\(\int \frac {1}{(\csc (x)-\sin (x))^{3/2}} \, dx\) [212]

Optimal result

Integrand size = 11, antiderivative size = 80 \[ \int \frac {1}{(\csc (x)-\sin (x))^{3/2}} \, dx=\frac {\sec (x)}{2 \sqrt {\cos (x) \cot (x)}}+\frac {\arctan \left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{4 \sqrt {\cos (x) \cot (x)}}+\frac {\text {arctanh}\left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{4 \sqrt {\cos (x) \cot (x)}} \] Output:

1/2*sec(x)/(cos(x)*cot(x))^(1/2)+1/4*arctan((-sin(x))^(1/2))*cot(x)*(-sin( 
x))^(1/2)/(cos(x)*cot(x))^(1/2)+1/4*arctanh((-sin(x))^(1/2))*cot(x)*(-sin( 
x))^(1/2)/(cos(x)*cot(x))^(1/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(\csc (x)-\sin (x))^{3/2}} \, dx=\frac {-\arctan \left (\sqrt [4]{\sin ^2(x)}\right ) \cos (x)-\text {arctanh}\left (\sqrt [4]{\sin ^2(x)}\right ) \cos (x)+2 \sec (x) \sqrt [4]{\sin ^2(x)}}{4 \sqrt {\cos (x) \cot (x)} \sqrt [4]{\sin ^2(x)}} \] Input:

Integrate[(Csc[x] - Sin[x])^(-3/2),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.49 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.273, Rules used = {3042, 4897, 3042, 4900, 3042, 3077, 3042, 3081, 3042, 3044, 266, 756, 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.

\(\displaystyle \int \frac {1}{(\csc (x)-\sin (x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(\csc (x)-\sin (x))^{3/2}}dx\)

\(\Big \downarrow \) 4897

\(\displaystyle \int \frac {1}{(\cos (x) \cot (x))^{3/2}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(\cos (x) \cot (x))^{3/2}}dx\)

\(\Big \downarrow \) 4900

\(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \int \frac {1}{\cos ^{\frac {3}{2}}(x) \cot ^{\frac {3}{2}}(x)}dx}{\sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \int \frac {1}{\sin \left (x+\frac {\pi }{2}\right )^{3/2} \left (-\tan \left (x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{\sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 3077

\(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}-\frac {1}{4} \int \frac {\sqrt {\cot (x)}}{\cos ^{\frac {3}{2}}(x)}dx\right )}{\sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}-\frac {1}{4} \int \frac {\sqrt {-\tan \left (x+\frac {\pi }{2}\right )}}{\sin \left (x+\frac {\pi }{2}\right )^{3/2}}dx\right )}{\sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 3081

\(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}-\frac {\sqrt {-\sin (x)} \sqrt {\cot (x)} \int \frac {\sec (x)}{\sqrt {-\sin (x)}}dx}{4 \sqrt {\cos (x)}}\right )}{\sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}-\frac {\sqrt {-\sin (x)} \sqrt {\cot (x)} \int \frac {1}{\cos (x) \sqrt {-\sin (x)}}dx}{4 \sqrt {\cos (x)}}\right )}{\sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 3044

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 756

\(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {\sqrt {-\sin (x)} \sqrt {\cot (x)} \left (\frac {1}{2} \int \frac {1}{1-\sin ^2(x)}d\sqrt {-\sin (x)}+\frac {1}{2} \int \frac {1}{\sin ^2(x)+1}d\sqrt {-\sin (x)}\right )}{2 \sqrt {\cos (x)}}+\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )}{\sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {\sqrt {-\sin (x)} \sqrt {\cot (x)} \left (\frac {1}{2} \int \frac {1}{1-\sin ^2(x)}d\sqrt {-\sin (x)}-\frac {1}{2} \arctan (\sin (x))\right )}{2 \sqrt {\cos (x)}}+\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )}{\sqrt {\cos (x) \cot (x)}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {\cos (x)} \sqrt {\cot (x)} \left (\frac {\sqrt {-\sin (x)} \sqrt {\cot (x)} \left (-\frac {1}{2} \arctan (\sin (x))-\frac {1}{2} \text {arctanh}(\sin (x))\right )}{2 \sqrt {\cos (x)}}+\frac {1}{2 \cos ^{\frac {3}{2}}(x) \sqrt {\cot (x)}}\right )}{\sqrt {\cos (x) \cot (x)}}\)

Input:

Int[(Csc[x] - Sin[x])^(-3/2),x]
 

