\(\int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx\) [445]

Optimal result

Integrand size = 20, antiderivative size = 54 \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a} \] Output:

-arctan((a^4-b^4*csc(x)^2)^(1/4)/a)/a+arctanh((a^4-b^4*csc(x)^2)^(1/4)/a)/ 
a
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(245\) vs. \(2(54)=108\).

Time = 0.20 (sec) , antiderivative size = 245, normalized size of antiderivative = 4.54 \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=\frac {\sqrt [4]{-a^4+2 b^4+a^4 \cos (2 x)} \left (-2 \arctan \left (1-\frac {\sqrt {2} a \sqrt {\sin (x)}}{\sqrt [4]{b^4-a^4 \sin ^2(x)}}\right )+2 \arctan \left (1+\frac {\sqrt {2} a \sqrt {\sin (x)}}{\sqrt [4]{b^4-a^4 \sin ^2(x)}}\right )-\log \left (1+\frac {a^2 \sin (x)}{\sqrt {b^4-a^4 \sin ^2(x)}}-\frac {\sqrt {2} a \sqrt {\sin (x)}}{\sqrt [4]{b^4-a^4 \sin ^2(x)}}\right )+\log \left (1+\frac {a^2 \sin (x)}{\sqrt {b^4-a^4 \sin ^2(x)}}+\frac {\sqrt {2} a \sqrt {\sin (x)}}{\sqrt [4]{b^4-a^4 \sin ^2(x)}}\right )\right )}{2\ 2^{3/4} a \sqrt [4]{a^4-b^4 \csc ^2(x)} \sqrt {\sin (x)}} \] Input:

Integrate[Cot[x]/(a^4 - b^4*Csc[x]^2)^(1/4),x]
 

Output:

((-a^4 + 2*b^4 + a^4*Cos[2*x])^(1/4)*(-2*ArcTan[1 - (Sqrt[2]*a*Sqrt[Sin[x] 
])/(b^4 - a^4*Sin[x]^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*a*Sqrt[Sin[x]])/(b^ 
4 - a^4*Sin[x]^2)^(1/4)] - Log[1 + (a^2*Sin[x])/Sqrt[b^4 - a^4*Sin[x]^2] - 
 (Sqrt[2]*a*Sqrt[Sin[x]])/(b^4 - a^4*Sin[x]^2)^(1/4)] + Log[1 + (a^2*Sin[x 
])/Sqrt[b^4 - a^4*Sin[x]^2] + (Sqrt[2]*a*Sqrt[Sin[x]])/(b^4 - a^4*Sin[x]^2 
)^(1/4)]))/(2*2^(3/4)*a*(a^4 - b^4*Csc[x]^2)^(1/4)*Sqrt[Sin[x]])
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3042, 25, 4627, 243, 73, 27, 827, 216, 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.

\(\displaystyle \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {\tan \left (x+\frac {\pi }{2}\right )}{\sqrt [4]{a^4-b^4 \sec \left (x+\frac {\pi }{2}\right )^2}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\tan \left (x+\frac {\pi }{2}\right )}{\sqrt [4]{a^4-b^4 \sec \left (x+\frac {\pi }{2}\right )^2}}dx\)

\(\Big \downarrow \) 4627

\(\displaystyle -\int \frac {\sin (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}}d\csc (x)\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {1}{2} \int \frac {\sin (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}}d\csc ^2(x)\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 \int \frac {b^4 \csc ^4(x)}{a^4-\csc ^8(x)}d\sqrt [4]{a^4-b^4 \csc ^2(x)}}{b^4}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {\csc ^4(x)}{a^4-\csc ^8(x)}d\sqrt [4]{a^4-b^4 \csc ^2(x)}\)

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 216

\(\displaystyle 2 \left (\frac {1}{2} \int \frac {1}{a^2-\csc ^4(x)}d\sqrt [4]{a^4-b^4 \csc ^2(x)}-\frac {\arctan \left (\frac {\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{2 a}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle 2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{2 a}-\frac {\arctan \left (\frac {\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{2 a}\right )\)

