3.4.18 \(\int \frac {1}{\sqrt {\csc (x)-\sin (x)}} \, dx\) [318]

3.4.18.1 Optimal result

Integrand size = 11, antiderivative size = 60 \[ \int \frac {1}{\sqrt {\csc (x)-\sin (x)}} \, dx=\frac {\arctan \left (\sqrt {-\sin (x)}\right ) \cos (x)}{\sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}}-\frac {\text {arctanh}\left (\sqrt {-\sin (x)}\right ) \cos (x)}{\sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}} \]

output

arctan((-sin(x))^(1/2))*cos(x)/(cos(x)*cot(x))^(1/2)/(-sin(x))^(1/2)-arcta 
nh((-sin(x))^(1/2))*cos(x)/(cos(x)*cot(x))^(1/2)/(-sin(x))^(1/2)
 

3.4.18.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\sqrt {\csc (x)-\sin (x)}} \, dx=-\frac {\left (\arctan \left (\sqrt [4]{\sin ^2(x)}\right )-\text {arctanh}\left (\sqrt [4]{\sin ^2(x)}\right )\right ) \sqrt {\cos (x) \cot (x)} \sin (x) \tan (x)}{\sin ^2(x)^{3/4}} \]

input

Integrate[1/Sqrt[Csc[x] - Sin[x]],x]
 

output

-(((ArcTan[(Sin[x]^2)^(1/4)] - ArcTanh[(Sin[x]^2)^(1/4)])*Sqrt[Cos[x]*Cot[ 
x]]*Sin[x]*Tan[x])/(Sin[x]^2)^(3/4))
 

3.4.18.3 Rubi [A] (warning: unable to verify)

Time = 0.38 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.60, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.091, Rules used = {3042, 4897, 3042, 4900, 3042, 3081, 3042, 3044, 266, 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.

\(\displaystyle \int \frac {1}{\sqrt {\csc (x)-\sin (x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt {\csc (x)-\sin (x)}}dx\)

\(\Big \downarrow \) 4897

\(\displaystyle \int \frac {1}{\sqrt {\cos (x) \cot (x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt {\cos (x) \cot (x)}}dx\)

\(\Big \downarrow \) 4900

\(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \int \frac {1}{\sqrt {\cos (x)} \sqrt {\cot (x)}}dx}{\sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3081

\(\displaystyle \frac {\cos (x) \int \sec (x) \sqrt {-\sin (x)}dx}{\sqrt {-\sin (x)} \sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\cos (x) \int \frac {\sqrt {-\sin (x)}}{\cos (x)}dx}{\sqrt {-\sin (x)} \sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 3044

\(\displaystyle -\frac {\cos (x) \int \frac {\sqrt {-\sin (x)}}{1-\sin ^2(x)}d(-\sin (x))}{\sqrt {-\sin (x)} \sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {2 \cos (x) \int \frac {\sin ^2(x)}{1-\sin ^4(x)}d\sqrt {-\sin (x)}}{\sqrt {-\sin (x)} \sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 \cos (x) \left (\frac {1}{2} \arctan (\sin (x))-\frac {1}{2} \text {arctanh}(\sin (x))\right )}{\sqrt {-\sin (x)} \sqrt {\cos (x) \cot (x)}}\)

input

Int[1/Sqrt[Csc[x] - Sin[x]],x]
 

output

(-2*(ArcTan[Sin[x]]/2 - ArcTanh[Sin[x]]/2)*Cos[x])/(Sqrt[Cos[x]*Cot[x]]*Sq 
rt[-Sin[x]])
 

3.4.18.3.1 Defintions of rubi rules used

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 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 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 3044

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ 
Symbol] :> Simp[1/(a*f)   Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a 
*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&  !(I 
ntegerQ[(m - 1)/2] && LtQ[0, m, n])
 

rule 3081

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[Cos[e + f*x]^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^ 
n)   Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b, e, 
 f, m, n}, x] &&  !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(- 
1)]) || IntegersQ[m - 1/2, n - 1/2])
 

rule 4897

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]
 

rule 4900

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTri 
g[u], vv = ActivateTrig[v], ww = ActivateTrig[w]}, Simp[(vv^m*ww^n)^FracPar 
t[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p]))   Int[uu*vv^(m*p)*ww^(n*p), x] 
, x]] /; FreeQ[{m, n, p}, x] &&  !IntegerQ[p] && ( !InertTrigFreeQ[v] ||  ! 
InertTrigFreeQ[w])
 

3.4.18.4 Maple [F]

\[\int \frac {1}{\sqrt {\csc \left (x \right )-\sin \left (x \right )}}d x\]

input

int(1/(csc(x)-sin(x))^(1/2),x)
 

output

int(1/(csc(x)-sin(x))^(1/2),x)
 

3.4.18.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (48) = 96\).

Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.07 \[ \int \frac {1}{\sqrt {\csc (x)-\sin (x)}} \, dx=\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{\cos \left (x\right ) \sin \left (x\right ) - \cos \left (x\right )}\right ) + \frac {1}{4} \, \log \left (\frac {\cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} + 6 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) + 4 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} - 2 \, \cos \left (x\right ) + 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right ) \]

input

integrate(1/(csc(x)-sin(x))^(1/2),x, algorithm="fricas")
 

output

1/2*arctan(2*sqrt(cos(x)^2/sin(x))*sin(x)/(cos(x)*sin(x) - cos(x))) + 1/4* 
log((cos(x)^3 - 5*cos(x)^2 - (cos(x)^2 + 6*cos(x) + 4)*sin(x) + 4*(cos(x)^ 
2 - (cos(x) + 1)*sin(x) - 1)*sqrt(cos(x)^2/sin(x)) - 2*cos(x) + 4)/(cos(x) 
^3 + 3*cos(x)^2 - (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4))
 

3.4.18.6 Sympy [F]

\[ \int \frac {1}{\sqrt {\csc (x)-\sin (x)}} \, dx=\int \frac {1}{\sqrt {- \sin {\left (x \right )} + \csc {\left (x \right )}}}\, dx \]

input

integrate(1/(csc(x)-sin(x))**(1/2),x)
 

output

Integral(1/sqrt(-sin(x) + csc(x)), x)
 

3.4.18.7 Maxima [F]

\[ \int \frac {1}{\sqrt {\csc (x)-\sin (x)}} \, dx=\int { \frac {1}{\sqrt {\csc \left (x\right ) - \sin \left (x\right )}} \,d x } \]

input

integrate(1/(csc(x)-sin(x))^(1/2),x, algorithm="maxima")
 

output

integrate(1/sqrt(csc(x) - sin(x)), x)
 

3.4.18.8 Giac [F]

\[ \int \frac {1}{\sqrt {\csc (x)-\sin (x)}} \, dx=\int { \frac {1}{\sqrt {\csc \left (x\right ) - \sin \left (x\right )}} \,d x } \]

input

integrate(1/(csc(x)-sin(x))^(1/2),x, algorithm="giac")
 

output

integrate(1/sqrt(csc(x) - sin(x)), x)
 

3.4.18.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {\csc (x)-\sin (x)}} \, dx=\int \frac {1}{\sqrt {\frac {1}{\sin \left (x\right )}-\sin \left (x\right )}} \,d x \]

input

int(1/(1/sin(x) - sin(x))^(1/2),x)
 

output

int(1/(1/sin(x) - sin(x))^(1/2), x)