3.52 \(\int \frac{1}{(a \csc ^2(x))^{3/2}} \, dx\)

Optimal. Leaf size=36 \[ -\frac{2 \cot (x)}{3 a \sqrt{a \csc ^2(x)}}-\frac{\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}} \]

[Out]

-Cot[x]/(3*(a*Csc[x]^2)^(3/2)) - (2*Cot[x])/(3*a*Sqrt[a*Csc[x]^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0183318, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4122, 192, 191} \[ -\frac{2 \cot (x)}{3 a \sqrt{a \csc ^2(x)}}-\frac{\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Cot[x]/(3*(a*Csc[x]^2)^(3/2)) - (2*Cot[x])/(3*a*Sqrt[a*Csc[x]^2])

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (a \csc ^2(x)\right )^{3/2}} \, dx &=-\left (a \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\right )\\ &=-\frac{\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}}-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{3 \left (a \csc ^2(x)\right )^{3/2}}-\frac{2 \cot (x)}{3 a \sqrt{a \csc ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0214471, size = 27, normalized size = 0.75 \[ \frac{(\cos (3 x)-9 \cos (x)) \csc ^3(x)}{12 \left (a \csc ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-9*Cos[x] + Cos[3*x])*Csc[x]^3)/(12*(a*Csc[x]^2)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.076, size = 31, normalized size = 0.9 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) \left ( \cos \left ( x \right ) -2 \right ) }{6\, \left ( -1+\cos \left ( x \right ) \right ) ^{2}} \left ( -{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*csc(x)^2)^(3/2),x)

[Out]

1/6*4^(1/2)*sin(x)*(cos(x)-2)/(-1+cos(x))^2/(-a/(cos(x)^2-1))^(3/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*csc(x)^2)^(-3/2), x)

________________________________________________________________________________________

Fricas [A]  time = 0.471213, size = 84, normalized size = 2.33 \begin{align*} \frac{{\left (\cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )} \sqrt{-\frac{a}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right )}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/3*(cos(x)^3 - 3*cos(x))*sqrt(-a/(cos(x)^2 - 1))*sin(x)/a^2

________________________________________________________________________________________

Sympy [A]  time = 3.57798, size = 39, normalized size = 1.08 \begin{align*} - \frac{2 \cot ^{3}{\left (x \right )}}{3 a^{\frac{3}{2}} \left (\csc ^{2}{\left (x \right )}\right )^{\frac{3}{2}}} - \frac{\cot{\left (x \right )}}{a^{\frac{3}{2}} \left (\csc ^{2}{\left (x \right )}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)**2)**(3/2),x)

[Out]

-2*cot(x)**3/(3*a**(3/2)*(csc(x)**2)**(3/2)) - cot(x)/(a**(3/2)*(csc(x)**2)**(3/2))

________________________________________________________________________________________

Giac [A]  time = 1.38485, size = 58, normalized size = 1.61 \begin{align*} -\frac{4 \,{\left (\frac{3 \, \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right ) \tan \left (\frac{1}{2} \, x\right )^{2} + \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )}{{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{3}} - \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )\right )\right )}}{3 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*csc(x)^2)^(3/2),x, algorithm="giac")

[Out]

-4/3*((3*sgn(tan(1/2*x))*tan(1/2*x)^2 + sgn(tan(1/2*x)))/(tan(1/2*x)^2 + 1)^3 - sgn(tan(1/2*x)))/a^(3/2)