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

Optimal. Leaf size=80 \[ \frac{\sec (x)}{2 \sqrt{\cos (x) \cot (x)}}+\frac{\sqrt{-\sin (x)} \cot (x) \tan ^{-1}\left (\sqrt{-\sin (x)}\right )}{4 \sqrt{\cos (x) \cot (x)}}+\frac{\sqrt{-\sin (x)} \cot (x) \tanh ^{-1}\left (\sqrt{-\sin (x)}\right )}{4 \sqrt{\cos (x) \cot (x)}} \]

[Out]

Sec[x]/(2*Sqrt[Cos[x]*Cot[x]]) + (ArcTan[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(4*Sqrt[Cos[x]*Cot[x]]) + (ArcTa
nh[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(4*Sqrt[Cos[x]*Cot[x]])

________________________________________________________________________________________

Rubi [A]  time = 0.11552, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.818, Rules used = {4397, 4400, 2597, 2601, 2564, 329, 212, 206, 203} \[ \frac{\sec (x)}{2 \sqrt{\cos (x) \cot (x)}}+\frac{\sqrt{-\sin (x)} \cot (x) \tan ^{-1}\left (\sqrt{-\sin (x)}\right )}{4 \sqrt{\cos (x) \cot (x)}}+\frac{\sqrt{-\sin (x)} \cot (x) \tanh ^{-1}\left (\sqrt{-\sin (x)}\right )}{4 \sqrt{\cos (x) \cot (x)}} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x] - Sin[x])^(-3/2),x]

[Out]

Sec[x]/(2*Sqrt[Cos[x]*Cot[x]]) + (ArcTan[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(4*Sqrt[Cos[x]*Cot[x]]) + (ArcTa
nh[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(4*Sqrt[Cos[x]*Cot[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 2597

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] - Dist[(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] && Integer
sQ[2*m, 2*n] &&  !(EqQ[n, -3/2] && EqQ[m, 1])

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 212

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

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])

Rubi steps

\begin{align*} \int \frac{1}{(\csc (x)-\sin (x))^{3/2}} \, dx &=\int \frac{1}{(\cos (x) \cot (x))^{3/2}} \, dx\\ &=\frac{\left (\sqrt{\cos (x)} \sqrt{\cot (x)}\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(x) \cot ^{\frac{3}{2}}(x)} \, dx}{\sqrt{\cos (x) \cot (x)}}\\ &=\frac{\sec (x)}{2 \sqrt{\cos (x) \cot (x)}}-\frac{\left (\sqrt{\cos (x)} \sqrt{\cot (x)}\right ) \int \frac{\sqrt{\cot (x)}}{\cos ^{\frac{3}{2}}(x)} \, dx}{4 \sqrt{\cos (x) \cot (x)}}\\ &=\frac{\sec (x)}{2 \sqrt{\cos (x) \cot (x)}}-\frac{\left (\cot (x) \sqrt{-\sin (x)}\right ) \int \frac{\sec (x)}{\sqrt{-\sin (x)}} \, dx}{4 \sqrt{\cos (x) \cot (x)}}\\ &=\frac{\sec (x)}{2 \sqrt{\cos (x) \cot (x)}}+\frac{\left (\cot (x) \sqrt{-\sin (x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-x^2\right )} \, dx,x,-\sin (x)\right )}{4 \sqrt{\cos (x) \cot (x)}}\\ &=\frac{\sec (x)}{2 \sqrt{\cos (x) \cot (x)}}+\frac{\left (\cot (x) \sqrt{-\sin (x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^4} \, dx,x,\sqrt{-\sin (x)}\right )}{2 \sqrt{\cos (x) \cot (x)}}\\ &=\frac{\sec (x)}{2 \sqrt{\cos (x) \cot (x)}}+\frac{\left (\cot (x) \sqrt{-\sin (x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{-\sin (x)}\right )}{4 \sqrt{\cos (x) \cot (x)}}+\frac{\left (\cot (x) \sqrt{-\sin (x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-\sin (x)}\right )}{4 \sqrt{\cos (x) \cot (x)}}\\ &=\frac{\sec (x)}{2 \sqrt{\cos (x) \cot (x)}}+\frac{\tan ^{-1}\left (\sqrt{-\sin (x)}\right ) \cot (x) \sqrt{-\sin (x)}}{4 \sqrt{\cos (x) \cot (x)}}+\frac{\tanh ^{-1}\left (\sqrt{-\sin (x)}\right ) \cot (x) \sqrt{-\sin (x)}}{4 \sqrt{\cos (x) \cot (x)}}\\ \end{align*}

Mathematica [A]  time = 0.154516, size = 60, normalized size = 0.75 \[ \frac{2 \sqrt [4]{\sin ^2(x)} \sec (x)+\cos (x) \left (-\tan ^{-1}\left (\sqrt [4]{\sin ^2(x)}\right )\right )-\cos (x) \tanh ^{-1}\left (\sqrt [4]{\sin ^2(x)}\right )}{4 \sqrt [4]{\sin ^2(x)} \sqrt{\cos (x) \cot (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x] - Sin[x])^(-3/2),x]

[Out]

(-(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[Co
s[x]*Cot[x]]*(Sin[x]^2)^(1/4))

________________________________________________________________________________________

Maple [C]  time = 0.209, size = 450, normalized size = 5.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*2^(1/2)*(-1+cos(x))*(2*I*sin(x)*cos(x)^2*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))
^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*EllipticF(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2*2^(1/2))-I*sin(x)*cos(x)
^2*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*Ellipt
icPi(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-I*sin(x)*cos(x)^2*((-I*cos(x)+sin(x)+I)/sin(x))
^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*EllipticPi(((I*cos(x)+sin(x)-I)/sin(x)
)^(1/2),1/2+1/2*I,1/2*2^(1/2))+sin(x)*cos(x)^2*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x)
)^(1/2)*(-I*(-1+cos(x))/sin(x))^(1/2)*EllipticPi(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-sin
(x)*cos(x)^2*((-I*cos(x)+sin(x)+I)/sin(x))^(1/2)*((I*cos(x)+sin(x)-I)/sin(x))^(1/2)*(-I*(-1+cos(x))/sin(x))^(1
/2)*EllipticPi(((I*cos(x)+sin(x)-I)/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-2*cos(x)*2^(1/2)+2*2^(1/2))*cos(x)*(1
+cos(x))^2/sin(x)^5/(cos(x)^2/sin(x))^(3/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 2.40963, size = 475, normalized size = 5.94 \begin{align*} \frac{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 ) \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*arctan(2*sqrt(cos(x)^2/sin(x))*sin(x)/(cos(x)*sin(x) - cos(x)))*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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \sin{\left (x \right )} + \csc{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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