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

Optimal. Leaf size=31 \[ \frac{2}{3} \cos (x) \sqrt{\cos (x) \cot (x)}-\frac{8}{3} \sec (x) \sqrt{\cos (x) \cot (x)} \]

[Out]

(2*Cos[x]*Sqrt[Cos[x]*Cot[x]])/3 - (8*Sqrt[Cos[x]*Cot[x]]*Sec[x])/3

________________________________________________________________________________________

Rubi [A]  time = 0.0812269, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4397, 4400, 2598, 2589} \[ \frac{2}{3} \cos (x) \sqrt{\cos (x) \cot (x)}-\frac{8}{3} \sec (x) \sqrt{\cos (x) \cot (x)} \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x] - Sin[x])^(3/2),x]

[Out]

(2*Cos[x]*Sqrt[Cos[x]*Cot[x]])/3 - (8*Sqrt[Cos[x]*Cot[x]]*Sec[x])/3

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 2598

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

Rule 2589

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

Rubi steps

\begin{align*} \int (\csc (x)-\sin (x))^{3/2} \, dx &=\int (\cos (x) \cot (x))^{3/2} \, dx\\ &=\frac{\sqrt{\cos (x) \cot (x)} \int \cos ^{\frac{3}{2}}(x) \cot ^{\frac{3}{2}}(x) \, dx}{\sqrt{\cos (x)} \sqrt{\cot (x)}}\\ &=\frac{2}{3} \cos (x) \sqrt{\cos (x) \cot (x)}+\frac{\left (4 \sqrt{\cos (x) \cot (x)}\right ) \int \frac{\cot ^{\frac{3}{2}}(x)}{\sqrt{\cos (x)}} \, dx}{3 \sqrt{\cos (x)} \sqrt{\cot (x)}}\\ &=\frac{2}{3} \cos (x) \sqrt{\cos (x) \cot (x)}-\frac{8}{3} \sqrt{\cos (x) \cot (x)} \sec (x)\\ \end{align*}

Mathematica [A]  time = 0.0402594, size = 21, normalized size = 0.68 \[ \frac{2}{3} \left (\cos ^2(x)-4\right ) \sec (x) \sqrt{\cos (x) \cot (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x] - Sin[x])^(3/2),x]

[Out]

(2*(-4 + Cos[x]^2)*Sqrt[Cos[x]*Cot[x]]*Sec[x])/3

________________________________________________________________________________________

Maple [A]  time = 0.088, size = 26, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\, \left ( \cos \left ( x \right ) \right ) ^{2}-8 \right ) \sin \left ( x \right ) }{3\, \left ( \cos \left ( x \right ) \right ) ^{3}} \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{\sin \left ( x \right ) }} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(cos(x)^2-4)*(cos(x)^2/sin(x))^(3/2)*sin(x)/cos(x)^3

________________________________________________________________________________________

Maxima [B]  time = 1.86962, size = 424, normalized size = 13.68 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(cos(x)^2 + sin(x)^2 + 2*cos(x) + 1)^(1/4)*(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1)^(1/4)*(((cos(9/2*x) - 15*c
os(5/2*x) - cos(3/2*x) + 15*cos(1/2*x) - sin(9/2*x) + 15*sin(5/2*x) - sin(3/2*x) - 15*sin(1/2*x))*cos(3/2*arct
an2(sin(x), cos(x) - 1)) + (cos(9/2*x) - 15*cos(5/2*x) - cos(3/2*x) + 15*cos(1/2*x) + sin(9/2*x) - 15*sin(5/2*
x) + sin(3/2*x) + 15*sin(1/2*x))*sin(3/2*arctan2(sin(x), cos(x) - 1)))*cos(3/2*arctan2(sin(x), cos(x) + 1)) +
((cos(9/2*x) - 15*cos(5/2*x) - cos(3/2*x) + 15*cos(1/2*x) + sin(9/2*x) - 15*sin(5/2*x) + sin(3/2*x) + 15*sin(1
/2*x))*cos(3/2*arctan2(sin(x), cos(x) - 1)) - (cos(9/2*x) - 15*cos(5/2*x) - cos(3/2*x) + 15*cos(1/2*x) - sin(9
/2*x) + 15*sin(5/2*x) - sin(3/2*x) - 15*sin(1/2*x))*sin(3/2*arctan2(sin(x), cos(x) - 1)))*sin(3/2*arctan2(sin(
x), cos(x) + 1)))/(cos(x)^4 + sin(x)^4 + 2*(cos(x)^2 + 1)*sin(x)^2 - 2*cos(x)^2 + 1)

________________________________________________________________________________________

Fricas [A]  time = 2.17473, size = 66, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (\cos \left (x\right )^{2} - 4\right )} \sqrt{\frac{\cos \left (x\right )^{2}}{\sin \left (x\right )}}}{3 \, \cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(cos(x)^2 - 4)*sqrt(cos(x)^2/sin(x))/cos(x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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((csc(x)-sin(x))^(3/2),x, algorithm="giac")

[Out]

integrate((csc(x) - sin(x))^(3/2), x)