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

Optimal. Leaf size=60 \[ \frac{\cos (x) \tan ^{-1}\left (\sqrt{-\sin (x)}\right )}{\sqrt{-\sin (x)} \sqrt{\cos (x) \cot (x)}}-\frac{\cos (x) \tanh ^{-1}\left (\sqrt{-\sin (x)}\right )}{\sqrt{-\sin (x)} \sqrt{\cos (x) \cot (x)}} \]

[Out]

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

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Rubi [A]  time = 0.0906745, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {4397, 4400, 2601, 2564, 329, 298, 203, 206} \[ \frac{\cos (x) \tan ^{-1}\left (\sqrt{-\sin (x)}\right )}{\sqrt{-\sin (x)} \sqrt{\cos (x) \cot (x)}}-\frac{\cos (x) \tanh ^{-1}\left (\sqrt{-\sin (x)}\right )}{\sqrt{-\sin (x)} \sqrt{\cos (x) \cot (x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Csc[x] - Sin[x]],x]

[Out]

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

Rule 4397

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

Rule 4400

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = Ac
tivateTrig[w]}, Dist[(vv^m*ww^n)^FracPart[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])

Rule 2601

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(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 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[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] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

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

Rule 203

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

Rule 206

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{\csc (x)-\sin (x)}} \, dx &=\int \frac{1}{\sqrt{\cos (x) \cot (x)}} \, dx\\ &=\frac{\left (\sqrt{\cos (x)} \sqrt{\cot (x)}\right ) \int \frac{1}{\sqrt{\cos (x)} \sqrt{\cot (x)}} \, dx}{\sqrt{\cos (x) \cot (x)}}\\ &=\frac{\cos (x) \int \sec (x) \sqrt{-\sin (x)} \, dx}{\sqrt{\cos (x) \cot (x)} \sqrt{-\sin (x)}}\\ &=-\frac{\cos (x) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1-x^2} \, dx,x,-\sin (x)\right )}{\sqrt{\cos (x) \cot (x)} \sqrt{-\sin (x)}}\\ &=-\frac{(2 \cos (x)) \operatorname{Subst}\left (\int \frac{x^2}{1-x^4} \, dx,x,\sqrt{-\sin (x)}\right )}{\sqrt{\cos (x) \cot (x)} \sqrt{-\sin (x)}}\\ &=-\frac{\cos (x) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{-\sin (x)}\right )}{\sqrt{\cos (x) \cot (x)} \sqrt{-\sin (x)}}+\frac{\cos (x) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-\sin (x)}\right )}{\sqrt{\cos (x) \cot (x)} \sqrt{-\sin (x)}}\\ &=\frac{\tan ^{-1}\left (\sqrt{-\sin (x)}\right ) \cos (x)}{\sqrt{\cos (x) \cot (x)} \sqrt{-\sin (x)}}-\frac{\tanh ^{-1}\left (\sqrt{-\sin (x)}\right ) \cos (x)}{\sqrt{\cos (x) \cot (x)} \sqrt{-\sin (x)}}\\ \end{align*}

Mathematica [A]  time = 0.266845, size = 44, normalized size = 0.73 \[ -\frac{\sin (x) \tan (x) \sqrt{\cos (x) \cot (x)} \left (\tan ^{-1}\left (\sqrt [4]{\sin ^2(x)}\right )-\tanh ^{-1}\left (\sqrt [4]{\sin ^2(x)}\right )\right )}{\sin ^2(x)^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Csc[x] - Sin[x]],x]

[Out]

-(((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))

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Maple [F]  time = 0.121, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{\csc \left ( x \right ) -\sin \left ( x \right ) }}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\csc \left (x\right ) - \sin \left (x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(csc(x) - sin(x)), x)

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Fricas [B]  time = 2.46269, size = 393, normalized size = 6.55 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \sin{\left (x \right )} + \csc{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-sin(x) + csc(x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\csc \left (x\right ) - \sin \left (x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(csc(x) - sin(x)), x)