3.43 \(\int \frac{1}{\sqrt{\csc ^2(x)}} \, dx\)

Optimal. Leaf size=12 \[ -\frac{\cot (x)}{\sqrt{\csc ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.0093475, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4122, 191} \[ -\frac{\cot (x)}{\sqrt{\csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Csc[x]^2],x]

[Out]

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

Mathematica [A]  time = 0.0072935, size = 12, normalized size = 1. \[ -\frac{\cot (x)}{\sqrt{\csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Csc[x]^2],x]

[Out]

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

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Maple [B]  time = 0.127, size = 26, normalized size = 2.2 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) }{-2+2\,\cos \left ( x \right ) }{\frac{1}{\sqrt{- \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.50187, size = 14, normalized size = 1.17 \begin{align*} -\frac{1}{\sqrt{\tan \left (x\right )^{2} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/sqrt(tan(x)^2 + 1)

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Fricas [A]  time = 0.463214, size = 12, normalized size = 1. \begin{align*} -\cos \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)

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Sympy [A]  time = 0.47557, size = 12, normalized size = 1. \begin{align*} - \frac{\cot{\left (x \right )}}{\sqrt{\csc ^{2}{\left (x \right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.32864, size = 15, normalized size = 1.25 \begin{align*} -\cos \left (x\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) + \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)^(1/2),x, algorithm="giac")

[Out]

-cos(x)*sgn(sin(x)) + sgn(sin(x))