Input:

Int[Cot[x]/(a^4 - b^4*Csc[x]^2)^(1/4),x]
 

Output:

2*(-1/2*ArcTan[(a^4 - b^4*Csc[x]^2)^(1/4)/a]/a + ArcTanh[(a^4 - b^4*Csc[x] 
^2)^(1/4)/a]/(2*a))
 

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 216

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

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 827

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

rule 3042

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

rule 4627

Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( 
f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Si 
mp[1/f   Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a + b*(c*ff*x)^n)^p/x), x] 
, x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[( 
m - 1)/2] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4] || IGtQ[p, 0] || Integers 
Q[2*n, p])
 

Maple [F]

\[\int \frac {\cot \left (x \right )}{\left (a^{4}-b^{4} \csc \left (x \right )^{2}\right )^{\frac {1}{4}}}d x\]

Input:

int(cot(x)/(a^4-b^4*csc(x)^2)^(1/4),x)
 

Output:

int(cot(x)/(a^4-b^4*csc(x)^2)^(1/4),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=\text {Timed out} \] Input:

integrate(cot(x)/(a^4-b^4*csc(x)^2)^(1/4),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(cot(x)/(a**4-b**4*csc(x)**2)**(1/4),x)
 

Output:

Integral(cot(x)/((a**2 - b**2*csc(x))*(a**2 + b**2*csc(x)))**(1/4), x)
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.37 \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=-\frac {\arctan \left (\frac {{\left (a^{4} - \frac {b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac {1}{4}}}{a}\right )}{a} + \frac {\log \left (a + {\left (a^{4} - \frac {b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac {1}{4}}\right )}{2 \, a} - \frac {\log \left (-a + {\left (a^{4} - \frac {b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac {1}{4}}\right )}{2 \, a} \] Input:

integrate(cot(x)/(a^4-b^4*csc(x)^2)^(1/4),x, algorithm="maxima")
 

Output:

-arctan((a^4 - b^4/sin(x)^2)^(1/4)/a)/a + 1/2*log(a + (a^4 - b^4/sin(x)^2) 
^(1/4))/a - 1/2*log(-a + (a^4 - b^4/sin(x)^2)^(1/4))/a
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.41 \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=-\frac {\arctan \left (\frac {{\left (a^{4} - \frac {b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac {1}{4}}}{a}\right )}{a} + \frac {\log \left ({\left | a + {\left (a^{4} - \frac {b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac {1}{4}} \right |}\right )}{2 \, a} - \frac {\log \left ({\left | -a + {\left (a^{4} - \frac {b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac {1}{4}} \right |}\right )}{2 \, a} \] Input:

integrate(cot(x)/(a^4-b^4*csc(x)^2)^(1/4),x, algorithm="giac")
 

Output:

-arctan((a^4 - b^4/sin(x)^2)^(1/4)/a)/a + 1/2*log(abs(a + (a^4 - b^4/sin(x 
)^2)^(1/4)))/a - 1/2*log(abs(-a + (a^4 - b^4/sin(x)^2)^(1/4)))/a
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=\int \frac {\mathrm {cot}\left (x\right )}{{\left (a^4-\frac {b^4}{{\sin \left (x\right )}^2}\right )}^{1/4}} \,d x \] Input:

int(cot(x)/(a^4 - b^4/sin(x)^2)^(1/4),x)
 

Output:

int(cot(x)/(a^4 - b^4/sin(x)^2)^(1/4), x)
 

Reduce [F]

\[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=\int \frac {\cot \left (x \right )}{\left (-\csc \left (x \right )^{2} b^{4}+a^{4}\right )^{\frac {1}{4}}}d x \] Input:

int(cot(x)/(a^4-b^4*csc(x)^2)^(1/4),x)
 

Output:

int(cot(x)/( - csc(x)**2*b**4 + a**4)**(1/4),x)