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

Optimal. Leaf size=29 \[ -\frac{2 \cot (x)}{3 \sqrt{\csc ^2(x)}}-\frac{\cot (x)}{3 \csc ^2(x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.011221, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4122, 192, 191} \[ -\frac{2 \cot (x)}{3 \sqrt{\csc ^2(x)}}-\frac{\cot (x)}{3 \csc ^2(x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0165732, size = 23, normalized size = 0.79 \[ \frac{(\cos (3 x)-9 \cos (x)) \csc (x)}{12 \sqrt{\csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.082, size = 30, normalized size = 1. \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 ( - \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-1} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.63545, size = 15, normalized size = 0.52 \begin{align*} \frac{1}{12} \, \cos \left (3 \, x\right ) - \frac{3}{4} \, \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*cos(3*x) - 3/4*cos(x)

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Fricas [A]  time = 0.46182, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*cos(x)^3 - cos(x)

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Sympy [A]  time = 1.47768, size = 29, normalized size = 1. \begin{align*} - \frac{2 \cot ^{3}{\left (x \right )}}{3 \left (\csc ^{2}{\left (x \right )}\right )^{\frac{3}{2}}} - \frac{\cot{\left (x \right )}}{\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/(csc(x)**2)**(3/2),x)

[Out]

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

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Giac [B]  time = 1.31127, size = 59, normalized size = 2.03 \begin{align*} -\frac{4 \,{\left (\frac{3 \,{\left (\cos \left (x\right ) - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} - \mathrm{sgn}\left (\sin \left (x\right )\right )\right )}}{3 \,{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{3}} + \frac{4}{3} \, \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(3*(cos(x) - 1)*sgn(sin(x))/(cos(x) + 1) - sgn(sin(x)))/((cos(x) - 1)/(cos(x) + 1) - 1)^3 + 4/3*sgn(sin(x
))