Output:

(Sqrt[Cos[x]]*Sqrt[Cot[x]]*(1/(2*Cos[x]^(3/2)*Sqrt[Cot[x]]) + ((-1/2*ArcTa 
n[Sin[x]] - ArcTanh[Sin[x]]/2)*Sqrt[Cot[x]]*Sqrt[-Sin[x]])/(2*Sqrt[Cos[x]] 
)))/Sqrt[Cos[x]*Cot[x]]
 

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 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   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 3077

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n + 1)/(b*f*(m 
 + n + 1))), x] - Simp[(n + 1)/(b^2*(m + n + 1))   Int[(a*Sin[e + f*x])^m*( 
b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] 
&& NeQ[m + n + 1, 0] && IntegersQ[2*m, 2*n] &&  !(EqQ[n, -3/2] && EqQ[m, 1] 
)
 

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])
 

Maple [A] (verified)

Time = 10.90 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.19

method	result	size
default	\(\frac {\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (2+2 \sec \left (x \right )\right )-\arctan \left (\frac {\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\cos \left (x \right )-1}\right ) \cos \left (x \right )+\operatorname {arctanh}\left (\frac {\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sin \left (x \right )}{\cos \left (x \right )-1}\right ) \cos \left (x \right )}{4 \left (\cos \left (x \right )+1\right ) \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {\cos \left (x \right ) \cot \left (x \right )}}\)	\(95\)

Input:

int(1/(csc(x)-sin(x))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/4/(cos(x)+1)/(sin(x)/(cos(x)+1)^2)^(1/2)/(cos(x)*cot(x))^(1/2)*((sin(x)/ 
(cos(x)+1)^2)^(1/2)*(2+2*sec(x))-arctan((sin(x)/(cos(x)+1)^2)^(1/2)*sin(x) 
/(cos(x)-1))*cos(x)+arctanh((sin(x)/(cos(x)+1)^2)^(1/2)*sin(x)/(cos(x)-1)) 
*cos(x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (60) = 120\).

Time = 0.11 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.94 \[ \int \frac {1}{(\csc (x)-\sin (x))^{3/2}} \, dx=-\frac {2 \, \arctan \left (-\frac {\sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} {\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )}}{2 \, {\left (\cos \left (x\right ) + \sin \left (x\right ) + 1\right )}}\right ) \cos \left (x\right )^{3} - \cos \left (x\right )^{3} \log \left (\frac {\cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} + 6 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) - 4 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} - 2 \, \cos \left (x\right ) + 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right ) - 8 \, \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{16 \, \cos \left (x\right )^{3}} \] Input:

integrate(1/(csc(x)-sin(x))^(3/2),x, algorithm="fricas")
 

Output:

-1/16*(2*arctan(-1/2*sqrt(cos(x)^2/sin(x))*(cos(x) - sin(x) + 1)/(cos(x) + 
 sin(x) + 1))*cos(x)^3 - cos(x)^3*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)) - 8*sqrt(cos(x)^2/sin(x))*sin(x))/cos(x)^3
 

Sympy [F]

\[ \int \frac {1}{(\csc (x)-\sin (x))^{3/2}} \, dx=\int \frac {1}{\left (- \sin {\left (x \right )} + \csc {\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(csc(x)-sin(x))**(3/2),x)
 

Output:

Integral((-sin(x) + csc(x))**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{(\csc (x)-\sin (x))^{3/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(csc(x)-sin(x))^(3/2),x, algorithm="maxima")
 

Output:

integrate((csc(x) - sin(x))^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{(\csc (x)-\sin (x))^{3/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(csc(x)-sin(x))^(3/2),x, algorithm="giac")
 

Output:

integrate((csc(x) - sin(x))^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(\csc (x)-\sin (x))^{3/2}} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (x\right )}-\sin \left (x\right )\right )}^{3/2}} \,d x \] Input:

int(1/(1/sin(x) - sin(x))^(3/2),x)
 

Output:

int(1/(1/sin(x) - sin(x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{(\csc (x)-\sin (x))^{3/2}} \, dx=\int \frac {\sqrt {\csc \left (x \right )-\sin \left (x \right )}}{\csc \left (x \right )^{2}-2 \csc \left (x \right ) \sin \left (x \right )+\sin \left (x \right )^{2}}d x \] Input:

int(1/(csc(x)-sin(x))^(3/2),x)
 

Output:

int(sqrt(csc(x) - sin(x))/(csc(x)**2 - 2*csc(x)*sin(x) + sin(x)**2),x)