\(\int \frac {\cot (x)}{(a+b \cot ^4(x))^{3/2}} \, dx\) [63]

Optimal result

Integrand size = 15, antiderivative size = 74 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{3/2}}-\frac {a+b \cot ^2(x)}{2 a (a+b) \sqrt {a+b \cot ^4(x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{3/2}} \, dx=\frac {1}{2} \left (\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{(a+b)^{3/2}}-\frac {a+b \cot ^2(x)}{a (a+b) \sqrt {a+b \cot ^4(x)}}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 25, 4153, 25, 1577, 496, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1577

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

\(\Big \downarrow \) 496

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(a + b)^(3/2 
) - (a + b*Cot[x]^2)/(a*(a + b)*Sqrt[a + b*Cot[x]^4]))/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 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. \(247\) vs. \(2(62)=124\).

Time = 1.39 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.35

method	result	size
derivativedivides	\(-\frac {b \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \left (\sqrt {-a b}+b \right ) \left (\sqrt {-a b}-b \right ) \sqrt {a +b}}-\frac {\sqrt {b \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}+b \right ) a \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}+\frac {\sqrt {b \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}-b \right ) a \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}\)	\(248\)
default	\(-\frac {b \ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \left (\sqrt {-a b}+b \right ) \left (\sqrt {-a b}-b \right ) \sqrt {a +b}}-\frac {\sqrt {b \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}+b \right ) a \left (\cot \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}+\frac {\sqrt {b \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}-b \right ) a \left (\cot \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}\)	\(248\)

Input:

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

Output:

-1/2*b/((-a*b)^(1/2)+b)/((-a*b)^(1/2)-b)/(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))-1/4/((-a*b)^(1/2)+b)/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)+1/4/((-a*b) 
^(1/2)-b)/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)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (64) = 128\).

Time = 0.23 (sec) , antiderivative size = 670, normalized size of antiderivative = 9.05 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/4*(((a^2 + a*b)*cos(2*x)^2 + a^2 + a*b - 2*(a^2 - a*b)*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)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - 
b^2)*cos(2*x)) - 2*((a^2 - b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 + 
a*b)*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^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + (a^4 + 3* 
a^3*b + 3*a^2*b^2 + a*b^3)*cos(2*x)^2 - 2*(a^4 + a^3*b - a^2*b^2 - a*b^3)* 
cos(2*x)), -1/2*(((a^2 + a*b)*cos(2*x)^2 + a^2 + a*b - 2*(a^2 - a*b)*cos(2 
*x))*sqrt(-a - b)*arctan(((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 + 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 
2*(a^2 - b^2)*cos(2*x))) + ((a^2 - b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2 
*(a^2 + a*b)*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^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + ( 
a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(2*x)^2 - 2*(a^4 + a^3*b - a^2*b^2 - 
 a*b^3)*cos(2*x))]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.50 \[ \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{3/2}} \, dx=-\frac {\frac {{\left (a - b\right )} \sin \left (x\right )^{2}}{a^{2} + a b} + \frac {b}{a^{2} + a b}}{2 \, \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}} - \frac {\log \left ({\left | -{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} \sqrt {a + b} + b \right |}\right )}{2 \, {\left (a + b\right )}^{\frac {3}{2}}} \] Input:

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

Output:

-1/2*((a - b)*sin(x)^2/(a^2 + a*b) + b/(a^2 + a*b))/sqrt(a*sin(x)^4 + b*si 
n(x)^4 - 2*b*sin(x)^2 + b) - 1/2*log(abs(-(sqrt(a + b)*sin(x)^2 - sqrt(a*s 
in(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))*sqrt(a + b) + b))/(a + b)^(3/2)
 

Mupad [F(-1)]

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

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

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

Output:

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