystyle \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\tan \left (x+\frac {\pi }{2}\right ) \left (a+b \tan \left (x+\frac {\pi }{2}\right )^4\right )^{3/2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \tan \left (x+\frac {\pi }{2}\right ) \left (b \tan \left (x+\frac {\pi }{2}\right )^4+a\right )^{3/2}dx\)

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1577

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

\(\Big \downarrow \) 493

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

\(\Big \downarrow \) 682

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

Int[Cot[x]*(a + b*Cot[x]^4)^(3/2),x]
 

Output:

((Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]] + 2 
*(a + b)^(3/2)*ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4]) 
])/2 - ((2*(a + b) - b*Cot[x]^2)*Sqrt[a + b*Cot[x]^4])/2 - (a + b*Cot[x]^4 
)^(3/2)/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 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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 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 493

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] + Simp[2*(p/(d*(n + 
 2*p + 1)))   Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*(a*d - b*c*x), x], x] /; 
FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && NeQ[n + 2*p + 1, 0] && ( !Rationa 
lQ[n] || LtQ[n, 1]) &&  !ILtQ[n + 2*p, 0] && IntQuadraticQ[a, 0, b, c, d, n 
, p, x]
 

rule 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

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. \(244\) vs. \(2(109)=218\).

Time = 0.18 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.79

method	result	size
derivativedivides	\(-\frac {b \sqrt {a +b \cot \left (x \right )^{4}}}{2}-\frac {b^{2} \left (\frac {\cot \left (x \right )^{4} \sqrt {a +b \cot \left (x \right )^{4}}}{3 b}-\frac {2 a \sqrt {a +b \cot \left (x \right )^{4}}}{3 b^{2}}\right )}{2}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \sqrt {a +b}}+\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \cot \left (x \right )^{2}+\sqrt {a +b \cot \left (x \right )^{4}}\right )}{2}+a \sqrt {b}\, \ln \left (\sqrt {b}\, \cot \left (x \right )^{2}+\sqrt {a +b \cot \left (x \right )^{4}}\right )+\frac {b^{2} \left (\frac {\cot \left (x \right )^{2} \sqrt {a +b \cot \left (x \right )^{4}}}{2 b}-\frac {a \ln \left (\sqrt {b}\, \cot \left (x \right )^{2}+\sqrt {a +b \cot \left (x \right )^{4}}\right )}{2 b^{\frac {3}{2}}}\right )}{2}-a \sqrt {a +b \cot \left (x \right )^{4}}\)	\(245\)
default	\(-\frac {b \sqrt {a +b \cot \left (x \right )^{4}}}{2}-\frac {b^{2} \left (\frac {\cot \left (x \right )^{4} \sqrt {a +b \cot \left (x \right )^{4}}}{3 b}-\frac {2 a \sqrt {a +b \cot \left (x \right )^{4}}}{3 b^{2}}\right )}{2}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \sqrt {a +b}}+\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \cot \left (x \right )^{2}+\sqrt {a +b \cot \left (x \right )^{4}}\right )}{2}+a \sqrt {b}\, \ln \left (\sqrt {b}\, \cot \left (x \right )^{2}+\sqrt {a +b \cot \left (x \right )^{4}}\right )+\frac {b^{2} \left (\frac {\cot \left (x \right )^{2} \sqrt {a +b \cot \left (x \right )^{4}}}{2 b}-\frac {a \ln \left (\sqrt {b}\, \cot \left (x \right )^{2}+\sqrt {a +b \cot \left (x \right )^{4}}\right )}{2 b^{\frac {3}{2}}}\right )}{2}-a \sqrt {a +b \cot \left (x \right )^{4}}\)	\(245\)

Input:

int(cot(x)*(a+b*cot(x)^4)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (111) = 222\).

Time = 0.24 (sec) , antiderivative size = 1486, normalized size of antiderivative = 10.85 \[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate(cot(x)*(a+b*cot(x)^4)^(3/2),x, algorithm="fricas")
 

Output:

[1/24*(6*((a + b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*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)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - b^2)*cos( 
2*x)) + 3*((3*a + 2*b)*cos(2*x)^2 - 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sq 
rt(b)*log(-((a + 2*b)*cos(2*x)^2 - 2*(cos(2*x)^2 - 1)*sqrt(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)) 
- 2*(a - 2*b)*cos(2*x) + a + 2*b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - 2*((8*a 
 + 11*b)*cos(2*x)^2 - 8*(2*a + b)*cos(2*x) + 8*a + 5*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)))/(cos( 
2*x)^2 - 2*cos(2*x) + 1), 1/12*(3*((3*a + 2*b)*cos(2*x)^2 - 2*(3*a + 2*b)* 
cos(2*x) + 3*a + 2*b)*sqrt(-b)*arctan(sqrt(-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))*(cos(2*x) - 1)/ 
(b*cos(2*x) + b)) + 3*((a + b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sq 
rt(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)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^ 
2 - b^2)*cos(2*x)) - ((8*a + 11*b)*cos(2*x)^2 - 8*(2*a + b)*cos(2*x) + 8*a 
 + 5*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)))/(cos(2*x)^2 - 2*cos(2*x) + 1), -1/24*(12*((a + b)*...
 

