\(\int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx\) [485]

Optimal result

Integrand size = 15, antiderivative size = 28 \[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sin ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sin ^3(x)}}{\sqrt {a}}\right )}{3 \sqrt {a}} \] Input:

Integrate[Cot[x]/Sqrt[a + b*Sin[x]^3],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3709, 798, 73, 221}

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.

\(\displaystyle \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (x) \sqrt {a+b \sin (x)^3}}dx\)

\(\Big \downarrow \) 3709

\(\displaystyle \int \frac {\csc (x)}{\sqrt {a+b \sin ^3(x)}}d\sin (x)\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {1}{3} \int \frac {\csc (x)}{\sqrt {b \sin ^3(x)+a}}d\sin ^3(x)\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 \int \frac {1}{\frac {\sin ^6(x)}{b}-\frac {a}{b}}d\sqrt {b \sin ^3(x)+a}}{3 b}\)

\(\Big \downarrow \) 221

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

Input:

Int[Cot[x]/Sqrt[a + b*Sin[x]^3],x]
 

Output:

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

Defintions of rubi rules used

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 221

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

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 3042

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

rule 3709

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( 
f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si 
mp[ff^(m + 1)/f   Subst[Int[x^m*((a + b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((m + 
1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && 
ILtQ[(m - 1)/2, 0]
 

Maple [F]

\[\int \frac {\cot \left (x \right )}{\sqrt {a +b \sin \left (x \right )^{3}}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   fail 
ed of mode Union(SparseUnivariatePolynomial(Expression(Complex(Integer))), 
failed) cannot be coerced to mode SparseUnivariatePolynomial(Expression(Co 
mplex(Int
 

Sympy [F]

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

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

Output:

Integral(cot(x)/sqrt(a + b*sin(x)**3), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=\frac {\log \left (\frac {\sqrt {b \sin \left (x\right )^{3} + a} - \sqrt {a}}{\sqrt {b \sin \left (x\right )^{3} + a} + \sqrt {a}}\right )}{3 \, \sqrt {a}} \] Input:

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

Output:

1/3*log((sqrt(b*sin(x)^3 + a) - sqrt(a))/(sqrt(b*sin(x)^3 + a) + sqrt(a))) 
/sqrt(a)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\cot (x)}{\sqrt {a+b \sin ^3(x)}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {b \sin \left (x\right )^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} \] Input:

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

Output:

2/3*arctan(sqrt(b*sin(x)^3 + a)/sqrt(-a))/sqrt(-a)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(sin(x)**3*b + a)*cot(x))/(sin(x)**3*b + a),x)