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

Optimal. Leaf size=14 \[ -\frac{\cot (x)}{\sqrt{a \csc ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.0134393, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4122, 191} \[ -\frac{\cot (x)}{\sqrt{a \csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0064168, size = 14, normalized size = 1. \[ -\frac{\cot (x)}{\sqrt{a \csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.132, size = 27, normalized size = 1.9 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) }{-2+2\,\cos \left ( x \right ) }{\frac{1}{\sqrt{-{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.49698, size = 18, normalized size = 1.29 \begin{align*} -\frac{1}{\sqrt{\tan \left (x\right )^{2} + 1} \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(sqrt(tan(x)^2 + 1)*sqrt(a))

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Fricas [A]  time = 0.468744, size = 57, normalized size = 4.07 \begin{align*} -\frac{\sqrt{-\frac{a}{\cos \left (x\right )^{2} - 1}} \cos \left (x\right ) \sin \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.789281, size = 17, normalized size = 1.21 \begin{align*} - \frac{\cot{\left (x \right )}}{\sqrt{a} \sqrt{\csc ^{2}{\left (x \right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.28229, size = 46, normalized size = 3.29 \begin{align*} \frac{2 \, \mathrm{sgn}\left (\sin \left (x\right )\right )}{\sqrt{a}} + \frac{2}{\sqrt{a}{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sgn(sin(x))/sqrt(a) + 2/(sqrt(a)*((cos(x) - 1)/(cos(x) + 1) - 1)*sgn(sin(x)))