Sympy [F]

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

integrate(cot(x)*(a+b*cot(x)**4)**(3/2),x)
 

Output:

Integral((a + b*cot(x)**4)**(3/2)*cot(x), x)
 

Maxima [F]

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

integrate(cot(x)*(a+b*cot(x)^4)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*cot(x)^4 + a)^(3/2)*cot(x), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (111) = 222\).

Time = 0.40 (sec) , antiderivative size = 445, normalized size of antiderivative = 3.25 \[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=-\frac {{\left (3 \, a b + 2 \, b^{2}\right )} \arctan \left (-\frac {\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b}} - \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left ({\left | -{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} {\left (a + b\right )} + \sqrt {a + b} b \right |}\right )}{2 \, \sqrt {a + b}} - \frac {3 \, {\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )}^{5} {\left (5 \, a b + 6 \, b^{2}\right )} + 8 \, {\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )}^{3} b^{3} - 12 \, {\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )}^{4} {\left (a b + 3 \, b^{2}\right )} \sqrt {a + b} + 12 \, {\left (a b^{2} + b^{3}\right )} {\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )}^{2} \sqrt {a + b} + 3 \, {\left (3 \, a b^{3} + 2 \, b^{4}\right )} {\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} - 8 \, {\left (a b^{3} + b^{4}\right )} \sqrt {a + b}}{6 \, {\left ({\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )}^{2} - b\right )}^{3}} \] Input:

integrate(cot(x)*(a+b*cot(x)^4)^(3/2),x, algorithm="giac")
 

Output:

-1/2*(3*a*b + 2*b^2)*arctan(-(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*s 
in(x)^4 - 2*b*sin(x)^2 + b))/sqrt(-b))/sqrt(-b) - 1/2*(a^2 + 2*a*b + b^2)* 
log(abs(-(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x) 
^2 + b))*(a + b) + sqrt(a + b)*b))/sqrt(a + b) - 1/6*(3*(sqrt(a + b)*sin(x 
)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))^5*(5*a*b + 6*b^2) 
+ 8*(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + 
b))^3*b^3 - 12*(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b* 
sin(x)^2 + b))^4*(a*b + 3*b^2)*sqrt(a + b) + 12*(a*b^2 + b^3)*(sqrt(a + b) 
*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))^2*sqrt(a + b 
) + 3*(3*a*b^3 + 2*b^4)*(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x) 
^4 - 2*b*sin(x)^2 + b)) - 8*(a*b^3 + b^4)*sqrt(a + b))/((sqrt(a + b)*sin(x 
)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))^2 - b)^3
 

Mupad [F(-1)]

Timed out. \[ \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx=\int \mathrm {cot}\left (x\right )\,{\left (b\,{\mathrm {cot}\left (x\right )}^4+a\right )}^{3/2} \,d x \] Input:

int(cot(x)*(a + b*cot(x)^4)^(3/2),x)
 

Output:

int(cot(x)*(a + b*cot(x)^4)^(3/2), x)
 

Reduce [F]

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

int(cot(x)*(a+b*cot(x)^4)^(3/2),x)
 

Output:

( - 2*sqrt(cot(x)**4*b + a)*cot(x)**4*b + 3*sqrt(cot(x)**4*b + a)*cot(x)** 
2*b - 8*sqrt(cot(x)**4*b + a)*a - 6*sqrt(cot(x)**4*b + a)*b + 12*int((sqrt 
(cot(x)**4*b + a)*cot(x))/(cot(x)**4*b + a),x)*a**2 + 6*int((sqrt(cot(x)** 
4*b + a)*cot(x))/(cot(x)**4*b + a),x)*a*b - 18*int((sqrt(cot(x)**4*b + a)* 
cot(x)**3)/(cot(x)**4*b + a),x)*a*b - 12*int((sqrt(cot(x)**4*b + a)*cot(x) 
**3)/(cot(x)**4*b + a),x)*b**2)/12